%------------------------------------------------------------------------------
% File     : ITP228_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Insert 00400_024466
% 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    : 0066_VEBT_Insert_00400_024466 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 13347 (3762 unt;2956 typ;   0 def)
%            Number of atoms       : 26531 (9310 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 17704 (2441   ~; 481   |;1419   &)
%                                         (2634 <=>;10729  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   15 (   2 avg)
%            Number of FOOLs       :  939 ( 590 fml; 349 var)
%            Number of X terms     :  775 (   0  []; 689 ite;  86 let)
%            Number of types       :  603 ( 602 usr)
%            Number of type conns  : 2795 (1866   >; 929   *;   0   +;   0  <<)
%            Number of predicates  :  165 ( 162 usr;   2 prp; 0-4 aty)
%            Number of functors    : 2205 (2205 usr; 501 con; 0-4 aty)
%            Number of variables   : 26263 (25597   !; 666   ?;26263   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 19:45:20.534
%------------------------------------------------------------------------------
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Int__Oint_J,type,
    fun_real_int: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

% Explicit typings (2354)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_complex > fun_complex_o ).

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

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__aaq____,type,
    aTP_Lamm_aaq: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aar____,type,
    aTP_Lamm_aar: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: fun_nat_o ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: fun_nat_o ).

tff(sy_c_ATP_058Lamm__aax____,type,
    aTP_Lamm_aax: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: nat > fun_na8563630686737515764_nat_o ).

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

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__abc____,type,
    aTP_Lamm_abc: complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__abh____,type,
    aTP_Lamm_abh: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__abs____,type,
    aTP_Lamm_abs: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__abt____,type,
    aTP_Lamm_abt: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__abu____,type,
    aTP_Lamm_abu: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__abv____,type,
    aTP_Lamm_abv: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__abw____,type,
    aTP_Lamm_abw: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__aca____,type,
    aTP_Lamm_aca: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__acb____,type,
    aTP_Lamm_acb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acc____,type,
    aTP_Lamm_acc: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__acf____,type,
    aTP_Lamm_acf: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__acg____,type,
    aTP_Lamm_acg: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ach____,type,
    aTP_Lamm_ach: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__aci____,type,
    aTP_Lamm_aci: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__acj____,type,
    aTP_Lamm_acj: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ack____,type,
    aTP_Lamm_ack: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__acl____,type,
    aTP_Lamm_acl: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__acm____,type,
    aTP_Lamm_acm: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__acn____,type,
    aTP_Lamm_acn: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__aco____,type,
    aTP_Lamm_aco: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__acp____,type,
    aTP_Lamm_acp: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__acq____,type,
    aTP_Lamm_acq: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__acr____,type,
    aTP_Lamm_acr: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__acs____,type,
    aTP_Lamm_acs: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__act____,type,
    aTP_Lamm_act: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__acu____,type,
    aTP_Lamm_acu: fun_real_real ).

tff(sy_c_ATP_058Lamm__acv____,type,
    aTP_Lamm_acv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__acw____,type,
    aTP_Lamm_acw: fun_real_real ).

tff(sy_c_ATP_058Lamm__acx____,type,
    aTP_Lamm_acx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__acy____,type,
    aTP_Lamm_acy: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__acz____,type,
    aTP_Lamm_acz: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ada____,type,
    aTP_Lamm_ada: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__adb____,type,
    aTP_Lamm_adb: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__adc____,type,
    aTP_Lamm_adc: fun_nat_real ).

tff(sy_c_ATP_058Lamm__add____,type,
    aTP_Lamm_add: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ade____,type,
    aTP_Lamm_ade: fun_nat_real ).

tff(sy_c_ATP_058Lamm__adf____,type,
    aTP_Lamm_adf: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adg____,type,
    aTP_Lamm_adg: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adh____,type,
    aTP_Lamm_adh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adi____,type,
    aTP_Lamm_adi: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adj____,type,
    aTP_Lamm_adj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adk____,type,
    aTP_Lamm_adk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adl____,type,
    aTP_Lamm_adl: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__adm____,type,
    aTP_Lamm_adm: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__adn____,type,
    aTP_Lamm_adn: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ado____,type,
    aTP_Lamm_ado: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__adp____,type,
    aTP_Lamm_adp: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__adq____,type,
    aTP_Lamm_adq: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__adr____,type,
    aTP_Lamm_adr: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ads____,type,
    aTP_Lamm_ads: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__adt____,type,
    aTP_Lamm_adt: fun_real_o > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__adu____,type,
    aTP_Lamm_adu: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__adv____,type,
    aTP_Lamm_adv: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__adw____,type,
    aTP_Lamm_adw: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__adx____,type,
    aTP_Lamm_adx: fun_real_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__ady____,type,
    aTP_Lamm_ady: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__adz____,type,
    aTP_Lamm_adz: fun_real_real > fun_fu3077872570260440615l_real ).

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

tff(sy_c_ATP_058Lamm__aea____,type,
    aTP_Lamm_aea: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aeb____,type,
    aTP_Lamm_aeb: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__aec____,type,
    aTP_Lamm_aec: fun_real_real ).

tff(sy_c_ATP_058Lamm__aed____,type,
    aTP_Lamm_aed: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__aee____,type,
    aTP_Lamm_aee: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__aef____,type,
    aTP_Lamm_aef: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__aeg____,type,
    aTP_Lamm_aeg: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__aeh____,type,
    aTP_Lamm_aeh: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aei____,type,
    aTP_Lamm_aei: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aej____,type,
    aTP_Lamm_aej: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__aek____,type,
    aTP_Lamm_aek: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__ael____,type,
    aTP_Lamm_ael: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__aem____,type,
    aTP_Lamm_aem: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aen____,type,
    aTP_Lamm_aen: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aeo____,type,
    aTP_Lamm_aeo: fun_real_o ).

tff(sy_c_ATP_058Lamm__aep____,type,
    aTP_Lamm_aep: fun_real_o ).

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

tff(sy_c_ATP_058Lamm__aer____,type,
    aTP_Lamm_aer: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__aes____,type,
    aTP_Lamm_aes: real > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__aet____,type,
    aTP_Lamm_aet: fun_re3844104906058862192omplex ).

tff(sy_c_ATP_058Lamm__aeu____,type,
    aTP_Lamm_aeu: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__aev____,type,
    aTP_Lamm_aev: fun_re5512370375968561900l_real ).

tff(sy_c_ATP_058Lamm__aew____,type,
    aTP_Lamm_aew: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__aex____,type,
    aTP_Lamm_aex: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__aey____,type,
    aTP_Lamm_aey: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__aez____,type,
    aTP_Lamm_aez: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_real_o > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__afa____,type,
    aTP_Lamm_afa: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__afb____,type,
    aTP_Lamm_afb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__afc____,type,
    aTP_Lamm_afc: fun_nat_real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__afd____,type,
    aTP_Lamm_afd: rat > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__afe____,type,
    aTP_Lamm_afe: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__aff____,type,
    aTP_Lamm_aff: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__afg____,type,
    aTP_Lamm_afg: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_Pr4796943543331071514_nat_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_real_o > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_Pr4796943543331071514_nat_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: set_complex > fun_set_complex_o ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: set_Pr1261947904930325089at_nat > fun_se4782206882984618170_nat_o ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: set_int > fun_set_int_o ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: set_list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: set_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: set_real > fun_fu4757868175734369219real_o ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: set_real > fun_fu8602227297639178947real_o ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: set_real > fun_fu4472351043248340291real_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: set_nat > fun_fu8304507701009772099_nat_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: set_nat > fun_fu3780232916559739075_nat_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: set_int > fun_fu7493154764834273859_int_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: set_int > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: set_int > fun_fu2620374121714387907_int_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: set_complex > fun_fu3074144567237359299plex_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_real_real ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_int_int ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: set_Pr1261947904930325089at_nat > fun_na5060536216179792737_nat_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: set_Pr1261947904930325089at_nat > fun_na5060536216179792737_nat_o ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_nat_fun_real_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: set_real > fun_fu9068519094125542729real_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: set_nat > fun_fu6176141739774794687_nat_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: set_nat > fun_fu6433034133342731997_nat_o ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: set_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: set_int > fun_fu3930813892005118153_int_o ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: set_complex > fun_fu2142003765521078979plex_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: set_real > fun_fu9068519094125542729real_o ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: set_nat > fun_fu6176141739774794687_nat_o ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: set_nat > fun_fu6433034133342731997_nat_o ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: set_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: set_int > fun_fu3930813892005118153_int_o ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: set_complex > fun_fu2142003765521078979plex_o ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: fun_int_int ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: fun_nat_real ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_o_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_o_int > fun_fu6453395667394816515nt_int ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_int_o > fun_fu1481857817119020493_int_o ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_int_int > fun_fu6092417073193843705nt_int ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_Pr982841511892225912nt_int > fun_fu717549361983437348nt_int ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_in4211310926729683942nt_int > fun_fu918106826131887214nt_int ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_Pr2478283820680522243nt_int > fun_fu3898734532474705027nt_int ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_fu3275390434952296369_nat_o > fun_fu6384561554008368101_nat_o ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_complex_complex > fun_fu5956788528830221615omplex ).

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

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: set_int > fun_fu7328815047503086531_int_o ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: set_int > fun_fu8404755328348021045al_int ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_real_fun_int_int > fun_int_fun_real_int ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: set_real > fun_fu6324116013759063093nt_int ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: set_int > fun_fu5646853623633916995_int_o ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: set_int > fun_fu4584101141160326545at_int ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_nat_fun_int_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: set_nat > fun_fu7802804735324696557nt_int ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: set_int > fun_fu810243669753863363_int_o ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: set_int > fun_fu8001260946965626035ex_int ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_co333780937270390912nt_int > fun_in3657390099950231424ex_int ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: set_complex > fun_fu2339071201290874307plex_o ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: set_complex > fun_fu7424888833511495217nt_int ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: set_complex > fun_fu3043630359712370243plex_o ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: set_complex > fun_fu7433039812794084273omplex ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: set_real > fun_fu1853463631764337203omplex ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: set_complex > fun_fu5273811730930669507plex_o ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: set_complex > fun_fu6938252901338612493omplex ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: set_nat > fun_fu3656911019502402539omplex ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: set_complex > fun_fu4149201653251398851plex_o ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: set_complex > fun_fu7724386853639975089omplex ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_in1307110748760623106omplex > fun_co7673900812888889602omplex ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: set_int > fun_fu260611630652282675omplex ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: set_nat > fun_fu5289434066392675523_nat_o ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: set_nat > fun_fu8695033588447641837al_nat ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: set_real > fun_fu5553606195750802321at_nat ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: set_nat > fun_fu927436805355353155_nat_o ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: set_nat > fun_fu1580565752866428867_nat_o ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: set_nat > fun_fu8020931516743756395ex_nat ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: set_complex > fun_fu3351286688164852109at_nat ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: set_nat > fun_fu5253574275612187245l_real ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: set_real > fun_fu5321668254738230545t_real ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_Pr2028404973800837934_nat_o > fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_Pr3238374608701176162_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_Pr982841511892225912nt_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_real_complex > fun_fu3532930458344041007omplex ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_nat_complex > fun_fu4302481349826706223omplex ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_int_complex > fun_fu3558165918884697647omplex ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_real_rat > fun_fu5551332503302594393al_rat ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_complex_rat > fun_fu4396998534562958809ex_rat ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_real_complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_int_complex > fun_int_o ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_int_real > fun_int_o ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_real_rat > fun_real_o ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_complex_rat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_real_nat > fun_real_o ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_nat_rat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_complex_nat > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: int > fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_nat_rat > fun_nat_rat ).

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

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

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_complex_real > fun_fu7714627026455246123x_real ).

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

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_complex_rat > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_nat_rat > fun_fu8328862064278946247at_rat ).

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

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_nat_int > fun_nat_int ).

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

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

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

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

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

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: $o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_complex_nat > fun_complex_complex ).

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

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_nat_nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_nat_nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_nat_nat > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_complex_int > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_nat_int > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_int_int > fun_int_real ).

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

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_int_int > fun_int_complex ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_nat_rat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_nat_int ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_complex_complex > fun_complex_real ).

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

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_nat_real ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_nat_real ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: set_Pr1200682188532975830nteger > fun_fu3422731660242565741eger_o ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: set_Pr3486312109254157132nteger > fun_fu7695273699059480439eger_o ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: set_Pr6700895561065273913nt_int > fun_fu6865642474282033866_int_o ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: set_Pr4166680992137330533nt_int > fun_fu2978539155105838238_int_o ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_nat_o > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_nat_o > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_nat_o > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: fun_nat_o > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: set_nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_nat_real ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: real > fun_fu3477889739401271834t_real ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: real > fun_fu4806068833880117316t_real ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_nat_real > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_complex_real > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: fun_int_int ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_real_real ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_se823489619503225858plex_o ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_nat_int > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_nat_int > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: set_nat > fun_fu8695033588447641837al_nat ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: set_real > fun_fu5553606195750802321at_nat ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: set_nat > fun_fu8020931516743756395ex_nat ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: set_complex > fun_fu3351286688164852109at_nat ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: set_nat > fun_fu3562227594288721901al_int ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_real_fun_nat_int > fun_nat_fun_real_int ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: set_real > fun_fu3518430294936654993at_int ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: set_nat > fun_fu2556788080963300715ex_int ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: fun_co7370231399432144932at_int > fun_na5967208032043182372ex_int ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: set_complex > fun_fu3636980702898289805at_int ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: set_int > fun_fu8404755328348021045al_int ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: set_real > fun_fu6324116013759063093nt_int ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: set_int > fun_fu8001260946965626035ex_int ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: set_complex > fun_fu7424888833511495217nt_int ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: set_real > fun_fu8783008166737851587real_o ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: set_real > fun_fu8018396694123807121at_nat ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_nat_fun_real_nat > fun_real_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: set_nat > fun_fu91202020541382083_nat_o ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: set_nat > fun_fu1169599456156577517al_nat ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: set_int > fun_fu9148247276982919825at_nat ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_nat_fun_int_nat > fun_int_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: set_nat > fun_fu3143578834292514029nt_nat ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: set_complex > fun_fu7903722123778649997at_nat ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_na8457291176974909128ex_nat > fun_co636942507509095880at_nat ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: set_nat > fun_fu8082241605084145771ex_nat ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: fun_real_complex > fun_fu3532930458344041007omplex ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_nat_complex > fun_fu4302481349826706223omplex ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_int_complex > fun_fu3558165918884697647omplex ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_complex_complex > fun_fu3719809307426920623omplex ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: fun_nat_real > fun_fu4002475321494201261t_real ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_real_rat > fun_fu5551332503302594393al_rat ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_real_complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_int_complex > fun_int_o ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_complex_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_int_real > fun_int_o ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: fun_real_rat > fun_real_o ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: fun_complex_rat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_real_nat > fun_real_o ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_int_real > fun_int_real ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_complex_real > fun_complex_real ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_Pr672294278902479040t_real > fun_Pr672294278902479040t_real ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_complex_complex > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_set_nat_real > fun_fu5354524256550068955t_real ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_complex_complex > fun_fu8968174646369908141x_real ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_real_complex > fun_fu6344110373156359337l_real ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_set_nat_complex > fun_fu1617006900395023197t_real ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_int_complex > fun_fu3594690983434520489t_real ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_nat_complex > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: code_integer > fun_na8135612474337863287nteger ).

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

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

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

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: nat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: real > fun_re4756235295228751087t_real ).

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

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: fun_nat_fun_nat_int ).

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

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

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: fun_nat_fun_nat_int ).

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

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

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: fun_nat_fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_set_nat_real > fun_set_nat_real ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: fun_int_real > fun_int_real ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: fun_complex_real > fun_complex_real ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: fun_Pr672294278902479040t_real > fun_Pr672294278902479040t_real ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: fun_nat_fun_nat_real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: fun_nat_complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: fun_nat_real > fun_nat_fun_real_o ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: fun_nat_real > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: nat > fun_re6135230694098607860t_real ).

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tff(sy_c_ATP_058Lamm__zv____,type,
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tff(sy_c_Code__Numeral_Odivmod__abs,type,
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tff(sy_c_Complex_OArg,type,
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tff(sy_c_Complex_Ocis,type,
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tff(sy_c_Complex_Ocnj,type,
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tff(sy_c_Complex_Ocomplex_OComplex,type,
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tff(sy_c_Complex_Ocomplex_OIm,type,
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tff(sy_c_Complex_Ocomplex_ORe,type,
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tff(sy_c_Complex_Ocsqrt,type,
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tff(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
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tff(sy_c_Filter_Ofiltermap_001t__Real__Oreal_001t__Real__Oreal,type,
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    fold_nat_nat: ( fun_nat_fun_nat_nat * list_nat ) > fun_nat_nat ).

tff(sy_c_List_Ogen__length_001t__Int__Oint,type,
    gen_length_int: nat > fun_list_int_nat ).

tff(sy_c_List_Ogen__length_001t__Nat__Onat,type,
    gen_length_nat: nat > fun_list_nat_nat ).

tff(sy_c_List_Ogen__length_001t__VEBT____Definitions__OVEBT,type,
    gen_length_VEBT_VEBT: nat > fun_li1940556759920501533BT_nat ).

tff(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
    linord1735203802627413978nt_int: ( fun_int_int * list_int ) > list_int ).

tff(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
    linord738340561235409698at_nat: ( fun_nat_nat * list_nat ) > list_nat ).

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

tff(sy_c_List_Olist_OCons_001t__Complex__Ocomplex,type,
    cons_complex: ( complex * list_complex ) > list_complex ).

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

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

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

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    cons_P9044669534377732177nteger: ( produc8923325533196201883nteger * list_P5578671422887162913nteger ) > list_P5578671422887162913nteger ).

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    cons_P3334398858971670639nt_int: ( product_prod_int_int * list_P5707943133018811711nt_int ) > list_P5707943133018811711nt_int ).

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    cons_P7512249878480867347nt_nat: ( product_prod_int_nat * list_P8198026277950538467nt_nat ) > list_P8198026277950538467nt_nat ).

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    cons_P2335045147070616083at_int: ( product_prod_nat_int * list_P3521021558325789923at_int ) > list_P3521021558325789923at_int ).

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

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    cons_P6847588673944434154nteger: ( produc7509114972854644512nteger * list_P1054633106646461488nteger ) > list_P1054633106646461488nteger ).

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    cons_P6394518999824463865nt_int: ( produc4788131091073775919nt_int * list_P932735248015874751nt_int ) > list_P932735248015874751nt_int ).

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    cons_P8208993393157215437nt_int: ( produc3043004577601930755nt_int * list_P2843741347166045587nt_int ) > list_P2843741347166045587nt_int ).

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

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

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

tff(sy_c_List_Olist_OCons_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    cons_f4064889718614483898e_term: ( fun_Co3460408610320954864e_term * list_f8851413139013959296e_term ) > list_f8851413139013959296e_term ).

tff(sy_c_List_Olist_OCons_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    cons_f6442722764679334921e_term: ( fun_in5730379987311621695e_term * list_f628289730585369423e_term ) > list_f628289730585369423e_term ).

tff(sy_c_List_Olist_OCons_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    cons_f7541796142091620804e_term: ( fun_Pr5098669352780829690e_term * list_f4306079564415238026e_term ) > list_f4306079564415238026e_term ).

tff(sy_c_List_Olist_OCons_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    cons_f7531370473056858165e_term: ( fun_Pr7763680713840462443e_term * list_f280343417242313083e_term ) > list_f280343417242313083e_term ).

tff(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

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

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

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

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

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

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

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__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__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    set_Pr8199049879907524818omplex: list_P7664491975274850627omplex > set_Pr5085853215250843933omplex ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Int__Oint_J,type,
    set_Pr4995810437751016784ex_int: list_P2206113689347244737ex_int > set_Pr2254670189886740123ex_int ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
    set_Pr9173661457260213492ex_nat: list_P4696196834278971493ex_nat > set_Pr4744753334818466879ex_nat ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Real__Oreal_J,type,
    set_Pr1225976482156248400x_real: list_P7647014805210017729x_real > set_Pr1133549439701694107x_real ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__VEBT____Definitions__OVEBT_J,type,
    set_Pr5158653123227461798T_VEBT: list_P7977503562704621835T_VEBT > set_Pr4085867452638698417T_VEBT ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Complex__Ocomplex_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    set_Pr6647972299459129970nt_nat: list_P8198026277950538467nt_nat > set_Pr3448869479623346877nt_nat ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
    set_Pr112895574167722958t_real: list_P6863124054624500543t_real > set_Pr3538720872664544793t_real ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
    set_Pr8714266321650254504T_VEBT: list_P7524865323317820941T_VEBT > set_Pr8044002425091019955T_VEBT ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
    set_Pr8536649499196266448omplex: list_P3881527313128557121omplex > set_Pr6591433984475009307omplex ).

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__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Complex__Ocomplex_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,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
    set_real2: list_real > set_real ).

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

tff(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Oset_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
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tff(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

tff(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
    list_update_complex: ( list_complex * nat * complex ) > list_complex ).

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__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    list_u5986614341521392946nteger: ( list_P1054633106646461488nteger * nat * produc7509114972854644512nteger ) > list_P1054633106646461488nteger ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    list_u4453756226229724993nt_int: ( list_P932735248015874751nt_int * nat * produc4788131091073775919nt_int ) > list_P932735248015874751nt_int ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    list_u5815421275070861992nteger: ( list_P2959130015091922598nteger * nat * produc2453742332425399446nteger ) > list_P2959130015091922598nteger ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    list_u7912136156437333013nt_int: ( list_P2843741347166045587nt_int * nat * produc3043004577601930755nt_int ) > list_P2843741347166045587nt_int ).

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

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

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

tff(sy_c_List_Olist__update_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    list_u2080314064485400322e_term: ( list_f8851413139013959296e_term * nat * fun_Co3460408610320954864e_term ) > list_f8851413139013959296e_term ).

tff(sy_c_List_Olist__update_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    list_u528859344425011025e_term: ( list_f628289730585369423e_term * nat * fun_in5730379987311621695e_term ) > list_f628289730585369423e_term ).

tff(sy_c_List_Olist__update_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    list_u3429793794183195916e_term: ( list_f4306079564415238026e_term * nat * fun_Pr5098669352780829690e_term ) > list_f4306079564415238026e_term ).

tff(sy_c_List_Olist__update_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    list_u7283322402012212093e_term: ( list_f280343417242313083e_term * nat * fun_Pr7763680713840462443e_term ) > list_f280343417242313083e_term ).

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    nth_Pr2854691652959048409nteger: ( list_P1054633106646461488nteger * nat ) > produc7509114972854644512nteger ).

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

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

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

tff(sy_c_List_Onth_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    nth_fu2298125310669257513e_term: ( list_f8851413139013959296e_term * nat ) > fun_Co3460408610320954864e_term ).

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

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

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

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Int__Oint,type,
    product_nat_int: ( list_nat * list_int ) > list_P3521021558325789923at_int ).

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__VEBT____Definitions__OVEBT,type,
    produc7156399406898700509T_VEBT: ( list_nat * list_VEBT_VEBT ) > list_P5647936690300460905T_VEBT ).

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

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

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

tff(sy_c_List_Oproduct_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc4042828510177926308nteger: ( list_f8851413139013959296e_term * list_P5578671422887162913nteger ) > list_P1054633106646461488nteger ).

tff(sy_c_List_Oremdups_001t__Nat__Onat,type,
    remdups_nat: list_nat > list_nat ).

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

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

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

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

tff(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
    replicate_set_nat: ( nat * set_nat ) > list_set_nat ).

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

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

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

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

tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: fun_int_fun_int_o > fun_list_int_o ).

tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
    sorted_wrt_nat: fun_nat_fun_nat_o > fun_list_nat_o ).

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

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

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

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

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    zip_complex_complex: ( list_complex * list_complex ) > list_P7664491975274850627omplex ).

tff(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__Int__Oint,type,
    zip_complex_int: ( list_complex * list_int ) > list_P2206113689347244737ex_int ).

tff(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    zip_complex_nat: ( list_complex * list_nat ) > list_P4696196834278971493ex_nat ).

tff(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    zip_complex_real: ( list_complex * list_real ) > list_P7647014805210017729x_real ).

tff(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__VEBT____Definitions__OVEBT,type,
    zip_co9157518722488180109T_VEBT: ( list_complex * list_VEBT_VEBT ) > list_P7977503562704621835T_VEBT ).

tff(sy_c_List_Ozip_001t__Int__Oint_001t__Complex__Ocomplex,type,
    zip_int_complex: ( list_int * list_complex ) > list_P1797514011394873281omplex ).

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

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

tff(sy_c_List_Ozip_001t__Int__Oint_001t__Real__Oreal,type,
    zip_int_real: ( list_int * list_real ) > list_P6863124054624500543t_real ).

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

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__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__Complex__Ocomplex,type,
    zip_real_complex: ( list_real * list_complex ) > list_P3881527313128557121omplex ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__Int__Oint,type,
    zip_real_int: ( list_real * list_int ) > list_P4344331454722006975al_int ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__Nat__Onat,type,
    zip_real_nat: ( list_real * list_nat ) > list_P6834414599653733731al_nat ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__Real__Oreal,type,
    zip_real_real: ( list_real * list_real ) > list_P8689742595348180415l_real ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    zip_real_VEBT_VEBT: ( list_real * list_VEBT_VEBT ) > list_P877281246627933069T_VEBT ).

tff(sy_c_List_Ozip_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    zip_set_nat_set_nat: ( list_set_nat * list_set_nat ) > list_P6254988961118846195et_nat ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Complex__Ocomplex,type,
    zip_VE2794733401258833515omplex: ( list_VEBT_VEBT * list_complex ) > list_P4108580160459392801omplex ).

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____Definitions__OVEBT,type,
    zip_VE537291747668921783T_VEBT: ( list_VEBT_VEBT * list_VEBT_VEBT ) > list_P7413028617227757229T_VEBT ).

tff(sy_c_List_Ozip_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    zip_fu7084642285130276018nteger: ( list_f8851413139013959296e_term * list_P5578671422887162913nteger ) > list_P1054633106646461488nteger ).

tff(sy_c_List_Ozip_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    zip_fu2910578966527623617nt_int: ( list_f628289730585369423e_term * list_P5707943133018811711nt_int ) > list_P932735248015874751nt_int ).

tff(sy_c_List_Ozip_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    zip_fu4451626411769283112nteger: ( list_f4306079564415238026e_term * list_P5578671422887162913nteger ) > list_P2959130015091922598nteger ).

tff(sy_c_List_Ozip_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    zip_fu5048005673116055189nt_int: ( list_f280343417242313083e_term * list_P5707943133018811711nt_int ) > list_P2843741347166045587nt_int ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

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

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

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

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

tff(sy_c_Nat_Onat_Opred,type,
    pred: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Complex__Ocomplex,type,
    semiri3842193898606819883omplex: set_complex ).

tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
    semiring_1_Nats_int: set_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: fun_nat_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: fun_nat_complex ).

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

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Code____Numeral__Ointeger,type,
    semiri4055485073559036834nteger: ( fun_Co8499238673658658103nteger * nat * code_integer ) > code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
    semiri2816024913162550771omplex: fun_complex_complex > fun_na1591161823130732966omplex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
    semiri8420488043553186161ux_int: fun_int_int > fun_nat_fun_int_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
    semiri8422978514062236437ux_nat: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
    semiri7787848453975740701ux_rat: ( fun_rat_rat * nat ) > fun_rat_rat ).

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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    size_s3254054031482475050et_nat: list_set_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat__Bijection_Oint__decode,type,
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tff(sy_c_Nat__Bijection_Oint__encode,type,
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tff(sy_c_Nat__Bijection_Olist__encode,type,
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tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
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tff(sy_c_Nat__Bijection_Oprod__encode,type,
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tff(sy_c_Nat__Bijection_Oset__decode,type,
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tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Num_OBitM,type,
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tff(sy_c_Num_Oinc,type,
    inc: fun_num_num ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
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tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
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tff(sy_c_Num_Onum_OOne,type,
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tff(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Num_Onum_Osize__num,type,
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tff(sy_c_Num_Onum__of__nat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
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tff(sy_c_Num_Opred__numeral,type,
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tff(sy_c_Option_Ooption_ONone_001t__Complex__Ocomplex,type,
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tff(sy_c_Option_Ooption_ONone_001t__Int__Oint,type,
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tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_ONone_001t__Real__Oreal,type,
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tff(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
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tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_le2510731241096832064er_nat: ( filter_nat * filter_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
    ord_le4104064031414453916r_real: ( filter_real * filter_real ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: fun_num_fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: fun_real_fun_real_o ).

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

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

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    ord_le6045566169113846134st_nat: ( set_list_nat * set_list_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: fun_se2170385732985038782_nat_o ).

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    ord_le2843351958646193337nt_int: ( set_Pr958786334691620121nt_int * set_Pr958786334691620121nt_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > fun_se4782206882984618170_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    ord_le2645283422117218614nteger: ( set_Pr1200682188532975830nteger * set_Pr1200682188532975830nteger ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    ord_le3197784583289274309nt_int: ( set_Pr4166680992137330533nt_int * set_Pr4166680992137330533nt_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    ord_le2932375828722672556nteger: ( set_Pr3486312109254157132nteger * set_Pr3486312109254157132nteger ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    ord_le3990897393281686169nt_int: ( set_Pr6700895561065273913nt_int * set_Pr6700895561065273913nt_int ) > $o ).

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

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    ord_le4403425263959731960et_int: ( set_set_int * set_set_int ) > $o ).

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le4337996190870823476T_VEBT: ( set_VEBT_VEBT * set_VEBT_VEBT ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    ord_le7644322562001619171plex_o: ( fun_complex_o * fun_complex_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_M_Eo_J,type,
    ord_le5418975059635765733_int_o: ( fun_int_o * fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    ord_le77980682512674322_int_o: ( fun_int_fun_int_o * fun_int_fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    ord_le4972187363741583025_nat_o: ( fun_list_nat_o * fun_list_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__List__Olist_It__Nat__Onat_J_Mt__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
    ord_le4470944457981725354_nat_o: ( fun_li1025265955262268170_nat_o * fun_li1025265955262268170_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Nat__Onat_M_Eo_J,type,
    ord_le7240017405068453697_nat_o: ( fun_nat_o * fun_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    ord_le768339454706049738_nat_o: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    ord_le14110778710936002_int_o: ( fun_Pr3238374608701176162_int_o * fun_Pr3238374608701176162_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    ord_le1416746762259324110_int_o: ( fun_Pr9018117161326028078_int_o * fun_Pr9018117161326028078_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    ord_le1572679713340831354_nat_o: fun_Pr4796943543331071514_nat_o > fun_fu6891447142836604759at_o_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    ord_le3650406611588909774_nat_o: ( fun_Pr2028404973800837934_nat_o * fun_Pr2028404973800837934_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_M_Eo_J,type,
    ord_le3107050023622517222_num_o: ( fun_Pr6331313853612757382_num_o * fun_Pr6331313853612757382_num_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_Mt__fun_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_M_Eo_J_J,type,
    ord_le6421466325430216142_num_o: ( fun_Pr4799464687642144302_num_o * fun_Pr4799464687642144302_num_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Real__Oreal_M_Eo_J,type,
    ord_le3605727316730466277real_o: ( fun_real_o * fun_real_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_le6468474429079055883_nat_o: ( fun_set_nat_o * fun_set_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
    ord_le7899862361585801933eger_o: ( fun_fu3422731660242565741eger_o * fun_fu3422731660242565741eger_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    ord_le2876595303158420734_int_o: ( fun_fu2978539155105838238_int_o * fun_fu2978539155105838238_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
    ord_le4825817444062779351eger_o: ( fun_fu7695273699059480439eger_o * fun_fu7695273699059480439eger_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    ord_le4806730579060245290_int_o: ( fun_fu6865642474282033866_int_o * fun_fu6865642474282033866_int_o ) > $o ).

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

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: fun_int_fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: num > fun_num_num ).

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: fun_re7681423839302404990l_real ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
    ord_max_set_int: ( set_int * set_int ) > set_int ).

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_ma7524802468073614006at_nat: ( set_Pr1261947904930325089at_nat * set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_max_set_real: ( set_real * set_real ) > set_real ).

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

tff(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
    ord_min_int: fun_int_fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_mono_nat_nat: fun_nat_nat > $o ).

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

tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_5726023648592871131at_nat: fun_nat_nat > $o ).

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

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    top_top_set_list_nat: set_list_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

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

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

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

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
    produc6677183202524767010eger_o: ( code_integer * $o ) > produc6271795597528267376eger_o ).

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

tff(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    produc101793102246108661omplex: ( complex * complex ) > produc4411394909380815293omplex ).

tff(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__Int__Oint,type,
    produc1367138851071493491ex_int: ( complex * int ) > produc6845221339535797307ex_int ).

tff(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    produc1369629321580543767ex_nat: ( complex * nat ) > produc1799700322190218207ex_nat ).

tff(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    produc1746590499379883635x_real: ( complex * real ) > produc8892588492097263291x_real ).

tff(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__VEBT____Definitions__OVEBT,type,
    produc2757191886755552429T_VEBT: ( complex * vEBT_VEBT ) > produc7151440242714718331T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Complex__Ocomplex,type,
    produc7948753499206759283omplex: ( int * complex ) > produc5838698208256999739omplex ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: fun_in1696471567231710611nt_int ).

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

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Real__Oreal,type,
    produc801115645435158769t_real: ( int * real ) > produc679980390762269497t_real ).

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

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__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 * num ) > product_prod_nat_num ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc487386426758144856at_nat: ( nat * product_prod_nat_nat ) > produc7248412053542808358at_nat ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc599794634098209291T_VEBT: ( nat * vEBT_VEBT ) > produc8025551001238799321T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
    product_Pair_num_num: ( num * num ) > product_prod_num_num ).

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

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    produc1693001998875562995omplex: ( real * complex ) > produc6979889472282505531omplex ).

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 * nat ) > produc3741383161447143261al_nat ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
    produc4511245868158468465l_real: ( real * real ) > produc2422161461964618553l_real ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    produc6931449550656315951T_VEBT: ( real * vEBT_VEBT ) > produc3757001726724277373T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    produc3790773574474814305omplex: ( set_complex * set_complex ) > produc8064648209034914857omplex ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
    produc6363374080413544029et_int: ( set_int * set_int ) > produc2115011035271226405et_int ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4532415448927165861et_nat: ( set_nat * set_nat ) > produc7819656566062154093et_nat ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc2922128104949294807at_nat: ( set_Pr1261947904930325089at_nat * set_Pr1261947904930325089at_nat ) > produc3843707927480180839at_nat ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    produc9060074326276436823at_nat: ( set_Pr4329608150637261639at_nat * set_Pr4329608150637261639at_nat ) > produc1319942482725812455at_nat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Complex__Ocomplex,type,
    produc5617778602380981643omplex: ( vEBT_VEBT * complex ) > produc8380087813684007313omplex ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc736041933913180425BT_int: ( vEBT_VEBT * int ) > produc4894624898956917775BT_int ).

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

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc8117437818029410057T_real: ( vEBT_VEBT * real ) > produc5170161368751668367T_real ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc537772716801021591T_VEBT: ( vEBT_VEBT * vEBT_VEBT ) > produc8243902056947475879T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc7264323995540650962nteger: ( fun_Co3460408610320954864e_term * produc8923325533196201883nteger ) > produc7509114972854644512nteger ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4729913460834754785nt_int: ( fun_in5730379987311621695e_term * product_prod_int_int ) > produc4788131091073775919nt_int ).

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__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc565448451787809096nteger: ( fun_Pr5098669352780829690e_term * produc8923325533196201883nteger ) > produc2453742332425399446nteger ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc7320969046608631221nt_int: ( fun_Pr7763680713840462443e_term * product_prod_int_int ) > produc3043004577601930755nt_int ).

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

tff(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc6499014454317279255nteger: fun_fu4800041728187187127nteger ).

tff(sy_c_Product__Type_Ointernal__case__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc8005341501107743676_int_o: fun_fu1345883967705819239_int_o ).

tff(sy_c_Product__Type_Ointernal__case__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc7926200574084438792nt_int: fun_fu5207212825597307859nt_int ).

tff(sy_c_Product__Type_Ointernal__case__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc297006045350968285nt_int: fun_fu6741006549690681129nt_int ).

tff(sy_c_Product__Type_Ointernal__case__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    produc9189008374811844670_nat_o: fun_fu2847332191311216467_nat_o ).

tff(sy_c_Product__Type_Ointernal__case__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc8104577556401574439at_nat: fun_fu2716251949043459987at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    produc1553301316500091796er_int: ( fun_Co1092983284983208416er_int * produc8923325533196201883nteger ) > int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    produc1555791787009142072er_nat: ( fun_Co3583066429914935172er_nat * produc8923325533196201883nteger ) > nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    produc7336495610019696514er_num: ( fun_Co8521658056209057742er_num * produc8923325533196201883nteger ) > num ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    produc9125791028180074456eger_o: ( fun_Co3544761417380347684eger_o * produc8923325533196201883nteger ) > produc6271795597528267376eger_o ).

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__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
    produc6771430404735790350plex_o: fun_co1148497455491383662plex_o > fun_Pr7716844367077540446plex_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc4947309494688390418_int_o: fun_fu1345883967705819239_int_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc8211389475949308722nt_int: fun_fu5207212825597307859nt_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_fu6741006549690681129nt_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    produc8580519160106071146omplex: ( fun_in230608924162954550omplex * product_prod_int_int ) > set_complex ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
    produc73460835934605544et_int: ( fun_in7040156458970095540et_int * product_prod_int_int ) > set_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4251311855443802252et_nat: ( fun_in2630332840903238488et_nat * product_prod_int_int ) > set_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Real__Oreal_J,type,
    produc6452406959799940328t_real: ( fun_in6553099857819052212t_real * product_prod_int_int ) > set_real ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    produc5233655623923918146et_nat: ( fun_in7207726656141197838et_nat * product_prod_int_int ) > set_set_nat ).

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
    produc1830744345554046123nteger: fun_na8135612474337863287nteger > fun_Pr6026251430685297nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    produc1917071388513777916omplex: ( fun_na1142814060743658952omplex * product_prod_nat_nat ) > complex ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
    produc6840382203811409530at_int: fun_nat_fun_nat_int > fun_Pr7647638413817135296at_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 > fun_Pr2118491707810114563at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
    produc6207742614233964070at_rat: ( fun_nat_fun_nat_rat * product_prod_nat_nat ) > rat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
    produc1703576794950452218t_real: fun_nat_fun_nat_real > fun_Pr672294278902479040t_real ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    produc6395135316500328296_nat_o: fun_fu2847332191311216467_nat_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc4528391901023682385at_nat: fun_fu2716251949043459987at_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_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
    produc5414030515140494994real_o: fun_real_fun_real_o > fun_Pr4814023175418636514real_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J_001_Eo,type,
    produc3914248068834153634plex_o: fun_se823489619503225858plex_o > fun_Pr6517966409240040690plex_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J_001_Eo,type,
    produc4109468873575309990_int_o: fun_se2051630309975736070_int_o > fun_Pr4126283446944868854_int_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
    produc6247414631856714078_nat_o: fun_se2170385732985038782_nat_o > fun_Pr3947133745332865710_nat_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
    produc410239310623530412_nat_o: fun_se5639961202347202350_nat_o > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
    produc7733317944277263833eger_o: fun_fu3422731660242565741eger_o > fun_Pr1347628893830317649eger_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    produc5772012072983024995omplex: ( fun_fu15390479812311035omplex * produc7509114972854644512nteger ) > set_complex ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Int__Oint_J,type,
    produc3783258407033660385et_int: ( fun_fu1569576118606662777et_int * produc7509114972854644512nteger ) > set_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc7961109426542857093et_nat: ( fun_fu6383124537394581533et_nat * produc7509114972854644512nteger ) > set_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Real__Oreal_J,type,
    produc5067746511659309921t_real: ( fun_fu7234941744988193529t_real * produc7509114972854644512nteger ) > set_real ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
    produc7271678899836001034_int_o: fun_fu2978539155105838238_int_o > fun_Pr1196100989263358594_int_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    produc4974371134869959794omplex: ( fun_fu5523889554619339274omplex * produc4788131091073775919nt_int ) > set_complex ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
    produc7373287791920631523eger_o: fun_fu7695273699059480439eger_o > fun_Pr7015759342434529115eger_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
    produc2207054017359051062_int_o: fun_fu6865642474282033866_int_o > fun_Pr3540825087398332590_int_o ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: fun_Pr982841511892225912nt_int ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
    product_snd_int_int: fun_Pr982841511892225912nt_int ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: fun_Pr2602117396471556196at_nat ).

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

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

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun_ra4503298312962071866nt_int ).

tff(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

tff(sy_c_Rat_Onormalize,type,
    normalize: fun_Pr2478283820680522243nt_int ).

tff(sy_c_Rat_Oof__int,type,
    of_int: fun_int_rat ).

tff(sy_c_Rat_Opositive,type,
    positive: fun_rat_o ).

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

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

tff(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

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

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
    real_V975177566351809787t_real: fun_re7681423839302404990l_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: fun_complex_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
    real_V7735802525324610683m_real: fun_real_real ).

tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: fun_real_complex ).

tff(sy_c_Relation_OField_001t__Nat__Onat,type,
    field_nat: set_Pr1261947904930325089at_nat > set_nat ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
    algebr932160517623751201me_int: fun_int_fun_int_o ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
    algebr934650988132801477me_nat: fun_nat_fun_nat_o ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
    divide6298287555418463151nteger: ( code_integer * code_integer ) > code_integer ).

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

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: 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: rat > fun_rat_rat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: fun_re7681423839302404990l_real ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
    dvd_dvd_Code_integer: fun_Co2081214437782444048eger_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
    dvd_dvd_complex: fun_co1148497455491383662plex_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
    dvd_dvd_rat: fun_rat_fun_rat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
    dvd_dvd_real: fun_real_fun_real_o ).

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

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: 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__Code____Numeral__Ointeger,type,
    zero_n356916108424825756nteger: $o > code_integer ).

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

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: fun_o_int ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: $o > nat ).

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

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

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

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

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

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

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

tff(sy_c_Series_Osummable_001t__Int__Oint,type,
    summable_int: fun_nat_int > $o ).

tff(sy_c_Series_Osummable_001t__Nat__Onat,type,
    summable_nat: fun_nat_nat > $o ).

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

tff(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
    sums_complex: fun_nat_complex > fun_complex_o ).

tff(sy_c_Series_Osums_001t__Int__Oint,type,
    sums_int: fun_nat_int > fun_int_o ).

tff(sy_c_Series_Osums_001t__Nat__Onat,type,
    sums_nat: fun_nat_nat > fun_nat_o ).

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

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

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

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

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

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

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

tff(sy_c_Set_OCollect_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: fun_nat_o > set_nat ).

tff(sy_c_Set_OCollect_001t__Num__Onum,type,
    collect_num: fun_num_o > set_num ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Complex__Ocomplex_J_Mt__Set__Oset_It__Complex__Ocomplex_J_J,type,
    collec5108298041176329748omplex: fun_Pr6517966409240040690plex_o > set_Pr6308028481084910985omplex ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    collec6662362479098859352et_nat: fun_Pr3947133745332865710_nat_o > set_Pr5488025237498180813et_nat ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    collec6321179662152712658at_nat: fun_Pr3059359313727237556_nat_o > set_Pr4329608150637261639at_nat ).

tff(sy_c_Set_OCollect_001t__Rat__Orat,type,
    collect_rat: fun_rat_o > set_rat ).

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

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    collect_set_complex: fun_set_complex_o > set_set_complex ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
    collect_set_int: fun_set_int_o > set_set_int ).

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

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Set_OPow_001t__Nat__Onat,type,
    pow_nat: set_nat > set_set_nat ).

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

tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
    image_int_nat: ( fun_int_nat * set_int ) > set_nat ).

tff(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    image_list_nat_nat: ( fun_list_nat_nat * set_list_nat ) > set_nat ).

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

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

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

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    image_5971271580939081552omplex: ( fun_re3844104906058862192omplex * set_real ) > set_fi4554929511873752355omplex ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
    image_real_real: ( fun_real_real * set_real ) > set_real ).

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

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

tff(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    insert8211810215607154385at_nat: ( product_prod_nat_nat * set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    insert9069300056098147895at_nat: ( produc3843707927480180839at_nat * set_Pr4329608150637261639at_nat ) > set_Pr4329608150637261639at_nat ).

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

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

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

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

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
    set_ord_atMost_int: 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_OatMost_001t__Num__Onum,type,
    set_ord_atMost_num: num > set_num ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
    set_ord_atMost_rat: rat > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
    set_or58775011639299419et_int: set_int > set_set_int ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4236626031148496127et_nat: set_nat > set_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
    set_or6656581121297822940st_int: fun_in7040156458970095540et_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: ( nat * nat ) > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
    set_or5832277885323065728an_int: fun_in7040156458970095540et_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
    set_ord_lessThan_num: num > set_num ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
    set_ord_lessThan_rat: rat > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or890127255671739683et_nat: set_nat > set_set_nat ).

tff(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
    topolo4902158794631467389eq_nat: fun_nat_nat > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
    topolo1459490580787246023eq_num: fun_nat_num > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
    topolo4267028734544971653eq_rat: fun_nat_rat > $o ).

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

tff(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
    topolo3100542954746470799et_int: fun_nat_set_int > $o ).

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

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

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

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Odiffs_001t__Code____Numeral__Ointeger,type,
    diffs_Code_integer: fun_nat_Code_integer > fun_nat_Code_integer ).

tff(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
    diffs_complex: fun_nat_complex > fun_nat_complex ).

tff(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
    diffs_int: fun_nat_int > fun_nat_int ).

tff(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
    diffs_rat: fun_nat_rat > fun_nat_rat ).

tff(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
    diffs_real: fun_nat_real > fun_nat_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

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

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

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

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

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr8877776288780692068_nat_o ).

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

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

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

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_o ).

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

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: fun_li1025265955262268170_nat_o > fun_list_nat_o ).

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

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: fun_Pr9018117161326028078_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: fun_Pr2028404973800837934_nat_o > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    accp_P3113834385874906142um_num: fun_Pr4799464687642144302_num_o > fun_Pr6331313853612757382_num_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: ( fun_Pr8877776288780692068_nat_o * produc9072475918466114483BT_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__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_Pr8463018485500633518_nat_o * produc9217583074051984497at_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
    accp_VEBT_VEBT: fun_VE2962686792283380846VEBT_o > fun_VEBT_VEBT_o ).

tff(sy_c_Wellfounded_Ofinite__psubset_001t__Complex__Ocomplex,type,
    finite8643634255014194347omplex: set_Pr6308028481084910985omplex ).

tff(sy_c_Wellfounded_Ofinite__psubset_001t__Int__Oint,type,
    finite_psubset_int: set_Pr2522554150109002629et_int ).

tff(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
    finite_psubset_nat: set_Pr5488025237498180813et_nat ).

tff(sy_c_Wellfounded_Ofinite__psubset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite469560695537375940at_nat: set_Pr4329608150637261639at_nat ).

tff(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

tff(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
    measure_int: fun_int_nat > set_Pr958786334691620121nt_int ).

tff(sy_c_Wellfounded_Omin__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Owf_001t__Int__Oint,type,
    wf_int: set_Pr958786334691620121nt_int > $o ).

tff(sy_c_aa_001_Eo_001_Eo,type,
    aa_o_o: ( fun_o_o * $o ) > $o ).

tff(sy_c_aa_001_Eo_001t__Int__Oint,type,
    aa_o_int: ( fun_o_int * $o ) > int ).

tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_o_P3134438092330336605nt_int: ( fun_o_8988054265482332260nt_int * $o ) > product_prod_int_int ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001_Eo,type,
    aa_Code_integer_o: ( fun_Code_integer_o * code_integer ) > $o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    aa_Cod1783974652533396728nteger: ( fun_Co8499238673658658103nteger * code_integer ) > code_integer ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    aa_Code_integer_int: ( fun_Code_integer_int * code_integer ) > int ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    aa_Code_integer_nat: ( fun_Code_integer_nat * code_integer ) > nat ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    aa_Code_integer_num: ( fun_Code_integer_num * code_integer ) > num ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    aa_Cod3168588076859124107eger_o: ( fun_Co1766037875893669450eger_o * code_integer ) > produc6271795597528267376eger_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod6977894654539884864nteger: ( fun_Co6597186368970283863nteger * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
    aa_Cod326357343816142457eger_o: ( fun_Co2081214437782444048eger_o * code_integer ) > fun_Code_integer_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod4161879620478718930nteger: ( fun_Co1981900195452921617nteger * code_integer ) > fun_Co8499238673658658103nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Int__Oint_J,type,
    aa_Cod1242632294114111009er_int: ( fun_Co1092983284983208416er_int * code_integer ) > fun_Code_integer_int ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J,type,
    aa_Cod5420483313623307717er_nat: ( fun_Co3583066429914935172er_nat * code_integer ) > fun_Code_integer_nat ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Num__Onum_J,type,
    aa_Cod6128727250572145423er_num: ( fun_Co8521658056209057742er_num * code_integer ) > fun_Code_integer_num ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    aa_Cod1096751537565085541eger_o: ( fun_Co3544761417380347684eger_o * code_integer ) > fun_Co1766037875893669450eger_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_Cod8380929501330563836nteger: ( fun_Co2470855966376689427nteger * code_integer ) > fun_Co6597186368970283863nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod1578151519121547077nteger: ( fun_Co6226381647958292740nteger * code_integer ) > fun_nat_Code_integer ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_Cod3140087502949435154nteger: ( fun_Co5080488584548667985nteger * code_integer ) > fun_na8135612474337863287nteger ).

tff(sy_c_aa_001t__Complex__Ocomplex_001_Eo,type,
    aa_complex_o: ( fun_complex_o * complex ) > $o ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Code____Numeral__Ointeger,type,
    aa_com4217951221296830281nteger: ( fun_co8153928105989009288nteger * complex ) > code_integer ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    aa_complex_complex: ( fun_complex_complex * complex ) > complex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Int__Oint,type,
    aa_complex_int: ( fun_complex_int * complex ) > int ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    aa_complex_nat: ( fun_complex_nat * complex ) > nat ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Num__Onum,type,
    aa_complex_num: ( fun_complex_num * complex ) > num ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    aa_complex_rat: ( fun_complex_rat * complex ) > rat ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    aa_complex_real: ( fun_complex_real * complex ) > real ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    aa_com5808801942499186519plex_o: ( fun_co1148497455491383662plex_o * complex ) > fun_complex_o ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_Mt__fun_It__Complex__Ocomplex_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J_J,type,
    aa_com177695874307951165omplex: ( fun_co3732688428756473980omplex * complex ) > fun_co6147893885205795665omplex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
    aa_com8177640530696437650omplex: ( fun_co6147893885205795665omplex * complex ) > fun_co3258106519541737638omplex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_complex_fun_int_o: ( fun_co3431063785546946928_int_o * complex ) > fun_int_o ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Int__Oint_Mt__Complex__Ocomplex_J,type,
    aa_com9156536935073639875omplex: ( fun_co7673900812888889602omplex * complex ) > fun_int_complex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_com3205223359959579713nt_int: ( fun_co333780937270390912nt_int * complex ) > fun_int_int ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_complex_fun_nat_o: ( fun_co5676208753217392844_nat_o * complex ) > fun_nat_o ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
    aa_com8181001469867462503omplex: ( fun_co3258106519541737638omplex * complex ) > fun_nat_complex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_com2205869648058525157at_int: ( fun_co7370231399432144932at_int * complex ) > fun_nat_int ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_com6383720667567721865at_nat: ( fun_co636942507509095880at_nat * complex ) > fun_nat_nat ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
    aa_com4347742694490471604omplex: ( fun_co6883864892131739507omplex * complex ) > fun_na1142814060743658952omplex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_com1951324360330529881real_o: ( fun_co4998825143550818160real_o * complex ) > fun_real_o ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
    aa_com4766338864777664963omplex: ( fun_co286204877037288194omplex * complex ) > fun_real_complex ).

tff(sy_c_aa_001t__Int__Oint_001_Eo,type,
    aa_int_o: ( fun_int_o * int ) > $o ).

tff(sy_c_aa_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    aa_int_Code_integer: ( fun_int_Code_integer * int ) > code_integer ).

tff(sy_c_aa_001t__Int__Oint_001t__Complex__Ocomplex,type,
    aa_int_complex: ( fun_int_complex * int ) > complex ).

tff(sy_c_aa_001t__Int__Oint_001t__Int__Oint,type,
    aa_int_int: ( fun_int_int * int ) > int ).

tff(sy_c_aa_001t__Int__Oint_001t__Nat__Onat,type,
    aa_int_nat: ( fun_int_nat * int ) > nat ).

tff(sy_c_aa_001t__Int__Oint_001t__Num__Onum,type,
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    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_xa____,type,
    xa: nat ).

tff(sy_v_ya____,type,
    ya: nat ).

% Relevant facts (10389)
tff(fact_0_True,axiom,
    xa = mi ).

% True
tff(fact_1_False,axiom,
    ~ ( ( ya = mi )
      | ( ya = ma ) ) ).

% False
tff(fact_2__C0_C,axiom,
    ( ( ya != mi )
    & ( ya != ma ) ) ).

% "0"
tff(fact_3__C5_Oprems_C_I3_J,axiom,
    aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),xa) ).

% "5.prems"(3)
tff(fact_4__C5_Ohyps_C_I7_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),ma) ).

% "5.hyps"(7)
tff(fact_5__C5_Ohyps_C_I6_J,axiom,
    ( ( mi = ma )
   => ! [X: vEBT_VEBT] :
        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList))
       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) ) ).

% "5.hyps"(6)
tff(fact_6__C001_C,axiom,
    aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),deg) ).

% "001"
tff(fact_7_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_12) ) ).

% not_min_Null_member
tff(fact_8_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] :
      ( ( vEBT_Node(X11,X12,X13,X14) = vEBT_Node(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_9_option_Oinject,axiom,
    ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X2) = some_P7363390416028606310at_nat(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_10_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( some_num(X2) = some_num(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_11_prod_Oinject,axiom,
    ! [X1: int,X2: int,Y1: int,Y2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_12_prod_Oinject,axiom,
    ! [X1: fun_Co3460408610320954864e_term,X2: produc8923325533196201883nteger,Y1: fun_Co3460408610320954864e_term,Y2: produc8923325533196201883nteger] :
      ( ( produc7264323995540650962nteger(X1,X2) = produc7264323995540650962nteger(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_13_prod_Oinject,axiom,
    ! [X1: fun_Pr5098669352780829690e_term,X2: produc8923325533196201883nteger,Y1: fun_Pr5098669352780829690e_term,Y2: produc8923325533196201883nteger] :
      ( ( produc565448451787809096nteger(X1,X2) = produc565448451787809096nteger(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_14_prod_Oinject,axiom,
    ! [X1: fun_Pr7763680713840462443e_term,X2: product_prod_int_int,Y1: fun_Pr7763680713840462443e_term,Y2: product_prod_int_int] :
      ( ( produc7320969046608631221nt_int(X1,X2) = produc7320969046608631221nt_int(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_15_prod_Oinject,axiom,
    ! [X1: fun_in5730379987311621695e_term,X2: product_prod_int_int,Y1: fun_in5730379987311621695e_term,Y2: product_prod_int_int] :
      ( ( produc4729913460834754785nt_int(X1,X2) = produc4729913460834754785nt_int(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_16_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_17_old_Oprod_Oinject,axiom,
    ! [A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger,A2: fun_Co3460408610320954864e_term,B2: produc8923325533196201883nteger] :
      ( ( produc7264323995540650962nteger(A,B) = produc7264323995540650962nteger(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_18_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,A2: fun_Pr5098669352780829690e_term,B2: produc8923325533196201883nteger] :
      ( ( produc565448451787809096nteger(A,B) = produc565448451787809096nteger(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_19_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,A2: fun_Pr7763680713840462443e_term,B2: product_prod_int_int] :
      ( ( produc7320969046608631221nt_int(A,B) = produc7320969046608631221nt_int(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_20_old_Oprod_Oinject,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,A2: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
      ( ( produc4729913460834754785nt_int(A,B) = produc4729913460834754785nt_int(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_21_prod__decode__aux_Ocases,axiom,
    ! [Xb: product_prod_nat_nat] :
      ~ ! [K: nat,M: nat] : Xb != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K),M) ).

% prod_decode_aux.cases
tff(fact_22__C5_Ohyps_C_I1_J,axiom,
    aa_nat_o(vEBT_invar_vebt(summary),m) ).

% "5.hyps"(1)
tff(fact_23_old_Oprod_Oexhaust,axiom,
    ! [Yb: product_prod_int_int] :
      ~ ! [A3: int,B3: int] : Yb != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) ).

% old.prod.exhaust
tff(fact_24_old_Oprod_Oexhaust,axiom,
    ! [Yb: produc7509114972854644512nteger] :
      ~ ! [A3: fun_Co3460408610320954864e_term,B3: produc8923325533196201883nteger] : Yb != produc7264323995540650962nteger(A3,B3) ).

% old.prod.exhaust
tff(fact_25_old_Oprod_Oexhaust,axiom,
    ! [Yb: produc2453742332425399446nteger] :
      ~ ! [A3: fun_Pr5098669352780829690e_term,B3: produc8923325533196201883nteger] : Yb != produc565448451787809096nteger(A3,B3) ).

% old.prod.exhaust
tff(fact_26_old_Oprod_Oexhaust,axiom,
    ! [Yb: produc3043004577601930755nt_int] :
      ~ ! [A3: fun_Pr7763680713840462443e_term,B3: product_prod_int_int] : Yb != produc7320969046608631221nt_int(A3,B3) ).

% old.prod.exhaust
tff(fact_27_old_Oprod_Oexhaust,axiom,
    ! [Yb: produc4788131091073775919nt_int] :
      ~ ! [A3: fun_in5730379987311621695e_term,B3: product_prod_int_int] : Yb != produc4729913460834754785nt_int(A3,B3) ).

% old.prod.exhaust
tff(fact_28_surj__pair,axiom,
    ! [P: product_prod_int_int] :
    ? [X3: int,Y: int] : P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) ).

% surj_pair
tff(fact_29_surj__pair,axiom,
    ! [P: produc7509114972854644512nteger] :
    ? [X3: fun_Co3460408610320954864e_term,Y: produc8923325533196201883nteger] : P = produc7264323995540650962nteger(X3,Y) ).

% surj_pair
tff(fact_30_surj__pair,axiom,
    ! [P: produc2453742332425399446nteger] :
    ? [X3: fun_Pr5098669352780829690e_term,Y: produc8923325533196201883nteger] : P = produc565448451787809096nteger(X3,Y) ).

% surj_pair
tff(fact_31_surj__pair,axiom,
    ! [P: produc3043004577601930755nt_int] :
    ? [X3: fun_Pr7763680713840462443e_term,Y: product_prod_int_int] : P = produc7320969046608631221nt_int(X3,Y) ).

% surj_pair
tff(fact_32_surj__pair,axiom,
    ! [P: produc4788131091073775919nt_int] :
    ? [X3: fun_in5730379987311621695e_term,Y: product_prod_int_int] : P = produc4729913460834754785nt_int(X3,Y) ).

% surj_pair
tff(fact_33_prod__cases,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,P: product_prod_int_int] :
      ( ! [A3: int,B3: int] : aa_Pro4592068669816069595_int_o(P2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3))
     => aa_Pro4592068669816069595_int_o(P2,P) ) ).

% prod_cases
tff(fact_34_prod__cases,axiom,
    ! [P2: fun_Pr1347628893830317649eger_o,P: produc7509114972854644512nteger] :
      ( ! [A3: fun_Co3460408610320954864e_term,B3: produc8923325533196201883nteger] : aa_Pro4652312591602923194eger_o(P2,produc7264323995540650962nteger(A3,B3))
     => aa_Pro4652312591602923194eger_o(P2,P) ) ).

% prod_cases
tff(fact_35_prod__cases,axiom,
    ! [P2: fun_Pr7015759342434529115eger_o,P: produc2453742332425399446nteger] :
      ( ! [A3: fun_Pr5098669352780829690e_term,B3: produc8923325533196201883nteger] : aa_Pro4148064003102580676eger_o(P2,produc565448451787809096nteger(A3,B3))
     => aa_Pro4148064003102580676eger_o(P2,P) ) ).

% prod_cases
tff(fact_36_prod__cases,axiom,
    ! [P2: fun_Pr3540825087398332590_int_o,P: produc3043004577601930755nt_int] :
      ( ! [A3: fun_Pr7763680713840462443e_term,B3: product_prod_int_int] : aa_Pro2601676399929874711_int_o(P2,produc7320969046608631221nt_int(A3,B3))
     => aa_Pro2601676399929874711_int_o(P2,P) ) ).

% prod_cases
tff(fact_37_prod__cases,axiom,
    ! [P2: fun_Pr1196100989263358594_int_o,P: produc4788131091073775919nt_int] :
      ( ! [A3: fun_in5730379987311621695e_term,B3: product_prod_int_int] : aa_Pro5451197003417746155_int_o(P2,produc4729913460834754785nt_int(A3,B3))
     => aa_Pro5451197003417746155_int_o(P2,P) ) ).

% prod_cases
tff(fact_38_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( Dega = Nb ) ) ).

% deg_deg_n
tff(fact_39_mi__eq__ma__no__ch,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
     => ( ( Mia = Maa )
       => ( ! [X: vEBT_VEBT] :
              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeLista))
             => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) )
          & ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_1) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_40_prod__induct3,axiom,
    ! [P2: fun_Pr1347628893830317649eger_o,Xb: produc7509114972854644512nteger] :
      ( ! [A3: fun_Co3460408610320954864e_term,B3: code_integer,C: code_integer] : aa_Pro4652312591602923194eger_o(P2,produc7264323995540650962nteger(A3,produc1086072967326762835nteger(B3,C)))
     => aa_Pro4652312591602923194eger_o(P2,Xb) ) ).

% prod_induct3
tff(fact_41_prod__induct3,axiom,
    ! [P2: fun_Pr7015759342434529115eger_o,Xb: produc2453742332425399446nteger] :
      ( ! [A3: fun_Pr5098669352780829690e_term,B3: code_integer,C: code_integer] : aa_Pro4148064003102580676eger_o(P2,produc565448451787809096nteger(A3,produc1086072967326762835nteger(B3,C)))
     => aa_Pro4148064003102580676eger_o(P2,Xb) ) ).

% prod_induct3
tff(fact_42_prod__induct3,axiom,
    ! [P2: fun_Pr3540825087398332590_int_o,Xb: produc3043004577601930755nt_int] :
      ( ! [A3: fun_Pr7763680713840462443e_term,B3: int,C: int] : aa_Pro2601676399929874711_int_o(P2,produc7320969046608631221nt_int(A3,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B3),C)))
     => aa_Pro2601676399929874711_int_o(P2,Xb) ) ).

% prod_induct3
tff(fact_43_prod__induct3,axiom,
    ! [P2: fun_Pr1196100989263358594_int_o,Xb: produc4788131091073775919nt_int] :
      ( ! [A3: fun_in5730379987311621695e_term,B3: int,C: int] : aa_Pro5451197003417746155_int_o(P2,produc4729913460834754785nt_int(A3,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B3),C)))
     => aa_Pro5451197003417746155_int_o(P2,Xb) ) ).

% prod_induct3
tff(fact_44_prod__cases3,axiom,
    ! [Yb: produc7509114972854644512nteger] :
      ~ ! [A3: fun_Co3460408610320954864e_term,B3: code_integer,C: code_integer] : Yb != produc7264323995540650962nteger(A3,produc1086072967326762835nteger(B3,C)) ).

% prod_cases3
tff(fact_45_prod__cases3,axiom,
    ! [Yb: produc2453742332425399446nteger] :
      ~ ! [A3: fun_Pr5098669352780829690e_term,B3: code_integer,C: code_integer] : Yb != produc565448451787809096nteger(A3,produc1086072967326762835nteger(B3,C)) ).

% prod_cases3
tff(fact_46_prod__cases3,axiom,
    ! [Yb: produc3043004577601930755nt_int] :
      ~ ! [A3: fun_Pr7763680713840462443e_term,B3: int,C: int] : Yb != produc7320969046608631221nt_int(A3,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B3),C)) ).

% prod_cases3
tff(fact_47_prod__cases3,axiom,
    ! [Yb: produc4788131091073775919nt_int] :
      ~ ! [A3: fun_in5730379987311621695e_term,B3: int,C: int] : Yb != produc4729913460834754785nt_int(A3,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B3),C)) ).

% prod_cases3
tff(fact_48_Pair__inject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_49_Pair__inject,axiom,
    ! [A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger,A2: fun_Co3460408610320954864e_term,B2: produc8923325533196201883nteger] :
      ( ( produc7264323995540650962nteger(A,B) = produc7264323995540650962nteger(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_50_Pair__inject,axiom,
    ! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,A2: fun_Pr5098669352780829690e_term,B2: produc8923325533196201883nteger] :
      ( ( produc565448451787809096nteger(A,B) = produc565448451787809096nteger(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_51_Pair__inject,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,A2: fun_Pr7763680713840462443e_term,B2: product_prod_int_int] :
      ( ( produc7320969046608631221nt_int(A,B) = produc7320969046608631221nt_int(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_52_Pair__inject,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,A2: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
      ( ( produc4729913460834754785nt_int(A,B) = produc4729913460834754785nt_int(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_53_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
       => aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_54_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
      <=> aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_55_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb,Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_56_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_57_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Ta),D)
    <=> aa_nat_o(vEBT_invar_vebt(Ta),D) ) ).

% valid_eq
tff(fact_58_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),D)
     => aa_nat_o(vEBT_VEBT_valid(Ta),D) ) ).

% valid_eq1
tff(fact_59_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Ta),D)
     => aa_nat_o(vEBT_invar_vebt(Ta),D) ) ).

% valid_eq2
tff(fact_60_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ~ aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ).

% min_Null_member
tff(fact_61_order__refl,axiom,
    ! [Xb: set_int] : aa_set_int_o(ord_less_eq_set_int(Xb),Xb) ).

% order_refl
tff(fact_62_order__refl,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Xb) ).

% order_refl
tff(fact_63_order__refl,axiom,
    ! [Xb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Xb) ).

% order_refl
tff(fact_64_order__refl,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Xb) ).

% order_refl
tff(fact_65_order__refl,axiom,
    ! [Xb: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xb) ).

% order_refl
tff(fact_66_dual__order_Orefl,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(A),A) ).

% dual_order.refl
tff(fact_67_dual__order_Orefl,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% dual_order.refl
tff(fact_68_dual__order_Orefl,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% dual_order.refl
tff(fact_69_dual__order_Orefl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% dual_order.refl
tff(fact_70_dual__order_Orefl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% dual_order.refl
tff(fact_71_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
     => ? [Info2: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Nb)),TreeList,S) ) ).

% deg_SUcn_Node
tff(fact_72_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
      <=> aa_set_nat_o(member_nat(Xb),vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_73__C5_Ohyps_C_I3_J,axiom,
    m = aa_nat_nat(suc,na) ).

% "5.hyps"(3)
tff(fact_74_mem__Collect__eq,axiom,
    ! [A: complex,P2: fun_complex_o] :
      ( aa_set_complex_o(member_complex(A),collect_complex(P2))
    <=> aa_complex_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_75_mem__Collect__eq,axiom,
    ! [A: real,P2: fun_real_o] :
      ( member_real(A,collect_real(P2))
    <=> aa_real_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_76_mem__Collect__eq,axiom,
    ! [A: list_nat,P2: fun_list_nat_o] :
      ( member_list_nat(A,collect_list_nat(P2))
    <=> aa_list_nat_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_77_mem__Collect__eq,axiom,
    ! [A: set_nat,P2: fun_set_nat_o] :
      ( member_set_nat(A,collect_set_nat(P2))
    <=> aa_set_nat_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_78_mem__Collect__eq,axiom,
    ! [A: nat,P2: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A),collect_nat(P2))
    <=> aa_nat_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_79_mem__Collect__eq,axiom,
    ! [A: int,P2: fun_int_o] :
      ( aa_set_int_o(member_int(A),collect_int(P2))
    <=> aa_int_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_80_Collect__mem__eq,axiom,
    ! [A4: set_complex] : collect_complex(aTP_Lamm_a(A4)) = A4 ).

% Collect_mem_eq
tff(fact_81_Collect__mem__eq,axiom,
    ! [A4: set_real] : collect_real(aTP_Lamm_aa(A4)) = A4 ).

% Collect_mem_eq
tff(fact_82_Collect__mem__eq,axiom,
    ! [A4: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A4)) = A4 ).

% Collect_mem_eq
tff(fact_83_Collect__mem__eq,axiom,
    ! [A4: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A4)) = A4 ).

% Collect_mem_eq
tff(fact_84_Collect__mem__eq,axiom,
    ! [A4: set_nat] : collect_nat(aTP_Lamm_ad(A4)) = A4 ).

% Collect_mem_eq
tff(fact_85_Collect__mem__eq,axiom,
    ! [A4: set_int] : collect_int(aTP_Lamm_ae(A4)) = A4 ).

% Collect_mem_eq
tff(fact_86_Collect__cong,axiom,
    ! [P2: fun_real_o,Q: fun_real_o] :
      ( ! [X3: real] :
          ( aa_real_o(P2,X3)
        <=> aa_real_o(Q,X3) )
     => ( collect_real(P2) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_87_Collect__cong,axiom,
    ! [P2: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X3: list_nat] :
          ( aa_list_nat_o(P2,X3)
        <=> aa_list_nat_o(Q,X3) )
     => ( collect_list_nat(P2) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_88_Collect__cong,axiom,
    ! [P2: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X3: set_nat] :
          ( aa_set_nat_o(P2,X3)
        <=> aa_set_nat_o(Q,X3) )
     => ( collect_set_nat(P2) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_89_Collect__cong,axiom,
    ! [P2: fun_nat_o,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(P2,X3)
        <=> aa_nat_o(Q,X3) )
     => ( collect_nat(P2) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_90_Collect__cong,axiom,
    ! [P2: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(P2,X3)
        <=> aa_int_o(Q,X3) )
     => ( collect_int(P2) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_91_order__antisym__conv,axiom,
    ! [Yb: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Yb),Xb)
     => ( aa_set_int_o(ord_less_eq_set_int(Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% order_antisym_conv
tff(fact_92_order__antisym__conv,axiom,
    ! [Yb: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% order_antisym_conv
tff(fact_93_order__antisym__conv,axiom,
    ! [Yb: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% order_antisym_conv
tff(fact_94_order__antisym__conv,axiom,
    ! [Yb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% order_antisym_conv
tff(fact_95_order__antisym__conv,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% order_antisym_conv
tff(fact_96_linorder__le__cases,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb) ) ).

% linorder_le_cases
tff(fact_97_linorder__le__cases,axiom,
    ! [Xb: num,Yb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb) ) ).

% linorder_le_cases
tff(fact_98_linorder__le__cases,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb) ) ).

% linorder_le_cases
tff(fact_99_linorder__le__cases,axiom,
    ! [Xb: int,Yb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb) ) ).

% linorder_le_cases
tff(fact_100_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_101_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_num(F,B) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_102_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_103_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_int(F,B) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_104_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_rat(F,B) = C2 )
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_105_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_num(F,B) = C2 )
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_106_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_nat(F,B) = C2 )
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_107_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_int(F,B) = C2 )
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_108_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_rat(F,B) = C2 )
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_109_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_num(F,B) = C2 )
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_110_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_111_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C2: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_112_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C2: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_113_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C2: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_rat_int(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_114_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( ( A = aa_num_rat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_115_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B: num,C2: num] :
      ( ( A = aa_num_num(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_116_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C2: num] :
      ( ( A = aa_num_nat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_117_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B: num,C2: num] :
      ( ( A = aa_num_int(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_num_int(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_118_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C2: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_119_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C2: nat] :
      ( ( A = aa_nat_num(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_120_linorder__linear,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb) ) ).

% linorder_linear
tff(fact_121_linorder__linear,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb) ) ).

% linorder_linear
tff(fact_122_linorder__linear,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb) ) ).

% linorder_linear
tff(fact_123_linorder__linear,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb) ) ).

% linorder_linear
tff(fact_124_order__eq__refl,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( ( Xb = Yb )
     => aa_set_int_o(ord_less_eq_set_int(Xb),Yb) ) ).

% order_eq_refl
tff(fact_125_order__eq__refl,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( Xb = Yb )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ).

% order_eq_refl
tff(fact_126_order__eq__refl,axiom,
    ! [Xb: num,Yb: num] :
      ( ( Xb = Yb )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb) ) ).

% order_eq_refl
tff(fact_127_order__eq__refl,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( Xb = Yb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ).

% order_eq_refl
tff(fact_128_order__eq__refl,axiom,
    ! [Xb: int,Yb: int] :
      ( ( Xb = Yb )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ).

% order_eq_refl
tff(fact_129_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_130_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_131_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_132_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_133_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_134_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,B)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_135_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,B)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_136_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,B)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_137_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_138_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,B)),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_139_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_140_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_141_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_142_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_143_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C2)) ) ) ) ).

% order_subst1
tff(fact_144_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C2)) ) ) ) ).

% order_subst1
tff(fact_145_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C2)) ) ) ) ).

% order_subst1
tff(fact_146_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B: int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,X3)),aa_int_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,C2)) ) ) ) ).

% order_subst1
tff(fact_147_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_148_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_149_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B)
        & aa_set_int_o(ord_less_eq_set_int(B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_150_Orderings_Oorder__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_151_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_152_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_153_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_154_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_155_antisym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_156_antisym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_157_antisym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_158_antisym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_159_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A)
     => ( aa_set_int_o(ord_less_eq_set_int(C2),B)
       => aa_set_int_o(ord_less_eq_set_int(C2),A) ) ) ).

% dual_order.trans
tff(fact_160_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% dual_order.trans
tff(fact_161_dual__order_Otrans,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A) ) ) ).

% dual_order.trans
tff(fact_162_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% dual_order.trans
tff(fact_163_dual__order_Otrans,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% dual_order.trans
tff(fact_164_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A)
     => ( aa_set_int_o(ord_less_eq_set_int(A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_165_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_166_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_167_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_168_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_169_dual__order_Oeq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( aa_set_int_o(ord_less_eq_set_int(B),A)
        & aa_set_int_o(ord_less_eq_set_int(A),B) ) ) ).

% dual_order.eq_iff
tff(fact_170_dual__order_Oeq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_171_dual__order_Oeq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_172_dual__order_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_173_dual__order_Oeq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_174_linorder__wlog,axiom,
    ! [P2: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A3),B3)
         => aa_rat_o(aa_rat_fun_rat_o(P2,A3),B3) )
     => ( ! [A3: rat,B3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(P2,B3),A3)
           => aa_rat_o(aa_rat_fun_rat_o(P2,A3),B3) )
       => aa_rat_o(aa_rat_fun_rat_o(P2,A),B) ) ) ).

% linorder_wlog
tff(fact_175_linorder__wlog,axiom,
    ! [P2: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A3),B3)
         => aa_num_o(aa_num_fun_num_o(P2,A3),B3) )
     => ( ! [A3: num,B3: num] :
            ( aa_num_o(aa_num_fun_num_o(P2,B3),A3)
           => aa_num_o(aa_num_fun_num_o(P2,A3),B3) )
       => aa_num_o(aa_num_fun_num_o(P2,A),B) ) ) ).

% linorder_wlog
tff(fact_176_linorder__wlog,axiom,
    ! [P2: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),B3)
         => aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3) )
     => ( ! [A3: nat,B3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P2,B3),A3)
           => aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3) )
       => aa_nat_o(aa_nat_fun_nat_o(P2,A),B) ) ) ).

% linorder_wlog
tff(fact_177_linorder__wlog,axiom,
    ! [P2: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B3)
         => aa_int_o(aa_int_fun_int_o(P2,A3),B3) )
     => ( ! [A3: int,B3: int] :
            ( aa_int_o(aa_int_fun_int_o(P2,B3),A3)
           => aa_int_o(aa_int_fun_int_o(P2,A3),B3) )
       => aa_int_o(aa_int_fun_int_o(P2,A),B) ) ) ).

% linorder_wlog
tff(fact_178_order__trans,axiom,
    ! [Xb: set_int,Yb: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Yb)
     => ( aa_set_int_o(ord_less_eq_set_int(Yb),Z)
       => aa_set_int_o(ord_less_eq_set_int(Xb),Z) ) ) ).

% order_trans
tff(fact_179_order__trans,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z) ) ) ).

% order_trans
tff(fact_180_order__trans,axiom,
    ! [Xb: num,Yb: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z) ) ) ).

% order_trans
tff(fact_181_order__trans,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z) ) ) ).

% order_trans
tff(fact_182_order__trans,axiom,
    ! [Xb: int,Yb: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z) ) ) ).

% order_trans
tff(fact_183_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),C2)
       => aa_set_int_o(ord_less_eq_set_int(A),C2) ) ) ).

% order.trans
tff(fact_184_order_Otrans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2) ) ) ).

% order.trans
tff(fact_185_order_Otrans,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C2) ) ) ).

% order.trans
tff(fact_186_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% order.trans
tff(fact_187_order_Otrans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% order.trans
tff(fact_188_order__antisym,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Yb)
     => ( aa_set_int_o(ord_less_eq_set_int(Yb),Xb)
       => ( Xb = Yb ) ) ) ).

% order_antisym
tff(fact_189_order__antisym,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb)
       => ( Xb = Yb ) ) ) ).

% order_antisym
tff(fact_190_order__antisym,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb)
       => ( Xb = Yb ) ) ) ).

% order_antisym
tff(fact_191_order__antisym,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
       => ( Xb = Yb ) ) ) ).

% order_antisym
tff(fact_192_order__antisym,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb)
       => ( Xb = Yb ) ) ) ).

% order_antisym
tff(fact_193_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( ( B = C2 )
       => aa_set_int_o(ord_less_eq_set_int(A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_194_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( B = C2 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_195_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( B = C2 )
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_196_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( B = C2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_197_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( B = C2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_198_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( A = B )
     => ( aa_set_int_o(ord_less_eq_set_int(B),C2)
       => aa_set_int_o(ord_less_eq_set_int(A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_199_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_200_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_201_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_202_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_203_order__class_Oorder__eq__iff,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( ( Xb = Yb )
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xb),Yb)
        & aa_set_int_o(ord_less_eq_set_int(Yb),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_204_order__class_Oorder__eq__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( Xb = Yb )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_205_order__class_Oorder__eq__iff,axiom,
    ! [Xb: num,Yb: num] :
      ( ( Xb = Yb )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_206_order__class_Oorder__eq__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( Xb = Yb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_207_order__class_Oorder__eq__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( ( Xb = Yb )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_208_le__cases3,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
       => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z) )
     => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z)
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Yb) )
         => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Yb)
             => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb) )
           => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z)
               => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ) ) ) ) ) ).

% le_cases3
tff(fact_209_le__cases3,axiom,
    ! [Xb: num,Yb: num,Z: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
       => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Z) )
     => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z) )
       => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z)
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Yb) )
         => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Yb)
             => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb) )
           => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Z)
               => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb) )
             => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb)
                 => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb) ) ) ) ) ) ) ).

% le_cases3
tff(fact_210_le__cases3,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Z) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Yb) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Yb)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Z)
               => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ) ) ) ) ) ).

% le_cases3
tff(fact_211_le__cases3,axiom,
    ! [Xb: int,Yb: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
       => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Z) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Yb) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Yb)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Z)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ) ) ) ) ) ).

% le_cases3
tff(fact_212_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_213_nle__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_214_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_215_nle__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_216_Suc__le__mono,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% Suc_le_mono
tff(fact_217_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N2))),aa_nat_set_int(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
       => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,N)),aa_nat_set_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_218_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_219_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N2))),aa_nat_num(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_220_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N2))),aa_nat_nat(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_221_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_222_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,N2)),aa_nat_set_int(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
       => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,Nb)),aa_nat_set_int(F,N)) ) ) ).

% lift_Suc_mono_le
tff(fact_223_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N)) ) ) ).

% lift_Suc_mono_le
tff(fact_224_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Nb)),aa_nat_num(F,N)) ) ) ).

% lift_Suc_mono_le
tff(fact_225_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N)) ) ) ).

% lift_Suc_mono_le
tff(fact_226_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Nb)),aa_nat_int(F,N)) ) ) ).

% lift_Suc_mono_le
tff(fact_227_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_228_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_229_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_230_transitive__stepwise__le,axiom,
    ! [Mb: nat,Nb: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X3),X3)
       => ( ! [X3: nat,Y: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X3),Y)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R,X3),Z2) ) )
         => ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(R,N2),aa_nat_nat(suc,N2))
           => aa_nat_o(aa_nat_fun_nat_o(R,Mb),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_231_nat__induct__at__least,axiom,
    ! [Mb: nat,Nb: nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(P2,Mb)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),N2)
             => ( aa_nat_o(P2,N2)
               => aa_nat_o(P2,aa_nat_nat(suc,N2)) ) )
         => aa_nat_o(P2,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_232_full__nat__induct,axiom,
    ! [P2: fun_nat_o,Nb: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),N2)
             => aa_nat_o(P2,M2) )
         => aa_nat_o(P2,N2) )
     => aa_nat_o(P2,Nb) ) ).

% full_nat_induct
tff(fact_233_not__less__eq__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% not_less_eq_eq
tff(fact_234_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_235__C5_Ohyps_C_I4_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "5.hyps"(4)
tff(fact_236_Suc__inject,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( aa_nat_nat(suc,Xb) = aa_nat_nat(suc,Yb) )
     => ( Xb = Yb ) ) ).

% Suc_inject
tff(fact_237_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa_nat_nat(suc,Nb) ).

% n_not_Suc_n
tff(fact_238_le__refl,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Nb) ).

% le_refl
tff(fact_239_le__trans,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2) ) ) ).

% le_trans
tff(fact_240_eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = Nb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% eq_imp_le
tff(fact_241_le__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% le_antisym
tff(fact_242_nat__le__linear,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% nat_le_linear
tff(fact_243_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(P2,K2)
     => ( ! [Y: nat] :
            ( aa_nat_o(P2,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B) )
       => ? [X3: nat] :
            ( aa_nat_o(P2,X3)
            & ! [Y3: nat] :
                ( aa_nat_o(P2,Y3)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),X3) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_244_Suc__leD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% Suc_leD
tff(fact_245_le__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_SucE
tff(fact_246_le__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_SucI
tff(fact_247_Suc__le__D,axiom,
    ! [Nb: nat,M3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M3)
     => ? [M: nat] : M3 = aa_nat_nat(suc,M) ) ).

% Suc_le_D
tff(fact_248_le__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        | ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_249_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_250_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,set_nat2(Xs)) ).

% List.finite_set
tff(fact_251_List_Ofinite__set,axiom,
    ! [Xs: list_int] : aa_set_int_o(finite_finite_int,set_int2(Xs)) ).

% List.finite_set
tff(fact_252_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : aa_set_complex_o(finite3207457112153483333omplex,set_complex2(Xs)) ).

% List.finite_set
tff(fact_253_List_Ofinite__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat] : aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,set_Pr5648618587558075414at_nat(Xs)) ).

% List.finite_set
tff(fact_254_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
    <=> ! [M4: nat] :
        ? [N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
          & aa_set_nat_o(member_nat(N3),S2) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_255_finite__nat__set__iff__bounded__le,axiom,
    ! [N4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
    <=> ? [M4: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),M4) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_256_finite__list,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A4)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A4 ) ).

% finite_list
tff(fact_257_finite__list,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A4 ) ).

% finite_list
tff(fact_258_finite__list,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ? [Xs2: list_int] : set_int2(Xs2) = A4 ) ).

% finite_list
tff(fact_259_finite__list,axiom,
    ! [A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A4 ) ).

% finite_list
tff(fact_260_finite__list,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ? [Xs2: list_P6011104703257516679at_nat] : set_Pr5648618587558075414at_nat(Xs2) = A4 ) ).

% finite_list
tff(fact_261_finite__has__maximal2,axiom,
    ! [A4: set_real,A: real] :
      ( finite_finite_real(A4)
     => ( member_real(A,A4)
       => ? [X3: real] :
            ( member_real(X3,A4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
            & ! [Xa: real] :
                ( member_real(Xa,A4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_262_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( member_set_nat(A,A4)
       => ? [X3: set_nat] :
            ( member_set_nat(X3,A4)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X3)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A4)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_263_finite__has__maximal2,axiom,
    ! [A4: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( member_set_int(A,A4)
       => ? [X3: set_int] :
            ( member_set_int(X3,A4)
            & aa_set_int_o(ord_less_eq_set_int(A),X3)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A4)
               => ( aa_set_int_o(ord_less_eq_set_int(X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_264_finite__has__maximal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( finite_finite_rat(A4)
     => ( member_rat(A,A4)
       => ? [X3: rat] :
            ( member_rat(X3,A4)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X3)
            & ! [Xa: rat] :
                ( member_rat(Xa,A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_265_finite__has__maximal2,axiom,
    ! [A4: set_num,A: num] :
      ( finite_finite_num(A4)
     => ( member_num(A,A4)
       => ? [X3: num] :
            ( member_num(X3,A4)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),X3)
            & ! [Xa: num] :
                ( member_num(Xa,A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_266_finite__has__maximal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X3)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_267_finite__has__maximal2,axiom,
    ! [A4: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(A),A4)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X3)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_268_finite__has__minimal2,axiom,
    ! [A4: set_real,A: real] :
      ( finite_finite_real(A4)
     => ( member_real(A,A4)
       => ? [X3: real] :
            ( member_real(X3,A4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),A)
            & ! [Xa: real] :
                ( member_real(Xa,A4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_269_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( member_set_nat(A,A4)
       => ? [X3: set_nat] :
            ( member_set_nat(X3,A4)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),A)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A4)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_270_finite__has__minimal2,axiom,
    ! [A4: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( member_set_int(A,A4)
       => ? [X3: set_int] :
            ( member_set_int(X3,A4)
            & aa_set_int_o(ord_less_eq_set_int(X3),A)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A4)
               => ( aa_set_int_o(ord_less_eq_set_int(Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_271_finite__has__minimal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( finite_finite_rat(A4)
     => ( member_rat(A,A4)
       => ? [X3: rat] :
            ( member_rat(X3,A4)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),A)
            & ! [Xa: rat] :
                ( member_rat(Xa,A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_272_finite__has__minimal2,axiom,
    ! [A4: set_num,A: num] :
      ( finite_finite_num(A4)
     => ( member_num(A,A4)
       => ? [X3: num] :
            ( member_num(X3,A4)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),A)
            & ! [Xa: num] :
                ( member_num(Xa,A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_273_finite__has__minimal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),A)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_274_finite__has__minimal2,axiom,
    ! [A4: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(A),A4)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),A)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_275_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xb)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya) ).

% vebt_insert.simps(4)
tff(fact_276_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),Xb) ).

% vebt_member.simps(4)
tff(fact_277_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_0_not
tff(fact_278_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_tree_deg_neq_0
tff(fact_279_even__odd__cases,axiom,
    ! [Xb: nat] :
      ( ! [N2: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)
     => ~ ! [N2: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) ) ).

% even_odd_cases
tff(fact_280_le0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% le0
tff(fact_281_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A) ).

% bot_nat_0.extremum
tff(fact_282_add__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc_right
tff(fact_283_add__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        & ( Nb = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_284_Nat_Oadd__0__right,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),zero_zero_nat) = Mb ).

% Nat.add_0_right
tff(fact_285_nat__add__left__cancel__le,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_286_not__None__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ( Xb != none_P5556105721700978146at_nat )
    <=> ? [Y4: product_prod_nat_nat] : Xb = some_P7363390416028606310at_nat(Y4) ) ).

% not_None_eq
tff(fact_287_not__None__eq,axiom,
    ! [Xb: option_num] :
      ( ( Xb != none_num )
    <=> ? [Y4: num] : Xb = some_num(Y4) ) ).

% not_None_eq
tff(fact_288_not__Some__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ! [Y4: product_prod_nat_nat] : Xb != some_P7363390416028606310at_nat(Y4)
    <=> ( Xb = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_289_not__Some__eq,axiom,
    ! [Xb: option_num] :
      ( ! [Y4: num] : Xb != some_num(Y4)
    <=> ( Xb = none_num ) ) ).

% not_Some_eq
tff(fact_290_plus__nat_Oadd__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Nb) = Nb ).

% plus_nat.add_0
tff(fact_291_add__eq__self__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = Mb )
     => ( Nb = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_292_add__is__1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_293_one__is__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_294_add__Suc__shift,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) ).

% add_Suc_shift
tff(fact_295_add__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc
tff(fact_296_nat__arith_Osuc1,axiom,
    ! [A4: nat,K2: nat,A: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),A) )
     => ( aa_nat_nat(suc,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_297_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X2) ).

% option.distinct(1)
tff(fact_298_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != some_num(X2) ).

% option.distinct(1)
tff(fact_299_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X2) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_300_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = some_num(X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_301_option_Oexhaust,axiom,
    ! [Yb: option4927543243414619207at_nat] :
      ( ( Yb != none_P5556105721700978146at_nat )
     => ~ ! [X22: product_prod_nat_nat] : Yb != some_P7363390416028606310at_nat(X22) ) ).

% option.exhaust
tff(fact_302_option_Oexhaust,axiom,
    ! [Yb: option_num] :
      ( ( Yb != none_num )
     => ~ ! [X22: num] : Yb != some_num(X22) ) ).

% option.exhaust
tff(fact_303_split__option__ex,axiom,
    ! [P2: fun_op8028897214545316308_nat_o] :
      ( ? [X_13: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P2,X_13)
    <=> ( aa_opt7112393969400857165_nat_o(P2,none_P5556105721700978146at_nat)
        | ? [X4: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P2,some_P7363390416028606310at_nat(X4)) ) ) ).

% split_option_ex
tff(fact_304_split__option__ex,axiom,
    ! [P2: fun_option_num_o] :
      ( ? [X_13: option_num] : aa_option_num_o(P2,X_13)
    <=> ( aa_option_num_o(P2,none_num)
        | ? [X4: num] : aa_option_num_o(P2,some_num(X4)) ) ) ).

% split_option_ex
tff(fact_305_split__option__all,axiom,
    ! [P2: fun_op8028897214545316308_nat_o] :
      ( ! [X_13: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P2,X_13)
    <=> ( aa_opt7112393969400857165_nat_o(P2,none_P5556105721700978146at_nat)
        & ! [X4: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P2,some_P7363390416028606310at_nat(X4)) ) ) ).

% split_option_all
tff(fact_306_split__option__all,axiom,
    ! [P2: fun_option_num_o] :
      ( ! [X_13: option_num] : aa_option_num_o(P2,X_13)
    <=> ( aa_option_num_o(P2,none_num)
        & ! [X4: num] : aa_option_num_o(P2,some_num(X4)) ) ) ).

% split_option_all
tff(fact_307_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P2: fun_op1221328669798608686_nat_o,Yb: option4927543243414619207at_nat] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P2,Xb),Yb) )
     => ( ( ( Yb = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P2,Xb),Yb) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A3) )
             => ( ( Yb = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P2,Xb),Yb) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P2,Xb),Yb) ) ) ) ).

% combine_options_cases
tff(fact_308_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P2: fun_op7653269905477231939_num_o,Yb: option_num] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_option_num_o(aa_opt7868583491048818492_num_o(P2,Xb),Yb) )
     => ( ( ( Yb = none_num )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P2,Xb),Yb) )
       => ( ! [A3: product_prod_nat_nat,B3: num] :
              ( ( Xb = some_P7363390416028606310at_nat(A3) )
             => ( ( Yb = some_num(B3) )
               => aa_option_num_o(aa_opt7868583491048818492_num_o(P2,Xb),Yb) ) )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P2,Xb),Yb) ) ) ) ).

% combine_options_cases
tff(fact_309_combine__options__cases,axiom,
    ! [Xb: option_num,P2: fun_op5373868528577402299_nat_o,Yb: option4927543243414619207at_nat] :
      ( ( ( Xb = none_num )
       => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P2,Xb),Yb) )
     => ( ( ( Yb = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P2,Xb),Yb) )
       => ( ! [A3: num,B3: product_prod_nat_nat] :
              ( ( Xb = some_num(A3) )
             => ( ( Yb = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P2,Xb),Yb) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P2,Xb),Yb) ) ) ) ).

% combine_options_cases
tff(fact_310_combine__options__cases,axiom,
    ! [Xb: option_num,P2: fun_op493924057549976310_num_o,Yb: option_num] :
      ( ( ( Xb = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(P2,Xb),Yb) )
     => ( ( ( Yb = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P2,Xb),Yb) )
       => ( ! [A3: num,B3: num] :
              ( ( Xb = some_num(A3) )
             => ( ( Yb = some_num(B3) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(P2,Xb),Yb) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P2,Xb),Yb) ) ) ) ).

% combine_options_cases
tff(fact_311_nat__le__iff__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3) ) ).

% nat_le_iff_add
tff(fact_312_trans__le__add2,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),J)) ) ).

% trans_le_add2
tff(fact_313_trans__le__add1,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Mb)) ) ).

% trans_le_add1
tff(fact_314_add__le__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) ) ).

% add_le_mono1
tff(fact_315_add__le__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_316_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L)
     => ? [N2: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N2) ) ).

% le_Suc_ex
tff(fact_317_add__leD2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ).

% add_leD2
tff(fact_318_add__leD1,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% add_leD1
tff(fact_319_le__add2,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% le_add2
tff(fact_320_le__add1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) ).

% le_add1
tff(fact_321_add__leE,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),Nb)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ) ).

% add_leE
tff(fact_322_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ~ ! [N2: nat] : Xb != aa_nat_nat(suc,N2) ) ).

% list_decode.cases
tff(fact_323_vebt__buildup_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( ( Xb != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va2: nat] : Xb != aa_nat_nat(suc,aa_nat_nat(suc,Va2)) ) ) ).

% vebt_buildup.cases
tff(fact_324_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ? [M: nat] : Nb = aa_nat_nat(suc,M) ) ).

% not0_implies_Suc
tff(fact_325_Zero__not__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_not_Suc
tff(fact_326_Zero__neq__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_neq_Suc
tff(fact_327_Suc__neq__Zero,axiom,
    ! [Mb: nat] : aa_nat_nat(suc,Mb) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_328_zero__induct,axiom,
    ! [P2: fun_nat_o,K2: nat] :
      ( aa_nat_o(P2,K2)
     => ( ! [N2: nat] :
            ( aa_nat_o(P2,aa_nat_nat(suc,N2))
           => aa_nat_o(P2,N2) )
       => aa_nat_o(P2,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_329_diff__induct,axiom,
    ! [P2: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
      ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,X3),zero_zero_nat)
     => ( ! [Y: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,zero_zero_nat),aa_nat_nat(suc,Y))
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P2,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(P2,aa_nat_nat(suc,X3)),aa_nat_nat(suc,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(P2,Mb),Nb) ) ) ) ).

% diff_induct
tff(fact_330_nat__induct,axiom,
    ! [P2: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P2,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(P2,N2)
           => aa_nat_o(P2,aa_nat_nat(suc,N2)) )
       => aa_nat_o(P2,Nb) ) ) ).

% nat_induct
tff(fact_331_old_Onat_Oexhaust,axiom,
    ! [Yb: nat] :
      ( ( Yb != zero_zero_nat )
     => ~ ! [Nat3: nat] : Yb != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_332_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_333_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_334_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_335_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_336_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_337_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_338_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_339_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% less_eq_nat.simps(1)
tff(fact_340_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),Xb) ).

% vebt_member.simps(2)
tff(fact_341_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_342_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts,S3),Xb) = vEBT_Node(Info,zero_zero_nat,Ts,S3) ).

% vebt_insert.simps(2)
tff(fact_343_rev__finite__subset,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% rev_finite_subset
tff(fact_344_rev__finite__subset,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => aa_set_complex_o(finite3207457112153483333omplex,A4) ) ) ).

% rev_finite_subset
tff(fact_345_rev__finite__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A4),B4)
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ) ).

% rev_finite_subset
tff(fact_346_rev__finite__subset,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => aa_set_int_o(finite_finite_int,A4) ) ) ).

% rev_finite_subset
tff(fact_347_infinite__super,axiom,
    ! [S2: set_nat,T: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
     => ( ~ aa_set_nat_o(finite_finite_nat,S2)
       => ~ aa_set_nat_o(finite_finite_nat,T) ) ) ).

% infinite_super
tff(fact_348_infinite__super,axiom,
    ! [S2: set_complex,T: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,T) ) ) ).

% infinite_super
tff(fact_349_infinite__super,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(S2),T)
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S2)
       => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,T) ) ) ).

% infinite_super
tff(fact_350_infinite__super,axiom,
    ! [S2: set_int,T: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(S2),T)
     => ( ~ aa_set_int_o(finite_finite_int,S2)
       => ~ aa_set_int_o(finite_finite_int,T) ) ) ).

% infinite_super
tff(fact_351_finite__subset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_subset
tff(fact_352_finite__subset,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => aa_set_complex_o(finite3207457112153483333omplex,A4) ) ) ).

% finite_subset
tff(fact_353_finite__subset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A4),B4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ) ).

% finite_subset
tff(fact_354_finite__subset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => aa_set_int_o(finite_finite_int,A4) ) ) ).

% finite_subset
tff(fact_355_bounded__Max__nat,axiom,
    ! [P2: fun_nat_o,Xb: nat,M5: nat] :
      ( aa_nat_o(P2,Xb)
     => ( ! [X3: nat] :
            ( aa_nat_o(P2,X3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M5) )
       => ~ ! [M: nat] :
              ( aa_nat_o(P2,M)
             => ~ ! [X: nat] :
                    ( aa_nat_o(P2,X)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),M) ) ) ) ) ).

% bounded_Max_nat
tff(fact_356_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),B4)
    <=> ! [X4: complex] :
          ( aa_set_complex_o(member_complex(X4),set_complex2(Xs))
         => aa_set_complex_o(member_complex(X4),B4) ) ) ).

% subset_code(1)
tff(fact_357_subset__code_I1_J,axiom,
    ! [Xs: list_real,B4: set_real] :
      ( ord_less_eq_set_real(set_real2(Xs),B4)
    <=> ! [X4: real] :
          ( member_real(X4,set_real2(Xs))
         => member_real(X4,B4) ) ) ).

% subset_code(1)
tff(fact_358_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(set_set_nat2(Xs),B4)
    <=> ! [X4: set_nat] :
          ( member_set_nat(X4,set_set_nat2(Xs))
         => member_set_nat(X4,B4) ) ) ).

% subset_code(1)
tff(fact_359_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B4)
    <=> ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => member_VEBT_VEBT(X4,B4) ) ) ).

% subset_code(1)
tff(fact_360_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Xs)),B4)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => aa_set_nat_o(member_nat(X4),B4) ) ) ).

% subset_code(1)
tff(fact_361_subset__code_I1_J,axiom,
    ! [Xs: list_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),B4)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(Xs))
         => aa_set_int_o(member_int(X4),B4) ) ) ).

% subset_code(1)
tff(fact_362_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S3),Xb) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S3) ).

% vebt_insert.simps(3)
tff(fact_363_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),Xb) ).

% vebt_member.simps(3)
tff(fact_364_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_365_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_366_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_367_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_368_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_369_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_370_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_371_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_372_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel1
tff(fact_373_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_374_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_375_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_376_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel2
tff(fact_377_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_378_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_379_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_380_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_381_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_382_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_383_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_384_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_385_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_386_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_387_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_388_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_389_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_390_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_391_add__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add_0
tff(fact_392_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_393_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_394_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Yb) )
    <=> ( ( Xb = zero_zero_nat )
        & ( Yb = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_395_add__eq__0__iff__both__eq__0,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Yb) = zero_zero_nat )
    <=> ( ( Xb = zero_zero_nat )
        & ( Yb = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_396_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(A),B) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_397_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_398_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_399_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_400_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_401_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_402_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_403_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_404_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_405_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_406_add__left__cancel,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_407_add__left__cancel,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_408_add__left__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_409_add__left__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_410_add__right__cancel,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_411_add__right__cancel,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_412_add__right__cancel,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_413_add__right__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_414_le__zero__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_415_add__le__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_left
tff(fact_416_add__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_417_add__le__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_418_add__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_419_add__le__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_right
tff(fact_420_add__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_421_add__le__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_422_add__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_423_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_424_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_425_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_426_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_427_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_428_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_429_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_430_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_431_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( aa_complex_complex(plus_plus_complex(B),A) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_432_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_433_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_434_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_435_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_436_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_437_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_438_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_439_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_440_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_441_zero__reorient,axiom,
    ! [Xb: complex] :
      ( ( zero_zero_complex = Xb )
    <=> ( Xb = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_442_zero__reorient,axiom,
    ! [Xb: real] :
      ( ( zero_zero_real = Xb )
    <=> ( Xb = zero_zero_real ) ) ).

% zero_reorient
tff(fact_443_zero__reorient,axiom,
    ! [Xb: rat] :
      ( ( zero_zero_rat = Xb )
    <=> ( Xb = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_444_zero__reorient,axiom,
    ! [Xb: nat] :
      ( ( zero_zero_nat = Xb )
    <=> ( Xb = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_445_zero__reorient,axiom,
    ! [Xb: int] :
      ( ( zero_zero_int = Xb )
    <=> ( Xb = zero_zero_int ) ) ).

% zero_reorient
tff(fact_446_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_447_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_448_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_449_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_450_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_451_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( aa_rat_rat(plus_plus_rat(I),K2) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_452_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_453_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_454_group__cancel_Oadd1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A4),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add1
tff(fact_455_group__cancel_Oadd1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4 = aa_rat_rat(plus_plus_rat(K2),A) )
     => ( aa_rat_rat(plus_plus_rat(A4),B) = aa_rat_rat(plus_plus_rat(K2),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add1
tff(fact_456_group__cancel_Oadd1,axiom,
    ! [A4: nat,K2: nat,A: nat,B: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_457_group__cancel_Oadd1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_458_group__cancel_Oadd2,axiom,
    ! [B4: real,K2: real,B: real,A: real] :
      ( ( B4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add2
tff(fact_459_group__cancel_Oadd2,axiom,
    ! [B4: rat,K2: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(plus_plus_rat(K2),B) )
     => ( aa_rat_rat(plus_plus_rat(A),B4) = aa_rat_rat(plus_plus_rat(K2),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add2
tff(fact_460_group__cancel_Oadd2,axiom,
    ! [B4: nat,K2: nat,B: nat,A: nat] :
      ( ( B4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_461_group__cancel_Oadd2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_462_add_Oassoc,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ).

% add.assoc
tff(fact_463_add_Oassoc,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) ).

% add.assoc
tff(fact_464_add_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% add.assoc
tff(fact_465_add_Oassoc,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% add.assoc
tff(fact_466_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_467_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_468_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_469_add_Oright__cancel,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_470_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_471_add_Oright__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_472_add_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) ).

% add.commute
tff(fact_473_add_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(B),A) ).

% add.commute
tff(fact_474_add_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) ).

% add.commute
tff(fact_475_add_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) ).

% add.commute
tff(fact_476_add_Oleft__commute,axiom,
    ! [B: real,A: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ).

% add.left_commute
tff(fact_477_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C2: rat] : aa_rat_rat(plus_plus_rat(B),aa_rat_rat(plus_plus_rat(A),C2)) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) ).

% add.left_commute
tff(fact_478_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% add.left_commute
tff(fact_479_add_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% add.left_commute
tff(fact_480_add__left__imp__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_481_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_482_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_483_add__left__imp__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_484_add__right__imp__eq,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_485_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_486_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_487_add__right__imp__eq,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_488_zero__le,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb) ).

% zero_le
tff(fact_489_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & ( K2 = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_490_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & ( K2 = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_491_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K2 = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_492_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K2 = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_493_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_494_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_495_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_496_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_497_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_498_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_499_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_500_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_501_add__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_mono
tff(fact_502_add__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_mono
tff(fact_503_add__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_mono
tff(fact_504_add__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_mono
tff(fact_505_add__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)) ) ).

% add_left_mono
tff(fact_506_add__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B)) ) ).

% add_left_mono
tff(fact_507_add__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)) ) ).

% add_left_mono
tff(fact_508_add__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% add_left_mono
tff(fact_509_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ) ).

% less_eqE
tff(fact_510_add__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ) ).

% add_right_mono
tff(fact_511_add__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2)) ) ).

% add_right_mono
tff(fact_512_add__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ) ).

% add_right_mono
tff(fact_513_add__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ) ).

% add_right_mono
tff(fact_514_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C3: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% le_iff_add
tff(fact_515_add__le__imp__le__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_left
tff(fact_516_add__le__imp__le__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_517_add__le__imp__le__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_518_add__le__imp__le__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_519_add__le__imp__le__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_right
tff(fact_520_add__le__imp__le__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_521_add__le__imp__le__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_522_add__le__imp__le__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_523_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_524_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_525_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_526_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_527_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_528_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_529_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_530_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_531_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_532_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_533_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_534_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_535_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_536_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_537_add__nonpos__eq__0__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),zero_zero_real)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Yb = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_538_add__nonpos__eq__0__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),zero_zero_rat)
       => ( ( aa_rat_rat(plus_plus_rat(Xb),Yb) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Yb = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_539_add__nonpos__eq__0__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Yb) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Yb = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_540_add__nonpos__eq__0__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Yb) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Yb = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_541_add__nonneg__eq__0__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Yb = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_542_add__nonneg__eq__0__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
       => ( ( aa_rat_rat(plus_plus_rat(Xb),Yb) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Yb = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_543_add__nonneg__eq__0__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Yb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Yb) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Yb = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_544_add__nonneg__eq__0__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Yb) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Yb = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_545_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_546_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_547_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_548_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_549_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_550_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_551_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_552_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_553_add__increasing2,axiom,
    ! [C2: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% add_increasing2
tff(fact_554_add__increasing2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C2)) ) ) ).

% add_increasing2
tff(fact_555_add__increasing2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_increasing2
tff(fact_556_add__increasing2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_increasing2
tff(fact_557_add__decreasing2,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B) ) ) ).

% add_decreasing2
tff(fact_558_add__decreasing2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),B) ) ) ).

% add_decreasing2
tff(fact_559_add__decreasing2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),B) ) ) ).

% add_decreasing2
tff(fact_560_add__decreasing2,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ) ) ).

% add_decreasing2
tff(fact_561_add__increasing,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% add_increasing
tff(fact_562_add__increasing,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C2)) ) ) ).

% add_increasing
tff(fact_563_add__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_increasing
tff(fact_564_add__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_increasing
tff(fact_565_add__decreasing,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B) ) ) ).

% add_decreasing
tff(fact_566_add__decreasing,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),B) ) ) ).

% add_decreasing
tff(fact_567_add__decreasing,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),B) ) ) ).

% add_decreasing
tff(fact_568_add__decreasing,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ) ) ).

% add_decreasing
tff(fact_569_option_Osize__gen_I2_J,axiom,
    ! [Xb: fun_Pr2602117396471556196at_nat,X2: product_prod_nat_nat] : size_o8335143837870341156at_nat(Xb,some_P7363390416028606310at_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(Xb,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_570_option_Osize__gen_I2_J,axiom,
    ! [Xb: fun_num_nat,X2: num] : size_option_num(Xb,some_num(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(Xb,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_571_subset__antisym,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
       => ( A4 = B4 ) ) ) ).

% subset_antisym
tff(fact_572_subsetI,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => aa_set_complex_o(member_complex(X3),B4) )
     => aa_set_complex_o(ord_le211207098394363844omplex(A4),B4) ) ).

% subsetI
tff(fact_573_subsetI,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ! [X3: real] :
          ( member_real(X3,A4)
         => member_real(X3,B4) )
     => ord_less_eq_set_real(A4,B4) ) ).

% subsetI
tff(fact_574_subsetI,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,A4)
         => member_set_nat(X3,B4) )
     => ord_le6893508408891458716et_nat(A4,B4) ) ).

% subsetI
tff(fact_575_subsetI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_set_nat_o(member_nat(X3),B4) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4) ) ).

% subsetI
tff(fact_576_subsetI,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_set_int_o(member_int(X3),B4) )
     => aa_set_int_o(ord_less_eq_set_int(A4),B4) ) ).

% subsetI
tff(fact_577_option_Osize__gen_I1_J,axiom,
    ! [Xb: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(Xb,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_578_option_Osize__gen_I1_J,axiom,
    ! [Xb: fun_num_nat] : size_option_num(Xb,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_579_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_580_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_581_option_Osize_I4_J,axiom,
    ! [X2: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_582_option_Osize_I4_J,axiom,
    ! [X2: num] : size_size_option_num(some_num(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_583_Euclid__induct,axiom,
    ! [P2: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3)
        <=> aa_nat_o(aa_nat_fun_nat_o(P2,B3),A3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,A3),zero_zero_nat)
       => ( ! [A3: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3)
             => aa_nat_o(aa_nat_fun_nat_o(P2,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(P2,A),B) ) ) ) ).

% Euclid_induct
tff(fact_584_triangle__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Nb)),aa_nat_nat(suc,Nb)) ).

% triangle_Suc
tff(fact_585_exists__least__lemma,axiom,
    ! [P2: fun_nat_o] :
      ( ~ aa_nat_o(P2,zero_zero_nat)
     => ( ? [X_1: nat] : aa_nat_o(P2,X_1)
       => ? [N2: nat] :
            ( ~ aa_nat_o(P2,N2)
            & aa_nat_o(P2,aa_nat_nat(suc,N2)) ) ) ) ).

% exists_least_lemma
tff(fact_586_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( A = zero_zero_complex ) ) ).

% add_0_iff
tff(fact_587_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_588_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_589_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_590_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_591_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_592_size__neq__size__imp__neq,axiom,
    ! [Xb: vEBT_VEBT,Yb: vEBT_VEBT] :
      ( ( aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,Xb) != aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,Yb) )
     => ( Xb != Yb ) ) ).

% size_neq_size_imp_neq
tff(fact_593_size__neq__size__imp__neq,axiom,
    ! [Xb: list_VEBT_VEBT,Yb: list_VEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xb) != aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Yb) )
     => ( Xb != Yb ) ) ).

% size_neq_size_imp_neq
tff(fact_594_size__neq__size__imp__neq,axiom,
    ! [Xb: num,Yb: num] :
      ( ( aa_num_nat(size_size_num,Xb) != aa_num_nat(size_size_num,Yb) )
     => ( Xb != Yb ) ) ).

% size_neq_size_imp_neq
tff(fact_595_size__neq__size__imp__neq,axiom,
    ! [Xb: list_int,Yb: list_int] :
      ( ( aa_list_int_nat(size_size_list_int,Xb) != aa_list_int_nat(size_size_list_int,Yb) )
     => ( Xb != Yb ) ) ).

% size_neq_size_imp_neq
tff(fact_596_size__neq__size__imp__neq,axiom,
    ! [Xb: list_nat,Yb: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xb) != aa_list_nat_nat(size_size_list_nat,Yb) )
     => ( Xb != Yb ) ) ).

% size_neq_size_imp_neq
tff(fact_597_in__mono,axiom,
    ! [A4: set_complex,B4: set_complex,Xb: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
     => ( aa_set_complex_o(member_complex(Xb),A4)
       => aa_set_complex_o(member_complex(Xb),B4) ) ) ).

% in_mono
tff(fact_598_in__mono,axiom,
    ! [A4: set_real,B4: set_real,Xb: real] :
      ( ord_less_eq_set_real(A4,B4)
     => ( member_real(Xb,A4)
       => member_real(Xb,B4) ) ) ).

% in_mono
tff(fact_599_in__mono,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,Xb: set_nat] :
      ( ord_le6893508408891458716et_nat(A4,B4)
     => ( member_set_nat(Xb,A4)
       => member_set_nat(Xb,B4) ) ) ).

% in_mono
tff(fact_600_in__mono,axiom,
    ! [A4: set_nat,B4: set_nat,Xb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_o(member_nat(Xb),A4)
       => aa_set_nat_o(member_nat(Xb),B4) ) ) ).

% in_mono
tff(fact_601_in__mono,axiom,
    ! [A4: set_int,B4: set_int,Xb: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(member_int(Xb),A4)
       => aa_set_int_o(member_int(Xb),B4) ) ) ).

% in_mono
tff(fact_602_subsetD,axiom,
    ! [A4: set_complex,B4: set_complex,C2: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
     => ( aa_set_complex_o(member_complex(C2),A4)
       => aa_set_complex_o(member_complex(C2),B4) ) ) ).

% subsetD
tff(fact_603_subsetD,axiom,
    ! [A4: set_real,B4: set_real,C2: real] :
      ( ord_less_eq_set_real(A4,B4)
     => ( member_real(C2,A4)
       => member_real(C2,B4) ) ) ).

% subsetD
tff(fact_604_subsetD,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,C2: set_nat] :
      ( ord_le6893508408891458716et_nat(A4,B4)
     => ( member_set_nat(C2,A4)
       => member_set_nat(C2,B4) ) ) ).

% subsetD
tff(fact_605_subsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_o(member_nat(C2),A4)
       => aa_set_nat_o(member_nat(C2),B4) ) ) ).

% subsetD
tff(fact_606_subsetD,axiom,
    ! [A4: set_int,B4: set_int,C2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(member_int(C2),A4)
       => aa_set_int_o(member_int(C2),B4) ) ) ).

% subsetD
tff(fact_607_equalityE,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
         => ~ aa_set_int_o(ord_less_eq_set_int(B4),A4) ) ) ).

% equalityE
tff(fact_608_subset__eq,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
    <=> ! [X4: complex] :
          ( aa_set_complex_o(member_complex(X4),A4)
         => aa_set_complex_o(member_complex(X4),B4) ) ) ).

% subset_eq
tff(fact_609_subset__eq,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ord_less_eq_set_real(A4,B4)
    <=> ! [X4: real] :
          ( member_real(X4,A4)
         => member_real(X4,B4) ) ) ).

% subset_eq
tff(fact_610_subset__eq,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A4,B4)
    <=> ! [X4: set_nat] :
          ( member_set_nat(X4,A4)
         => member_set_nat(X4,B4) ) ) ).

% subset_eq
tff(fact_611_subset__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_set_nat_o(member_nat(X4),B4) ) ) ).

% subset_eq
tff(fact_612_subset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A4)
         => aa_set_int_o(member_int(X4),B4) ) ) ).

% subset_eq
tff(fact_613_equalityD1,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
     => aa_set_int_o(ord_less_eq_set_int(A4),B4) ) ).

% equalityD1
tff(fact_614_equalityD2,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
     => aa_set_int_o(ord_less_eq_set_int(B4),A4) ) ).

% equalityD2
tff(fact_615_subset__iff,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
    <=> ! [T2: complex] :
          ( aa_set_complex_o(member_complex(T2),A4)
         => aa_set_complex_o(member_complex(T2),B4) ) ) ).

% subset_iff
tff(fact_616_subset__iff,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ord_less_eq_set_real(A4,B4)
    <=> ! [T2: real] :
          ( member_real(T2,A4)
         => member_real(T2,B4) ) ) ).

% subset_iff
tff(fact_617_subset__iff,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A4,B4)
    <=> ! [T2: set_nat] :
          ( member_set_nat(T2,A4)
         => member_set_nat(T2,B4) ) ) ).

% subset_iff
tff(fact_618_subset__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
    <=> ! [T2: nat] :
          ( aa_set_nat_o(member_nat(T2),A4)
         => aa_set_nat_o(member_nat(T2),B4) ) ) ).

% subset_iff
tff(fact_619_subset__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
    <=> ! [T2: int] :
          ( aa_set_int_o(member_int(T2),A4)
         => aa_set_int_o(member_int(T2),B4) ) ) ).

% subset_iff
tff(fact_620_subset__refl,axiom,
    ! [A4: set_int] : aa_set_int_o(ord_less_eq_set_int(A4),A4) ).

% subset_refl
tff(fact_621_Collect__mono,axiom,
    ! [P2: fun_real_o,Q: fun_real_o] :
      ( ! [X3: real] :
          ( aa_real_o(P2,X3)
         => aa_real_o(Q,X3) )
     => ord_less_eq_set_real(collect_real(P2),collect_real(Q)) ) ).

% Collect_mono
tff(fact_622_Collect__mono,axiom,
    ! [P2: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X3: list_nat] :
          ( aa_list_nat_o(P2,X3)
         => aa_list_nat_o(Q,X3) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P2),collect_list_nat(Q)) ) ).

% Collect_mono
tff(fact_623_Collect__mono,axiom,
    ! [P2: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X3: set_nat] :
          ( aa_set_nat_o(P2,X3)
         => aa_set_nat_o(Q,X3) )
     => ord_le6893508408891458716et_nat(collect_set_nat(P2),collect_set_nat(Q)) ) ).

% Collect_mono
tff(fact_624_Collect__mono,axiom,
    ! [P2: fun_nat_o,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(P2,X3)
         => aa_nat_o(Q,X3) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(P2)),collect_nat(Q)) ) ).

% Collect_mono
tff(fact_625_Collect__mono,axiom,
    ! [P2: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(P2,X3)
         => aa_int_o(Q,X3) )
     => aa_set_int_o(ord_less_eq_set_int(collect_int(P2)),collect_int(Q)) ) ).

% Collect_mono
tff(fact_626_subset__trans,axiom,
    ! [A4: set_int,B4: set_int,C4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),C4)
       => aa_set_int_o(ord_less_eq_set_int(A4),C4) ) ) ).

% subset_trans
tff(fact_627_set__eq__subset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
    <=> ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
        & aa_set_int_o(ord_less_eq_set_int(B4),A4) ) ) ).

% set_eq_subset
tff(fact_628_Collect__mono__iff,axiom,
    ! [P2: fun_real_o,Q: fun_real_o] :
      ( ord_less_eq_set_real(collect_real(P2),collect_real(Q))
    <=> ! [X4: real] :
          ( aa_real_o(P2,X4)
         => aa_real_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_629_Collect__mono__iff,axiom,
    ! [P2: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(collect_list_nat(P2),collect_list_nat(Q))
    <=> ! [X4: list_nat] :
          ( aa_list_nat_o(P2,X4)
         => aa_list_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_630_Collect__mono__iff,axiom,
    ! [P2: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ord_le6893508408891458716et_nat(collect_set_nat(P2),collect_set_nat(Q))
    <=> ! [X4: set_nat] :
          ( aa_set_nat_o(P2,X4)
         => aa_set_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_631_Collect__mono__iff,axiom,
    ! [P2: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(P2)),collect_nat(Q))
    <=> ! [X4: nat] :
          ( aa_nat_o(P2,X4)
         => aa_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_632_Collect__mono__iff,axiom,
    ! [P2: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(P2)),collect_int(Q))
    <=> ! [X4: int] :
          ( aa_int_o(P2,X4)
         => aa_int_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_633_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_634_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_635_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_636_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_637_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_638_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% deg_not_0
tff(fact_639_le__numeral__extra_I3_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_zero_real) ).

% le_numeral_extra(3)
tff(fact_640_le__numeral__extra_I3_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_zero_rat) ).

% le_numeral_extra(3)
tff(fact_641_le__numeral__extra_I3_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_zero_nat) ).

% le_numeral_extra(3)
tff(fact_642_le__numeral__extra_I3_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% le_numeral_extra(3)
tff(fact_643_vebt__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X3)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X3)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),X3)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X3)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),X3) ) ) ) ) ).

% vebt_member.cases
tff(fact_644_vebt__insert_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X3)
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S),X3)
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S),X3)
         => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary),X3)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),X3) ) ) ) ) ).

% vebt_insert.cases
tff(fact_645_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),X3)
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),X3)
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_646_count__notin,axiom,
    ! [Xb: complex,Xs: list_complex] :
      ( ~ aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
     => ( aa_complex_nat(count_list_complex(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_647_count__notin,axiom,
    ! [Xb: real,Xs: list_real] :
      ( ~ member_real(Xb,set_real2(Xs))
     => ( aa_real_nat(count_list_real(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_648_count__notin,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( ~ member_set_nat(Xb,set_set_nat2(Xs))
     => ( aa_set_nat_nat(count_list_set_nat(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_649_count__notin,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
     => ( aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_650_count__notin,axiom,
    ! [Xb: int,Xs: list_int] :
      ( ~ aa_set_int_o(member_int(Xb),set_int2(Xs))
     => ( aa_int_nat(count_list_int(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_651_count__notin,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( ~ aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
     => ( aa_nat_nat(count_list_nat(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_652_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),zero_zero_nat,Va,Vb)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_653_Leaf__0__not,axiom,
    ! [A: $o,B: $o] : ~ aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_654__092_060open_062x_A_061_Ami_A_092_060or_062_Ax_A_061_Ama_A_092_060or_062_Aboth__member__options_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_092_060close_062,axiom,
    ( ( xa = mi )
    | ( xa = ma )
    | aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)) ) ).

% \<open>x = mi \<or> x = ma \<or> both_member_options (treeList ! high x n) (low x n)\<close>
tff(fact_655_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Yb: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
      <=> (Yb) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Yb) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => (Yb) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => (Yb) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ~ (Yb) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => (Yb) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_656_inthall,axiom,
    ! [Xs: list_complex,P2: fun_complex_o,Nb: nat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),set_complex2(Xs))
         => aa_complex_o(P2,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
       => aa_complex_o(P2,aa_nat_complex(nth_complex(Xs),Nb)) ) ) ).

% inthall
tff(fact_657_inthall,axiom,
    ! [Xs: list_real,P2: fun_real_o,Nb: nat] :
      ( ! [X3: real] :
          ( member_real(X3,set_real2(Xs))
         => aa_real_o(P2,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
       => aa_real_o(P2,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% inthall
tff(fact_658_inthall,axiom,
    ! [Xs: list_set_nat,P2: fun_set_nat_o,Nb: nat] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,set_set_nat2(Xs))
         => aa_set_nat_o(P2,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
       => aa_set_nat_o(P2,nth_set_nat(Xs,Nb)) ) ) ).

% inthall
tff(fact_659_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_o,Nb: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P2,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
       => aa_VEBT_VEBT_o(P2,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% inthall
tff(fact_660_inthall,axiom,
    ! [Xs: list_int,P2: fun_int_o,Nb: nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),set_int2(Xs))
         => aa_int_o(P2,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_list_int_nat(size_size_list_int,Xs))
       => aa_int_o(P2,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% inthall
tff(fact_661_inthall,axiom,
    ! [Xs: list_nat,P2: fun_nat_o,Nb: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_nat_o(P2,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_list_nat_nat(size_size_list_nat,Xs))
       => aa_nat_o(P2,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% inthall
tff(fact_662_bit__split__inv,axiom,
    ! [Xb: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),D)),D) = Xb ).

% bit_split_inv
tff(fact_663_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leaf((X21),(X222)) = vEBT_Leaf((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X222)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_664_not__gr__zero,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_665_add__less__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_left
tff(fact_666_add__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_667_add__less__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_668_add__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_669_add__less__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_right
tff(fact_670_add__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_671_add__less__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_672_add__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_673_lessI,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Nb)) ).

% lessI
tff(fact_674_Suc__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) ) ).

% Suc_mono
tff(fact_675_Suc__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_eq
tff(fact_676_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_677_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% neq0_conv
tff(fact_678_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_679_nat__add__left__cancel__less,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_680_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_681_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_682_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_683_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_684_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_685_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_686_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_687_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_688_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel1
tff(fact_689_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_690_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_691_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_692_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel2
tff(fact_693_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_694_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_695_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_696_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_697_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_698_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_699_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_700_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_701_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_702_zero__less__Suc,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) ).

% zero_less_Suc
tff(fact_703_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_704_add__gr__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% add_gr_0
tff(fact_705_length__induct,axiom,
    ! [P2: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys: list_VEBT_VEBT] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs2))
             => aa_list_VEBT_VEBT_o(P2,Ys) )
         => aa_list_VEBT_VEBT_o(P2,Xs2) )
     => aa_list_VEBT_VEBT_o(P2,Xs) ) ).

% length_induct
tff(fact_706_length__induct,axiom,
    ! [P2: fun_list_int_o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_int_nat(size_size_list_int,Ys)),aa_list_int_nat(size_size_list_int,Xs2))
             => aa_list_int_o(P2,Ys) )
         => aa_list_int_o(P2,Xs2) )
     => aa_list_int_o(P2,Xs) ) ).

% length_induct
tff(fact_707_length__induct,axiom,
    ! [P2: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_nat_nat(size_size_list_nat,Ys)),aa_list_nat_nat(size_size_list_nat,Xs2))
             => aa_list_nat_o(P2,Ys) )
         => aa_list_nat_o(P2,Xs2) )
     => aa_list_nat_o(P2,Xs) ) ).

% length_induct
tff(fact_708_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys2,I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_709_nth__equalityI,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys2) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(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(Ys2),I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_710_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys2) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys2),I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_711_Skolem__list__nth,axiom,
    ! [K2: nat,P2: fun_na6887940067898170868VEBT_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
         => ? [X_13: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P2,I3),X_13) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs3) = K2 )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P2,I3),nth_VEBT_VEBT(Xs3,I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_712_Skolem__list__nth,axiom,
    ! [K2: nat,P2: fun_nat_fun_int_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
         => ? [X_13: int] : aa_int_o(aa_nat_fun_int_o(P2,I3),X_13) )
    <=> ? [Xs3: list_int] :
          ( ( aa_list_int_nat(size_size_list_int,Xs3) = K2 )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
             => aa_int_o(aa_nat_fun_int_o(P2,I3),aa_nat_int(nth_int(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_713_Skolem__list__nth,axiom,
    ! [K2: nat,P2: fun_nat_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
         => ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,I3),X_13) )
    <=> ? [Xs3: list_nat] :
          ( ( aa_list_nat_nat(size_size_list_nat,Xs3) = K2 )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
             => aa_nat_o(aa_nat_fun_nat_o(P2,I3),aa_nat_nat(nth_nat(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_714_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( Xs = Ys2 )
    <=> ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys2,I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_715_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( Xs = Ys2 )
    <=> ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys2) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(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(Ys2),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_716_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( Xs = Ys2 )
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys2) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys2),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_717_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_o] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P2,X4) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
         => aa_VEBT_VEBT_o(P2,nth_VEBT_VEBT(Xs,I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_718_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P2: fun_int_o] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(Xs))
         => aa_int_o(P2,X4) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,Xs))
         => aa_int_o(P2,aa_nat_int(nth_int(Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_719_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P2: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => aa_nat_o(P2,X4) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(P2,aa_nat_nat(nth_nat(Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_720_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P2: fun_complex_o,Xb: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3451745648224563538omplex(Xs))
         => aa_complex_o(P2,aa_nat_complex(nth_complex(Xs),I2)) )
     => ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
       => aa_complex_o(P2,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_721_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P2: fun_real_o,Xb: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
         => aa_real_o(P2,aa_nat_real(nth_real(Xs),I2)) )
     => ( member_real(Xb,set_real2(Xs))
       => aa_real_o(P2,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_722_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P2: fun_set_nat_o,Xb: set_nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3254054031482475050et_nat(Xs))
         => aa_set_nat_o(P2,nth_set_nat(Xs,I2)) )
     => ( member_set_nat(Xb,set_set_nat2(Xs))
       => aa_set_nat_o(P2,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_723_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
         => aa_VEBT_VEBT_o(P2,nth_VEBT_VEBT(Xs,I2)) )
     => ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P2,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_724_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P2: fun_int_o,Xb: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_int_nat(size_size_list_int,Xs))
         => aa_int_o(P2,aa_nat_int(nth_int(Xs),I2)) )
     => ( aa_set_int_o(member_int(Xb),set_int2(Xs))
       => aa_int_o(P2,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_725_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P2: fun_nat_o,Xb: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(P2,aa_nat_nat(nth_nat(Xs),I2)) )
     => ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
       => aa_nat_o(P2,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_726_in__set__conv__nth,axiom,
    ! [Xb: complex,Xs: list_complex] :
      ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3451745648224563538omplex(Xs))
          & ( aa_nat_complex(nth_complex(Xs),I3) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_727_in__set__conv__nth,axiom,
    ! [Xb: real,Xs: list_real] :
      ( member_real(Xb,set_real2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I3) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_728_in__set__conv__nth,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( member_set_nat(Xb,set_set_nat2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3254054031482475050et_nat(Xs))
          & ( nth_set_nat(Xs,I3) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_729_in__set__conv__nth,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
          & ( nth_VEBT_VEBT(Xs,I3) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_730_in__set__conv__nth,axiom,
    ! [Xb: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xb),set_int2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,Xs))
          & ( aa_nat_int(nth_int(Xs),I3) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_731_in__set__conv__nth,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs))
          & ( aa_nat_nat(nth_nat(Xs),I3) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_732_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( ! [X3: vEBT_VEBT] :
            ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P2,X3) )
       => aa_VEBT_VEBT_o(P2,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% list_ball_nth
tff(fact_733_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_int,P2: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_list_int_nat(size_size_list_int,Xs))
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),set_int2(Xs))
           => aa_int_o(P2,X3) )
       => aa_int_o(P2,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_734_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
           => aa_nat_o(P2,X3) )
       => aa_nat_o(P2,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_735_nth__mem,axiom,
    ! [Nb: nat,Xs: list_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => aa_set_complex_o(member_complex(aa_nat_complex(nth_complex(Xs),Nb)),set_complex2(Xs)) ) ).

% nth_mem
tff(fact_736_nth__mem,axiom,
    ! [Nb: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => member_real(aa_nat_real(nth_real(Xs),Nb),set_real2(Xs)) ) ).

% nth_mem
tff(fact_737_nth__mem,axiom,
    ! [Nb: nat,Xs: list_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
     => member_set_nat(nth_set_nat(Xs,Nb),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_738_nth__mem,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => member_VEBT_VEBT(nth_VEBT_VEBT(Xs,Nb),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_739_nth__mem,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_list_int_nat(size_size_list_int,Xs))
     => aa_set_int_o(member_int(aa_nat_int(nth_int(Xs),Nb)),set_int2(Xs)) ) ).

% nth_mem
tff(fact_740_nth__mem,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_list_nat_nat(size_size_list_nat,Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Nb)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_741_finite__maxlen,axiom,
    ! [M5: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M5)
     => ? [N2: nat] :
        ! [X: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X,M5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,X)),N2) ) ) ).

% finite_maxlen
tff(fact_742_finite__maxlen,axiom,
    ! [M5: set_list_int] :
      ( finite3922522038869484883st_int(M5)
     => ? [N2: nat] :
        ! [X: list_int] :
          ( member_list_int(X,M5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_int_nat(size_size_list_int,X)),N2) ) ) ).

% finite_maxlen
tff(fact_743_finite__maxlen,axiom,
    ! [M5: set_list_nat] :
      ( finite8100373058378681591st_nat(M5)
     => ? [N2: nat] :
        ! [X: list_nat] :
          ( member_list_nat(X,M5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_nat_nat(size_size_list_nat,X)),N2) ) ) ).

% finite_maxlen
tff(fact_744_order__less__imp__not__less,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ).

% order_less_imp_not_less
tff(fact_745_order__less__imp__not__less,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ).

% order_less_imp_not_less
tff(fact_746_order__less__imp__not__less,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ).

% order_less_imp_not_less
tff(fact_747_order__less__imp__not__less,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ).

% order_less_imp_not_less
tff(fact_748_order__less__imp__not__less,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ).

% order_less_imp_not_less
tff(fact_749_order__less__imp__not__eq2,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( Yb != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_750_order__less__imp__not__eq2,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( Yb != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_751_order__less__imp__not__eq2,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( Yb != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_752_order__less__imp__not__eq2,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( Yb != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_753_order__less__imp__not__eq2,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( Yb != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_754_order__less__imp__not__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( Xb != Yb ) ) ).

% order_less_imp_not_eq
tff(fact_755_order__less__imp__not__eq,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( Xb != Yb ) ) ).

% order_less_imp_not_eq
tff(fact_756_order__less__imp__not__eq,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( Xb != Yb ) ) ).

% order_less_imp_not_eq
tff(fact_757_order__less__imp__not__eq,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( Xb != Yb ) ) ).

% order_less_imp_not_eq
tff(fact_758_order__less__imp__not__eq,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( Xb != Yb ) ) ).

% order_less_imp_not_eq
tff(fact_759_linorder__less__linear,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
      | ( Xb = Yb )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ).

% linorder_less_linear
tff(fact_760_linorder__less__linear,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
      | ( Xb = Yb )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ).

% linorder_less_linear
tff(fact_761_linorder__less__linear,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
      | ( Xb = Yb )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ).

% linorder_less_linear
tff(fact_762_linorder__less__linear,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
      | ( Xb = Yb )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ).

% linorder_less_linear
tff(fact_763_linorder__less__linear,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
      | ( Xb = Yb )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ).

% linorder_less_linear
tff(fact_764_order__less__imp__triv,axiom,
    ! [Xb: real,Yb: real,P2: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb)
       => (P2) ) ) ).

% order_less_imp_triv
tff(fact_765_order__less__imp__triv,axiom,
    ! [Xb: rat,Yb: rat,P2: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb)
       => (P2) ) ) ).

% order_less_imp_triv
tff(fact_766_order__less__imp__triv,axiom,
    ! [Xb: num,Yb: num,P2: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb)
       => (P2) ) ) ).

% order_less_imp_triv
tff(fact_767_order__less__imp__triv,axiom,
    ! [Xb: nat,Yb: nat,P2: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb)
       => (P2) ) ) ).

% order_less_imp_triv
tff(fact_768_order__less__imp__triv,axiom,
    ! [Xb: int,Yb: int,P2: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb)
       => (P2) ) ) ).

% order_less_imp_triv
tff(fact_769_order__less__not__sym,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ).

% order_less_not_sym
tff(fact_770_order__less__not__sym,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ).

% order_less_not_sym
tff(fact_771_order__less__not__sym,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ).

% order_less_not_sym
tff(fact_772_order__less__not__sym,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ).

% order_less_not_sym
tff(fact_773_order__less__not__sym,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ).

% order_less_not_sym
tff(fact_774_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_775_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,B)),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_776_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,B)),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X3)),aa_real_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_777_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,B)),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_778_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,B)),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X3)),aa_real_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_779_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_780_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_781_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_782_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_783_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_784_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_785_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_786_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X3)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_787_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_788_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_789_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_790_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_791_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_792_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_793_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_794_order__less__irrefl,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Xb) ).

% order_less_irrefl
tff(fact_795_order__less__irrefl,axiom,
    ! [Xb: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Xb) ).

% order_less_irrefl
tff(fact_796_order__less__irrefl,axiom,
    ! [Xb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Xb) ).

% order_less_irrefl
tff(fact_797_order__less__irrefl,axiom,
    ! [Xb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xb) ).

% order_less_irrefl
tff(fact_798_order__less__irrefl,axiom,
    ! [Xb: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Xb) ).

% order_less_irrefl
tff(fact_799_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,B) = C2 )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_800_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_rat(F,B) = C2 )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_801_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_num(F,B) = C2 )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X3)),aa_real_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_802_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_nat(F,B) = C2 )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_803_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_int(F,B) = C2 )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X3)),aa_real_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_804_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_real(F,B) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_805_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_806_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_num(F,B) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_807_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_808_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_int(F,B) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_809_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C2: real] :
      ( ( A = aa_real_real(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_810_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C2: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_811_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C2: real] :
      ( ( A = aa_real_num(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X3)),aa_real_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_real_num(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_812_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C2: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_real_nat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_813_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C2: real] :
      ( ( A = aa_real_int(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X3)),aa_real_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_real_int(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_814_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C2: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_815_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_816_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C2: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_817_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C2: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_818_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C2: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_819_order__less__trans,axiom,
    ! [Xb: real,Yb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_less_trans
tff(fact_820_order__less__trans,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_less_trans
tff(fact_821_order__less__trans,axiom,
    ! [Xb: num,Yb: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_less_trans
tff(fact_822_order__less__trans,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_less_trans
tff(fact_823_order__less__trans,axiom,
    ! [Xb: int,Yb: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_less_trans
tff(fact_824_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% order_less_asym'
tff(fact_825_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order_less_asym'
tff(fact_826_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order_less_asym'
tff(fact_827_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order_less_asym'
tff(fact_828_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order_less_asym'
tff(fact_829_linorder__neq__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( ( Xb != Yb )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ) ).

% linorder_neq_iff
tff(fact_830_linorder__neq__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( Xb != Yb )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ) ).

% linorder_neq_iff
tff(fact_831_linorder__neq__iff,axiom,
    ! [Xb: num,Yb: num] :
      ( ( Xb != Yb )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ) ).

% linorder_neq_iff
tff(fact_832_linorder__neq__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( Xb != Yb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ) ).

% linorder_neq_iff
tff(fact_833_linorder__neq__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( ( Xb != Yb )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ) ).

% linorder_neq_iff
tff(fact_834_order__less__asym,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ).

% order_less_asym
tff(fact_835_order__less__asym,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ).

% order_less_asym
tff(fact_836_order__less__asym,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ).

% order_less_asym
tff(fact_837_order__less__asym,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ).

% order_less_asym
tff(fact_838_order__less__asym,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ).

% order_less_asym
tff(fact_839_linorder__neqE,axiom,
    ! [Xb: real,Yb: real] :
      ( ( Xb != Yb )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ) ).

% linorder_neqE
tff(fact_840_linorder__neqE,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( Xb != Yb )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ) ).

% linorder_neqE
tff(fact_841_linorder__neqE,axiom,
    ! [Xb: num,Yb: num] :
      ( ( Xb != Yb )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ) ).

% linorder_neqE
tff(fact_842_linorder__neqE,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( Xb != Yb )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ) ).

% linorder_neqE
tff(fact_843_linorder__neqE,axiom,
    ! [Xb: int,Yb: int] :
      ( ( Xb != Yb )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ) ).

% linorder_neqE
tff(fact_844_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_845_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_846_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_847_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_848_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_849_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_850_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_851_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_852_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_853_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_854_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_855_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_856_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_857_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_858_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_859_not__less__iff__gr__or__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb)
        | ( Xb = Yb ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_860_not__less__iff__gr__or__eq,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb)
        | ( Xb = Yb ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_861_not__less__iff__gr__or__eq,axiom,
    ! [Xb: num,Yb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb)
        | ( Xb = Yb ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_862_not__less__iff__gr__or__eq,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb)
        | ( Xb = Yb ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_863_not__less__iff__gr__or__eq,axiom,
    ! [Xb: int,Yb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb)
        | ( Xb = Yb ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_864_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2) ) ) ).

% order.strict_trans
tff(fact_865_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% order.strict_trans
tff(fact_866_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% order.strict_trans
tff(fact_867_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% order.strict_trans
tff(fact_868_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% order.strict_trans
tff(fact_869_linorder__less__wlog,axiom,
    ! [P2: fun_real_fun_real_o,A: real,B: real] :
      ( ! [A3: real,B3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,A3),B3)
         => aa_real_o(aa_real_fun_real_o(P2,A3),B3) )
     => ( ! [A3: real] : aa_real_o(aa_real_fun_real_o(P2,A3),A3)
       => ( ! [A3: real,B3: real] :
              ( aa_real_o(aa_real_fun_real_o(P2,B3),A3)
             => aa_real_o(aa_real_fun_real_o(P2,A3),B3) )
         => aa_real_o(aa_real_fun_real_o(P2,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_870_linorder__less__wlog,axiom,
    ! [P2: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A3),B3)
         => aa_rat_o(aa_rat_fun_rat_o(P2,A3),B3) )
     => ( ! [A3: rat] : aa_rat_o(aa_rat_fun_rat_o(P2,A3),A3)
       => ( ! [A3: rat,B3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(P2,B3),A3)
             => aa_rat_o(aa_rat_fun_rat_o(P2,A3),B3) )
         => aa_rat_o(aa_rat_fun_rat_o(P2,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_871_linorder__less__wlog,axiom,
    ! [P2: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,A3),B3)
         => aa_num_o(aa_num_fun_num_o(P2,A3),B3) )
     => ( ! [A3: num] : aa_num_o(aa_num_fun_num_o(P2,A3),A3)
       => ( ! [A3: num,B3: num] :
              ( aa_num_o(aa_num_fun_num_o(P2,B3),A3)
             => aa_num_o(aa_num_fun_num_o(P2,A3),B3) )
         => aa_num_o(aa_num_fun_num_o(P2,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_872_linorder__less__wlog,axiom,
    ! [P2: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
         => aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,A3),A3)
       => ( ! [A3: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P2,B3),A3)
             => aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3) )
         => aa_nat_o(aa_nat_fun_nat_o(P2,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_873_linorder__less__wlog,axiom,
    ! [P2: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
         => aa_int_o(aa_int_fun_int_o(P2,A3),B3) )
     => ( ! [A3: int] : aa_int_o(aa_int_fun_int_o(P2,A3),A3)
       => ( ! [A3: int,B3: int] :
              ( aa_int_o(aa_int_fun_int_o(P2,B3),A3)
             => aa_int_o(aa_int_fun_int_o(P2,A3),B3) )
         => aa_int_o(aa_int_fun_int_o(P2,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_874_exists__least__iff,axiom,
    ! [P2: fun_nat_o] :
      ( ? [X_13: nat] : aa_nat_o(P2,X_13)
    <=> ? [N3: nat] :
          ( aa_nat_o(P2,N3)
          & ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N3)
             => ~ aa_nat_o(P2,M4) ) ) ) ).

% exists_least_iff
tff(fact_875_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% dual_order.irrefl
tff(fact_876_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% dual_order.irrefl
tff(fact_877_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% dual_order.irrefl
tff(fact_878_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_879_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_880_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% dual_order.asym
tff(fact_881_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% dual_order.asym
tff(fact_882_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% dual_order.asym
tff(fact_883_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% dual_order.asym
tff(fact_884_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% dual_order.asym
tff(fact_885_linorder__cases,axiom,
    ! [Xb: real,Yb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( ( Xb != Yb )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ) ).

% linorder_cases
tff(fact_886_linorder__cases,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( ( Xb != Yb )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ) ).

% linorder_cases
tff(fact_887_linorder__cases,axiom,
    ! [Xb: num,Yb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( ( Xb != Yb )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ) ).

% linorder_cases
tff(fact_888_linorder__cases,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( ( Xb != Yb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ) ).

% linorder_cases
tff(fact_889_linorder__cases,axiom,
    ! [Xb: int,Yb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( ( Xb != Yb )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ) ).

% linorder_cases
tff(fact_890_antisym__conv3,axiom,
    ! [Yb: real,Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv3
tff(fact_891_antisym__conv3,axiom,
    ! [Yb: rat,Xb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv3
tff(fact_892_antisym__conv3,axiom,
    ! [Yb: num,Xb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv3
tff(fact_893_antisym__conv3,axiom,
    ! [Yb: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv3
tff(fact_894_antisym__conv3,axiom,
    ! [Yb: int,Xb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv3
tff(fact_895_less__induct,axiom,
    ! [P2: fun_nat_o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),X3)
             => aa_nat_o(P2,Y3) )
         => aa_nat_o(P2,X3) )
     => aa_nat_o(P2,A) ) ).

% less_induct
tff(fact_896_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( B = C2 )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_897_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( B = C2 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_898_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( ( B = C2 )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_899_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( B = C2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_900_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( B = C2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_901_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( A = B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_902_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_903_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_904_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_905_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_906_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% order.asym
tff(fact_907_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order.asym
tff(fact_908_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order.asym
tff(fact_909_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order.asym
tff(fact_910_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order.asym
tff(fact_911_less__imp__neq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( Xb != Yb ) ) ).

% less_imp_neq
tff(fact_912_less__imp__neq,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( Xb != Yb ) ) ).

% less_imp_neq
tff(fact_913_less__imp__neq,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( Xb != Yb ) ) ).

% less_imp_neq
tff(fact_914_less__imp__neq,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( Xb != Yb ) ) ).

% less_imp_neq
tff(fact_915_less__imp__neq,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( Xb != Yb ) ) ).

% less_imp_neq
tff(fact_916_dense,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ? [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),Yb) ) ) ).

% dense
tff(fact_917_dense,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ? [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),Yb) ) ) ).

% dense
tff(fact_918_gt__ex,axiom,
    ! [Xb: real] :
    ? [X_12: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),X_12) ).

% gt_ex
tff(fact_919_gt__ex,axiom,
    ! [Xb: rat] :
    ? [X_12: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),X_12) ).

% gt_ex
tff(fact_920_gt__ex,axiom,
    ! [Xb: nat] :
    ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),X_12) ).

% gt_ex
tff(fact_921_gt__ex,axiom,
    ! [Xb: int] :
    ? [X_12: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),X_12) ).

% gt_ex
tff(fact_922_lt__ex,axiom,
    ! [Xb: real] :
    ? [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ).

% lt_ex
tff(fact_923_lt__ex,axiom,
    ! [Xb: rat] :
    ? [Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ).

% lt_ex
tff(fact_924_lt__ex,axiom,
    ! [Xb: int] :
    ? [Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ).

% lt_ex
tff(fact_925_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Leaf((Uu),(Uv))),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_926_linorder__neqE__nat,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( Xb != Yb )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ) ).

% linorder_neqE_nat
tff(fact_927_infinite__descent,axiom,
    ! [P2: fun_nat_o,Nb: nat] :
      ( ! [N2: nat] :
          ( ~ aa_nat_o(P2,N2)
         => ? [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N2)
              & ~ aa_nat_o(P2,M2) ) )
     => aa_nat_o(P2,Nb) ) ).

% infinite_descent
tff(fact_928_nat__less__induct,axiom,
    ! [P2: fun_nat_o,Nb: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N2)
             => aa_nat_o(P2,M2) )
         => aa_nat_o(P2,N2) )
     => aa_nat_o(P2,Nb) ) ).

% nat_less_induct
tff(fact_929_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_930_less__not__refl3,axiom,
    ! [S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S3),Ta)
     => ( S3 != Ta ) ) ).

% less_not_refl3
tff(fact_931_less__not__refl2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( Mb != Nb ) ) ).

% less_not_refl2
tff(fact_932_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_not_refl
tff(fact_933_nat__neq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb != Nb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% nat_neq_iff
tff(fact_934_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Nb: nat,Mb: nat] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_935_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Nb: nat,Mb: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_936_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Nb: nat,Mb: nat] :
      ( ! [N2: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_937_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Nb: nat,Mb: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_938_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Nb: nat,Mb: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_939_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,N)) ) ) ).

% lift_Suc_mono_less
tff(fact_940_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N)) ) ) ).

% lift_Suc_mono_less
tff(fact_941_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,N)) ) ) ).

% lift_Suc_mono_less
tff(fact_942_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N)) ) ) ).

% lift_Suc_mono_less
tff(fact_943_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Nb: nat,N: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,N)) ) ) ).

% lift_Suc_mono_less
tff(fact_944_verit__comp__simplify1_I3_J,axiom,
    ! [B2: real,A2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_945_verit__comp__simplify1_I3_J,axiom,
    ! [B2: rat,A2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_946_verit__comp__simplify1_I3_J,axiom,
    ! [B2: num,A2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_947_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_948_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_949_less__numeral__extra_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_zero_real) ).

% less_numeral_extra(3)
tff(fact_950_less__numeral__extra_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_zero_rat) ).

% less_numeral_extra(3)
tff(fact_951_less__numeral__extra_I3_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_zero_nat) ).

% less_numeral_extra(3)
tff(fact_952_less__numeral__extra_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_numeral_extra(3)
tff(fact_953_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_954_count__le__length,axiom,
    ! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),Xb)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)) ).

% count_le_length
tff(fact_955_count__le__length,axiom,
    ! [Xs: list_int,Xb: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(count_list_int(Xs),Xb)),aa_list_int_nat(size_size_list_int,Xs)) ).

% count_le_length
tff(fact_956_count__le__length,axiom,
    ! [Xs: list_nat,Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(count_list_nat(Xs),Xb)),aa_list_nat_nat(size_size_list_nat,Xs)) ).

% count_le_length
tff(fact_957_length__pos__if__in__set,axiom,
    ! [Xb: complex,Xs: list_complex] :
      ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs)) ) ).

% length_pos_if_in_set
tff(fact_958_length__pos__if__in__set,axiom,
    ! [Xb: real,Xs: list_real] :
      ( member_real(Xb,set_real2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_real(Xs)) ) ).

% length_pos_if_in_set
tff(fact_959_length__pos__if__in__set,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( member_set_nat(Xb,set_set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_960_length__pos__if__in__set,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)) ) ).

% length_pos_if_in_set
tff(fact_961_length__pos__if__in__set,axiom,
    ! [Xb: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xb),set_int2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_list_int_nat(size_size_list_int,Xs)) ) ).

% length_pos_if_in_set
tff(fact_962_length__pos__if__in__set,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs)) ) ).

% length_pos_if_in_set
tff(fact_963_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),D2)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Deg2) ) ).

% VEBT_internal.valid'.cases
tff(fact_964_VEBT_Oexhaust,axiom,
    ! [Yb: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Yb != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X223: $o] : Yb != vEBT_Leaf((X212),(X223)) ) ).

% VEBT.exhaust
tff(fact_965_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X222)) ).

% VEBT.distinct(1)
tff(fact_966_leD,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ).

% leD
tff(fact_967_leD,axiom,
    ! [Yb: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Yb),Xb)
     => ~ aa_set_int_o(ord_less_set_int(Xb),Yb) ) ).

% leD
tff(fact_968_leD,axiom,
    ! [Yb: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb) ) ).

% leD
tff(fact_969_leD,axiom,
    ! [Yb: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb) ) ).

% leD
tff(fact_970_leD,axiom,
    ! [Yb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb) ) ).

% leD
tff(fact_971_leD,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb) ) ).

% leD
tff(fact_972_leI,axiom,
    ! [Xb: real,Yb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb) ) ).

% leI
tff(fact_973_leI,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb) ) ).

% leI
tff(fact_974_leI,axiom,
    ! [Xb: num,Yb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb) ) ).

% leI
tff(fact_975_leI,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb) ) ).

% leI
tff(fact_976_leI,axiom,
    ! [Xb: int,Yb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb) ) ).

% leI
tff(fact_977_nless__le,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_978_nless__le,axiom,
    ! [A: set_int,B: set_int] :
      ( ~ aa_set_int_o(ord_less_set_int(A),B)
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_979_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_980_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_981_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_982_nless__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_983_antisym__conv1,axiom,
    ! [Xb: real,Yb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv1
tff(fact_984_antisym__conv1,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( ~ aa_set_int_o(ord_less_set_int(Xb),Yb)
     => ( aa_set_int_o(ord_less_eq_set_int(Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv1
tff(fact_985_antisym__conv1,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv1
tff(fact_986_antisym__conv1,axiom,
    ! [Xb: num,Yb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv1
tff(fact_987_antisym__conv1,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv1
tff(fact_988_antisym__conv1,axiom,
    ! [Xb: int,Yb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv1
tff(fact_989_antisym__conv2,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv2
tff(fact_990_antisym__conv2,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Yb)
     => ( ~ aa_set_int_o(ord_less_set_int(Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv2
tff(fact_991_antisym__conv2,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv2
tff(fact_992_antisym__conv2,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv2
tff(fact_993_antisym__conv2,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv2
tff(fact_994_antisym__conv2,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv2
tff(fact_995_dense__ge,axiom,
    ! [Z: real,Yb: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),X3) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Z) ) ).

% dense_ge
tff(fact_996_dense__ge,axiom,
    ! [Z: rat,Yb: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),X3) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z) ) ).

% dense_ge
tff(fact_997_dense__le,axiom,
    ! [Yb: real,Z: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Yb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Z) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Z) ) ).

% dense_le
tff(fact_998_dense__le,axiom,
    ! [Yb: rat,Z: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Yb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z) ) ).

% dense_le
tff(fact_999_less__le__not__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb) ) ) ).

% less_le_not_le
tff(fact_1000_less__le__not__le,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xb),Yb)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xb),Yb)
        & ~ aa_set_int_o(ord_less_eq_set_int(Yb),Xb) ) ) ).

% less_le_not_le
tff(fact_1001_less__le__not__le,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb) ) ) ).

% less_le_not_le
tff(fact_1002_less__le__not__le,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb) ) ) ).

% less_le_not_le
tff(fact_1003_less__le__not__le,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb) ) ) ).

% less_le_not_le
tff(fact_1004_less__le__not__le,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb) ) ) ).

% less_le_not_le
tff(fact_1005_not__le__imp__less,axiom,
    ! [Yb: real,Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ).

% not_le_imp_less
tff(fact_1006_not__le__imp__less,axiom,
    ! [Yb: rat,Xb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb) ) ).

% not_le_imp_less
tff(fact_1007_not__le__imp__less,axiom,
    ! [Yb: num,Xb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb) ) ).

% not_le_imp_less
tff(fact_1008_not__le__imp__less,axiom,
    ! [Yb: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb) ) ).

% not_le_imp_less
tff(fact_1009_not__le__imp__less,axiom,
    ! [Yb: int,Xb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb) ) ).

% not_le_imp_less
tff(fact_1010_order_Oorder__iff__strict,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1011_order_Oorder__iff__strict,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
    <=> ( aa_set_int_o(ord_less_set_int(A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1012_order_Oorder__iff__strict,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1013_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1014_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1015_order_Oorder__iff__strict,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1016_order_Ostrict__iff__order,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1017_order_Ostrict__iff__order,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1018_order_Ostrict__iff__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1019_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1020_order_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1021_order_Ostrict__iff__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1022_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2) ) ) ).

% order.strict_trans1
tff(fact_1023_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( aa_set_int_o(ord_less_set_int(B),C2)
       => aa_set_int_o(ord_less_set_int(A),C2) ) ) ).

% order.strict_trans1
tff(fact_1024_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% order.strict_trans1
tff(fact_1025_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% order.strict_trans1
tff(fact_1026_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% order.strict_trans1
tff(fact_1027_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% order.strict_trans1
tff(fact_1028_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2) ) ) ).

% order.strict_trans2
tff(fact_1029_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),C2)
       => aa_set_int_o(ord_less_set_int(A),C2) ) ) ).

% order.strict_trans2
tff(fact_1030_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% order.strict_trans2
tff(fact_1031_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% order.strict_trans2
tff(fact_1032_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% order.strict_trans2
tff(fact_1033_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% order.strict_trans2
tff(fact_1034_order_Ostrict__iff__not,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1035_order_Ostrict__iff__not,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B)
        & ~ aa_set_int_o(ord_less_eq_set_int(B),A) ) ) ).

% order.strict_iff_not
tff(fact_1036_order_Ostrict__iff__not,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1037_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1038_order_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1039_order_Ostrict__iff__not,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1040_dense__ge__bounded,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xb)
     => ( ! [W: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),W)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Xb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),W) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Z) ) ) ).

% dense_ge_bounded
tff(fact_1041_dense__ge__bounded,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xb)
     => ( ! [W: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Xb)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),W) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z) ) ) ).

% dense_ge_bounded
tff(fact_1042_dense__le__bounded,axiom,
    ! [Xb: real,Yb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( ! [W: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),W)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Yb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Z) ) ) ).

% dense_le_bounded
tff(fact_1043_dense__le__bounded,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( ! [W: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),W)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Yb)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z) ) ) ).

% dense_le_bounded
tff(fact_1044_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1045_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A)
    <=> ( aa_set_int_o(ord_less_set_int(B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1046_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1047_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1048_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1049_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1050_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1051_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),A)
    <=> ( aa_set_int_o(ord_less_eq_set_int(B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1052_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1053_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1054_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1055_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1056_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1057_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A)
     => ( aa_set_int_o(ord_less_set_int(C2),B)
       => aa_set_int_o(ord_less_set_int(C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1058_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1059_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1060_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1061_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1062_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1063_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),A)
     => ( aa_set_int_o(ord_less_eq_set_int(C2),B)
       => aa_set_int_o(ord_less_set_int(C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1064_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1065_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1066_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1067_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1068_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1069_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),A)
    <=> ( aa_set_int_o(ord_less_eq_set_int(B),A)
        & ~ aa_set_int_o(ord_less_eq_set_int(A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1070_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1071_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1072_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1073_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1074_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% order.strict_implies_order
tff(fact_1075_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
     => aa_set_int_o(ord_less_eq_set_int(A),B) ) ).

% order.strict_implies_order
tff(fact_1076_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% order.strict_implies_order
tff(fact_1077_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% order.strict_implies_order
tff(fact_1078_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% order.strict_implies_order
tff(fact_1079_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% order.strict_implies_order
tff(fact_1080_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1081_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),A)
     => aa_set_int_o(ord_less_eq_set_int(B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1082_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1083_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1084_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1085_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1086_order__le__less,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_less
tff(fact_1087_order__le__less,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Yb)
    <=> ( aa_set_int_o(ord_less_set_int(Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_less
tff(fact_1088_order__le__less,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_less
tff(fact_1089_order__le__less,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_less
tff(fact_1090_order__le__less,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_less
tff(fact_1091_order__le__less,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_less
tff(fact_1092_order__less__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
        & ( Xb != Yb ) ) ) ).

% order_less_le
tff(fact_1093_order__less__le,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xb),Yb)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xb),Yb)
        & ( Xb != Yb ) ) ) ).

% order_less_le
tff(fact_1094_order__less__le,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
        & ( Xb != Yb ) ) ) ).

% order_less_le
tff(fact_1095_order__less__le,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
        & ( Xb != Yb ) ) ) ).

% order_less_le
tff(fact_1096_order__less__le,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
        & ( Xb != Yb ) ) ) ).

% order_less_le
tff(fact_1097_order__less__le,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
        & ( Xb != Yb ) ) ) ).

% order_less_le
tff(fact_1098_linorder__not__le,axiom,
    ! [Xb: real,Yb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ).

% linorder_not_le
tff(fact_1099_linorder__not__le,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ).

% linorder_not_le
tff(fact_1100_linorder__not__le,axiom,
    ! [Xb: num,Yb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ).

% linorder_not_le
tff(fact_1101_linorder__not__le,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ).

% linorder_not_le
tff(fact_1102_linorder__not__le,axiom,
    ! [Xb: int,Yb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ).

% linorder_not_le
tff(fact_1103_linorder__not__less,axiom,
    ! [Xb: real,Yb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb) ) ).

% linorder_not_less
tff(fact_1104_linorder__not__less,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb) ) ).

% linorder_not_less
tff(fact_1105_linorder__not__less,axiom,
    ! [Xb: num,Yb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb) ) ).

% linorder_not_less
tff(fact_1106_linorder__not__less,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb) ) ).

% linorder_not_less
tff(fact_1107_linorder__not__less,axiom,
    ! [Xb: int,Yb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb) ) ).

% linorder_not_less
tff(fact_1108_order__less__imp__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ).

% order_less_imp_le
tff(fact_1109_order__less__imp__le,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xb),Yb)
     => aa_set_int_o(ord_less_eq_set_int(Xb),Yb) ) ).

% order_less_imp_le
tff(fact_1110_order__less__imp__le,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ).

% order_less_imp_le
tff(fact_1111_order__less__imp__le,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb) ) ).

% order_less_imp_le
tff(fact_1112_order__less__imp__le,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ).

% order_less_imp_le
tff(fact_1113_order__less__imp__le,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ).

% order_less_imp_le
tff(fact_1114_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ( A != B )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1115_order__le__neq__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( ( A != B )
       => aa_set_int_o(ord_less_set_int(A),B) ) ) ).

% order_le_neq_trans
tff(fact_1116_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( A != B )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1117_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( A != B )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1118_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( A != B )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1119_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( A != B )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1120_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1121_order__neq__le__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A != B )
     => ( aa_set_int_o(ord_less_eq_set_int(A),B)
       => aa_set_int_o(ord_less_set_int(A),B) ) ) ).

% order_neq_le_trans
tff(fact_1122_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1123_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1124_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1125_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1126_order__le__less__trans,axiom,
    ! [Xb: real,Yb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1127_order__le__less__trans,axiom,
    ! [Xb: set_int,Yb: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Yb)
     => ( aa_set_int_o(ord_less_set_int(Yb),Z)
       => aa_set_int_o(ord_less_set_int(Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1128_order__le__less__trans,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1129_order__le__less__trans,axiom,
    ! [Xb: num,Yb: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1130_order__le__less__trans,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1131_order__le__less__trans,axiom,
    ! [Xb: int,Yb: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1132_order__less__le__trans,axiom,
    ! [Xb: real,Yb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1133_order__less__le__trans,axiom,
    ! [Xb: set_int,Yb: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xb),Yb)
     => ( aa_set_int_o(ord_less_eq_set_int(Yb),Z)
       => aa_set_int_o(ord_less_set_int(Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1134_order__less__le__trans,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1135_order__less__le__trans,axiom,
    ! [Xb: num,Yb: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1136_order__less__le__trans,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1137_order__less__le__trans,axiom,
    ! [Xb: int,Yb: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1138_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1139_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1140_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X3)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1141_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1142_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1143_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1144_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1145_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1146_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1147_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1148_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1149_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1150_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1151_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1152_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1153_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C2: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,B)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X3)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1154_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,B)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1155_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,B)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1156_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,B)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1157_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,B)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1158_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1159_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1160_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1161_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1162_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C2: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1163_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X3)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1164_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1165_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1166_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1167_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B: num,C2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1168_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1169_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1170_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C2: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,B)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X3)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1171_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,B)),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1172_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C2: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,B)),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1173_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,B)),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1174_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1175_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1176_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1177_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C2: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,B)),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1178_linorder__le__less__linear,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ).

% linorder_le_less_linear
tff(fact_1179_linorder__le__less__linear,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ).

% linorder_le_less_linear
tff(fact_1180_linorder__le__less__linear,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ).

% linorder_le_less_linear
tff(fact_1181_linorder__le__less__linear,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ).

% linorder_le_less_linear
tff(fact_1182_linorder__le__less__linear,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ).

% linorder_le_less_linear
tff(fact_1183_order__le__imp__less__or__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1184_order__le__imp__less__or__eq,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Yb)
     => ( aa_set_int_o(ord_less_set_int(Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1185_order__le__imp__less__or__eq,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1186_order__le__imp__less__or__eq,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1187_order__le__imp__less__or__eq,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1188_order__le__imp__less__or__eq,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1189_gr__zeroI,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr_zeroI
tff(fact_1190_not__less__zero,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less_zero
tff(fact_1191_gr__implies__not__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_1192_zero__less__iff__neq__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_1193_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1194_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1195_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1196_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1197_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1198_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1199_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1200_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1201_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & ( K2 = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1202_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & ( K2 = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1203_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( K2 = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1204_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & ( K2 = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1205_add__strict__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_strict_mono
tff(fact_1206_add__strict__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_strict_mono
tff(fact_1207_add__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_strict_mono
tff(fact_1208_add__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_strict_mono
tff(fact_1209_add__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)) ) ).

% add_strict_left_mono
tff(fact_1210_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B)) ) ).

% add_strict_left_mono
tff(fact_1211_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)) ) ).

% add_strict_left_mono
tff(fact_1212_add__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% add_strict_left_mono
tff(fact_1213_add__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ) ).

% add_strict_right_mono
tff(fact_1214_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2)) ) ).

% add_strict_right_mono
tff(fact_1215_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ) ).

% add_strict_right_mono
tff(fact_1216_add__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ) ).

% add_strict_right_mono
tff(fact_1217_add__less__imp__less__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_left
tff(fact_1218_add__less__imp__less__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_1219_add__less__imp__less__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_1220_add__less__imp__less__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_1221_add__less__imp__less__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_right
tff(fact_1222_add__less__imp__less__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_1223_add__less__imp__less__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_1224_add__less__imp__less__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_1225_Nat_OlessE,axiom,
    ! [I: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
     => ( ( K2 != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
             => ( K2 != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_1226_Suc__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_lessD
tff(fact_1227_Suc__lessE,axiom,
    ! [I: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K2)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
           => ( K2 != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_1228_Suc__lessI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( ( aa_nat_nat(suc,Mb) != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb) ) ) ).

% Suc_lessI
tff(fact_1229_less__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( Mb = Nb ) ) ) ).

% less_SucE
tff(fact_1230_less__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% less_SucI
tff(fact_1231_Ex__less__Suc,axiom,
    ! [Nb: nat,P2: fun_nat_o] :
      ( ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Nb))
          & aa_nat_o(P2,I3) )
    <=> ( aa_nat_o(P2,Nb)
        | ? [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
            & aa_nat_o(P2,I3) ) ) ) ).

% Ex_less_Suc
tff(fact_1232_less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% less_Suc_eq
tff(fact_1233_not__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb)) ) ).

% not_less_eq
tff(fact_1234_All__less__Suc,axiom,
    ! [Nb: nat,P2: fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Nb))
         => aa_nat_o(P2,I3) )
    <=> ( aa_nat_o(P2,Nb)
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
           => aa_nat_o(P2,I3) ) ) ) ).

% All_less_Suc
tff(fact_1235_Suc__less__eq2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb)
    <=> ? [M6: nat] :
          ( ( Mb = aa_nat_nat(suc,M6) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M6) ) ) ).

% Suc_less_eq2
tff(fact_1236_less__antisym,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
       => ( Mb = Nb ) ) ) ).

% less_antisym
tff(fact_1237_Suc__less__SucD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_SucD
tff(fact_1238_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K2) ) ) ).

% less_trans_Suc
tff(fact_1239_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P2: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,I2),aa_nat_nat(suc,I2))
       => ( ! [I2: nat,J2: nat,K: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K)
               => ( aa_nat_o(aa_nat_fun_nat_o(P2,I2),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P2,J2),K)
                   => aa_nat_o(aa_nat_fun_nat_o(P2,I2),K) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P2,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_1240_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => aa_nat_o(P2,I2) )
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
             => ( aa_nat_o(P2,aa_nat_nat(suc,I2))
               => aa_nat_o(P2,I2) ) )
         => aa_nat_o(P2,I) ) ) ) ).

% strict_inc_induct
tff(fact_1241_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% not_less_less_Suc_eq
tff(fact_1242_infinite__descent0,axiom,
    ! [P2: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P2,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( ~ aa_nat_o(P2,N2)
             => ? [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N2)
                  & ~ aa_nat_o(P2,M2) ) ) )
       => aa_nat_o(P2,Nb) ) ) ).

% infinite_descent0
tff(fact_1243_gr__implies__not0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_1244_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_zeroE
tff(fact_1245_not__less0,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less0
tff(fact_1246_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr0
tff(fact_1247_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr0I
tff(fact_1248_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_1249_nat__less__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & ( Mb != Nb ) ) ) ).

% nat_less_le
tff(fact_1250_less__imp__le__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_imp_le_nat
tff(fact_1251_le__eq__less__or__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_1252_less__or__eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_or_eq_imp_le
tff(fact_1253_le__neq__implies__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ( Mb != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% le_neq_implies_less
tff(fact_1254_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_1255_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% less_add_eq_less
tff(fact_1256_trans__less__add2,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),J)) ) ).

% trans_less_add2
tff(fact_1257_trans__less__add1,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Mb)) ) ).

% trans_less_add1
tff(fact_1258_add__less__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) ) ).

% add_less_mono1
tff(fact_1259_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I) ).

% not_add_less2
tff(fact_1260_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I) ).

% not_add_less1
tff(fact_1261_add__less__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_1262_add__lessD1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2) ) ).

% add_lessD1
tff(fact_1263_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf((Uu),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_1264_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($true,(Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_1265_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_1266_unbounded__k__infinite,axiom,
    ! [K2: nat,S2: set_nat] :
      ( ! [M: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),M)
         => ? [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N5)
              & aa_set_nat_o(member_nat(N5),S2) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S2) ) ).

% unbounded_k_infinite
tff(fact_1267_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,Nb: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Nb) )
     => aa_set_nat_o(finite_finite_nat,N4) ) ).

% bounded_nat_set_is_finite
tff(fact_1268_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
    <=> ! [M4: nat] :
        ? [N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N3)
          & aa_set_nat_o(member_nat(N3),S2) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_1269_finite__nat__set__iff__bounded,axiom,
    ! [N4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
    <=> ? [M4: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),M4) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1270_add__less__le__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_less_le_mono
tff(fact_1271_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_less_le_mono
tff(fact_1272_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_1273_add__less__le__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_less_le_mono
tff(fact_1274_add__le__less__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_le_less_mono
tff(fact_1275_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_le_less_mono
tff(fact_1276_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_1277_add__le__less__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_le_less_mono
tff(fact_1278_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1279_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1280_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1281_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1282_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1283_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1284_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1285_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1286_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_1287_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_1288_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_1289_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_1290_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_pos
tff(fact_1291_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_pos_pos
tff(fact_1292_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_pos
tff(fact_1293_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_pos
tff(fact_1294_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1295_pos__add__strict,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% pos_add_strict
tff(fact_1296_pos__add__strict,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C2)) ) ) ).

% pos_add_strict
tff(fact_1297_pos__add__strict,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% pos_add_strict
tff(fact_1298_pos__add__strict,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% pos_add_strict
tff(fact_1299_less__Suc__eq__0__disj,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( ( Mb = zero_zero_nat )
        | ? [J3: nat] :
            ( ( Mb = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1300_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ? [M: nat] : Nb = aa_nat_nat(suc,M) ) ).

% gr0_implies_Suc
tff(fact_1301_All__less__Suc2,axiom,
    ! [Nb: nat,P2: fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Nb))
         => aa_nat_o(P2,I3) )
    <=> ( aa_nat_o(P2,zero_zero_nat)
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
           => aa_nat_o(P2,aa_nat_nat(suc,I3)) ) ) ) ).

% All_less_Suc2
tff(fact_1302_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ? [M4: nat] : Nb = aa_nat_nat(suc,M4) ) ).

% gr0_conv_Suc
tff(fact_1303_Ex__less__Suc2,axiom,
    ! [Nb: nat,P2: fun_nat_o] :
      ( ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Nb))
          & aa_nat_o(P2,I3) )
    <=> ( aa_nat_o(P2,zero_zero_nat)
        | ? [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
            & aa_nat_o(P2,aa_nat_nat(suc,I3)) ) ) ) ).

% Ex_less_Suc2
tff(fact_1304_le__imp__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_1305_less__eq__Suc__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% less_eq_Suc_le
tff(fact_1306_less__Suc__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_Suc_eq_le
tff(fact_1307_le__less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% le_less_Suc_eq
tff(fact_1308_Suc__le__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_lessD
tff(fact_1309_inc__induct,axiom,
    ! [I: nat,J: nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P2,J)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),J)
               => ( aa_nat_o(P2,aa_nat_nat(suc,N2))
                 => aa_nat_o(P2,N2) ) ) )
         => aa_nat_o(P2,I) ) ) ) ).

% inc_induct
tff(fact_1310_dec__induct,axiom,
    ! [I: nat,J: nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P2,I)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),J)
               => ( aa_nat_o(P2,N2)
                 => aa_nat_o(P2,aa_nat_nat(suc,N2)) ) ) )
         => aa_nat_o(P2,J) ) ) ) ).

% dec_induct
tff(fact_1311_Suc__le__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_eq
tff(fact_1312_Suc__leI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb) ) ).

% Suc_leI
tff(fact_1313_ex__least__nat__le,axiom,
    ! [P2: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P2,Nb)
     => ( ~ aa_nat_o(P2,zero_zero_nat)
       => ? [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
               => ~ aa_nat_o(P2,I4) )
            & aa_nat_o(P2,K) ) ) ) ).

% ex_least_nat_le
tff(fact_1314_less__imp__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ? [K: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)) ) ).

% less_imp_Suc_add
tff(fact_1315_less__iff__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3)) ) ).

% less_iff_Suc_add
tff(fact_1316_less__add__Suc2,axiom,
    ! [I: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),I))) ).

% less_add_Suc2
tff(fact_1317_less__add__Suc1,axiom,
    ! [I: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb))) ).

% less_add_Suc1
tff(fact_1318_less__natE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ~ ! [Q2: nat] : Nb != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)) ) ).

% less_natE
tff(fact_1319_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ? [K: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1320_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,Mb: nat,K2: nat] :
      ( ! [M: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M)),aa_nat_nat(F,N2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,Mb)),K2)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2))) ) ).

% mono_nat_linear_lb
tff(fact_1321_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X3)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_1322_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_1323_add__strict__increasing2,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_1324_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_1325_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_1326_add__strict__increasing2,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_1327_add__strict__increasing,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_1328_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C2)) ) ) ).

% add_strict_increasing
tff(fact_1329_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_1330_add__strict__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_1331_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_1332_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_1333_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_1334_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_1335_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_1336_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_1337_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_1338_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_1339_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_1340_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_1341_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_1342_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_1343_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_1344_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_1345_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_1346_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_1347_VEBT__internal_OminNull_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ( Xb != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_1348_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(A),A) ).

% verit_comp_simplify1(2)
tff(fact_1349_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1350_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1351_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1352_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1353_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% verit_la_disequality
tff(fact_1354_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% verit_la_disequality
tff(fact_1355_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% verit_la_disequality
tff(fact_1356_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% verit_la_disequality
tff(fact_1357_ex__least__nat__less,axiom,
    ! [P2: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P2,Nb)
     => ( ~ aa_nat_o(P2,zero_zero_nat)
       => ? [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),Nb)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),K)
               => ~ aa_nat_o(P2,I4) )
            & aa_nat_o(P2,aa_nat_nat(suc,K)) ) ) ) ).

% ex_least_nat_less
tff(fact_1358_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ).

% is_num_normalize(1)
tff(fact_1359_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) ).

% is_num_normalize(1)
tff(fact_1360_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% is_num_normalize(1)
tff(fact_1361_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_1362_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_1363_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_1364_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list_VEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Nb,TreeLista),Xb)
    <=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)) ) ).

% in_children_def
tff(fact_1365_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Tree),Xb)
       => ( aa_nat_o(vEBT_V5719532721284313246member(Tree),Xb)
          | aa_nat_o(vEBT_VEBT_membermima(Tree),Xb) ) ) ) ).

% member_valid_both_member_options
tff(fact_1366_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_buildup(Nb)),Nb) ) ).

% buildup_gives_valid
tff(fact_1367_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
    <=> ( aa_nat_o(vEBT_V5719532721284313246member(Ta),Xb)
        | aa_nat_o(vEBT_VEBT_membermima(Ta),Xb) ) ) ).

% both_member_options_def
tff(fact_1368_field__le__epsilon,axiom,
    ! [Xb: real,Yb: real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Yb),E)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ).

% field_le_epsilon
tff(fact_1369_field__le__epsilon,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(plus_plus_rat(Yb),E)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ).

% field_le_epsilon
tff(fact_1370_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_min_max
tff(fact_1371_add__less__zeroD,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_1372_add__less__zeroD,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Xb),Yb)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_1373_add__less__zeroD,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Yb)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_1374_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
    <=> ? [A5: $o,B5: $o] : Ta = vEBT_Leaf((A5),(B5)) ) ).

% deg1Leaf
tff(fact_1375_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
     => ? [A3: $o,B3: $o] : Ta = vEBT_Leaf((A3),(B3)) ) ).

% deg_1_Leaf
tff(fact_1376_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( Nb = one_one_nat )
       => ? [A3: $o,B3: $o] : Ta = vEBT_Leaf((A3),(B3)) ) ) ).

% deg_1_Leafy
tff(fact_1377_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_leaf
tff(fact_1378_psubsetI,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( ( A4 != B4 )
       => aa_set_int_o(ord_less_set_int(A4),B4) ) ) ).

% psubsetI
tff(fact_1379_less__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),one_one_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% less_one
tff(fact_1380_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_VEBT_VEBT] : aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs2) = Nb ).

% Ex_list_of_length
tff(fact_1381_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_int] : aa_list_int_nat(size_size_list_int,Xs2) = Nb ).

% Ex_list_of_length
tff(fact_1382_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_nat] : aa_list_nat_nat(size_size_list_nat,Xs2) = Nb ).

% Ex_list_of_length
tff(fact_1383_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) != aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2) )
     => ( Xs != Ys2 ) ) ).

% neq_if_length_neq
tff(fact_1384_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) != aa_list_int_nat(size_size_list_int,Ys2) )
     => ( Xs != Ys2 ) ) ).

% neq_if_length_neq
tff(fact_1385_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) != aa_list_nat_nat(size_size_list_nat,Ys2) )
     => ( Xs != Ys2 ) ) ).

% neq_if_length_neq
tff(fact_1386_one__reorient,axiom,
    ! [Xb: complex] :
      ( ( one_one_complex = Xb )
    <=> ( Xb = one_one_complex ) ) ).

% one_reorient
tff(fact_1387_one__reorient,axiom,
    ! [Xb: real] :
      ( ( one_one_real = Xb )
    <=> ( Xb = one_one_real ) ) ).

% one_reorient
tff(fact_1388_one__reorient,axiom,
    ! [Xb: rat] :
      ( ( one_one_rat = Xb )
    <=> ( Xb = one_one_rat ) ) ).

% one_reorient
tff(fact_1389_one__reorient,axiom,
    ! [Xb: nat] :
      ( ( one_one_nat = Xb )
    <=> ( Xb = one_one_nat ) ) ).

% one_reorient
tff(fact_1390_one__reorient,axiom,
    ! [Xb: int] :
      ( ( one_one_int = Xb )
    <=> ( Xb = one_one_int ) ) ).

% one_reorient
tff(fact_1391_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_1392_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_1393_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1394_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1395_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1396_finite__psubset__induct,axiom,
    ! [A4: set_nat,P2: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [A6: set_nat] :
            ( aa_set_nat_o(finite_finite_nat,A6)
           => ( ! [B6: set_nat] :
                  ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B6),A6)
                 => aa_set_nat_o(P2,B6) )
             => aa_set_nat_o(P2,A6) ) )
       => aa_set_nat_o(P2,A4) ) ) ).

% finite_psubset_induct
tff(fact_1397_finite__psubset__induct,axiom,
    ! [A4: set_int,P2: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [A6: set_int] :
            ( aa_set_int_o(finite_finite_int,A6)
           => ( ! [B6: set_int] :
                  ( aa_set_int_o(ord_less_set_int(B6),A6)
                 => aa_set_int_o(P2,B6) )
             => aa_set_int_o(P2,A6) ) )
       => aa_set_int_o(P2,A4) ) ) ).

% finite_psubset_induct
tff(fact_1398_finite__psubset__induct,axiom,
    ! [A4: set_complex,P2: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [A6: set_complex] :
            ( aa_set_complex_o(finite3207457112153483333omplex,A6)
           => ( ! [B6: set_complex] :
                  ( aa_set_complex_o(ord_less_set_complex(B6),A6)
                 => aa_set_complex_o(P2,B6) )
             => aa_set_complex_o(P2,A6) ) )
       => aa_set_complex_o(P2,A4) ) ) ).

% finite_psubset_induct
tff(fact_1399_finite__psubset__induct,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,P2: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( ! [A6: set_Pr1261947904930325089at_nat] :
            ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A6)
           => ( ! [B6: set_Pr1261947904930325089at_nat] :
                  ( aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(B6),A6)
                 => aa_set3702741264012733747_nat_o(P2,B6) )
             => aa_set3702741264012733747_nat_o(P2,A6) ) )
       => aa_set3702741264012733747_nat_o(P2,A4) ) ) ).

% finite_psubset_induct
tff(fact_1400_psubsetE,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
         => aa_set_int_o(ord_less_eq_set_int(B4),A4) ) ) ).

% psubsetE
tff(fact_1401_psubset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
        & ( A4 != B4 ) ) ) ).

% psubset_eq
tff(fact_1402_psubset__imp__subset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
     => aa_set_int_o(ord_less_eq_set_int(A4),B4) ) ).

% psubset_imp_subset
tff(fact_1403_psubset__subset__trans,axiom,
    ! [A4: set_int,B4: set_int,C4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),C4)
       => aa_set_int_o(ord_less_set_int(A4),C4) ) ) ).

% psubset_subset_trans
tff(fact_1404_subset__not__subset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
        & ~ aa_set_int_o(ord_less_eq_set_int(B4),A4) ) ) ).

% subset_not_subset_eq
tff(fact_1405_subset__psubset__trans,axiom,
    ! [A4: set_int,B4: set_int,C4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(ord_less_set_int(B4),C4)
       => aa_set_int_o(ord_less_set_int(A4),C4) ) ) ).

% subset_psubset_trans
tff(fact_1406_subset__iff__psubset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
    <=> ( aa_set_int_o(ord_less_set_int(A4),B4)
        | ( A4 = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1407_le__numeral__extra_I4_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),one_one_real) ).

% le_numeral_extra(4)
tff(fact_1408_le__numeral__extra_I4_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),one_one_rat) ).

% le_numeral_extra(4)
tff(fact_1409_le__numeral__extra_I4_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),one_one_nat) ).

% le_numeral_extra(4)
tff(fact_1410_le__numeral__extra_I4_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),one_one_int) ).

% le_numeral_extra(4)
tff(fact_1411_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Leaf((A),(B))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A),
          $ite(Xb = one_one_nat,(B),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_1412_zero__less__one__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% zero_less_one_class.zero_le_one
tff(fact_1413_zero__less__one__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one_class.zero_le_one
tff(fact_1414_zero__less__one__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one_class.zero_le_one
tff(fact_1415_zero__less__one__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% zero_less_one_class.zero_le_one
tff(fact_1416_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1417_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1418_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1419_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1420_not__one__le__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),zero_zero_real) ).

% not_one_le_zero
tff(fact_1421_not__one__le__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),zero_zero_rat) ).

% not_one_le_zero
tff(fact_1422_not__one__le__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),zero_zero_nat) ).

% not_one_le_zero
tff(fact_1423_not__one__le__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),zero_zero_int) ).

% not_one_le_zero
tff(fact_1424_not__one__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),zero_zero_real) ).

% not_one_less_zero
tff(fact_1425_not__one__less__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),zero_zero_rat) ).

% not_one_less_zero
tff(fact_1426_not__one__less__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),zero_zero_nat) ).

% not_one_less_zero
tff(fact_1427_not__one__less__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),zero_zero_int) ).

% not_one_less_zero
tff(fact_1428_zero__less__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% zero_less_one
tff(fact_1429_zero__less__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one
tff(fact_1430_zero__less__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one
tff(fact_1431_zero__less__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% zero_less_one
tff(fact_1432_add__mono1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),one_one_real)) ) ).

% add_mono1
tff(fact_1433_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_rat_rat(plus_plus_rat(B),one_one_rat)) ) ).

% add_mono1
tff(fact_1434_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat)) ) ).

% add_mono1
tff(fact_1435_add__mono1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)) ) ).

% add_mono1
tff(fact_1436_less__add__one,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)) ).

% less_add_one
tff(fact_1437_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),one_one_rat)) ).

% less_add_one
tff(fact_1438_less__add__one,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)) ).

% less_add_one
tff(fact_1439_less__add__one,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)) ).

% less_add_one
tff(fact_1440_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_1441_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_1442_zero__less__two,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)) ).

% zero_less_two
tff(fact_1443_zero__less__two,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)) ).

% zero_less_two
tff(fact_1444_less__numeral__extra_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% less_numeral_extra(1)
tff(fact_1445_less__numeral__extra_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% less_numeral_extra(1)
tff(fact_1446_less__numeral__extra_I1_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% less_numeral_extra(1)
tff(fact_1447_less__numeral__extra_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% less_numeral_extra(1)
tff(fact_1448_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_1449_Suc__eq__plus1,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat) ).

% Suc_eq_plus1
tff(fact_1450_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_1451_Suc__eq__plus1__left,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Nb) ).

% Suc_eq_plus1_left
tff(fact_1452_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_1453_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_1454_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_1455_nat__induct__non__zero,axiom,
    ! [Nb: nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(P2,one_one_nat)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => ( aa_nat_o(P2,N2)
               => aa_nat_o(P2,aa_nat_nat(suc,N2)) ) )
         => aa_nat_o(P2,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_1456_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Leaf((A),(B))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A),
          $ite(Xb = one_one_nat,(B),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_1457_vebt__insert_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A),(B)),Xb) = $ite(
        Xb = zero_zero_nat,
        vEBT_Leaf($true,(B)),
        $ite(Xb = one_one_nat,vEBT_Leaf((A),$true),vEBT_Leaf((A),(B))) ) ).

% vebt_insert.simps(1)
tff(fact_1458_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(2)
tff(fact_1459_discrete,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).

% discrete
tff(fact_1460_discrete,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B) ) ).

% discrete
tff(fact_1461_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_1462_nth__enumerate__eq,axiom,
    ! [Mb: nat,Xs: list_VEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( nth_Pr744662078594809490T_VEBT(enumerate_VEBT_VEBT(Nb,Xs),Mb) = produc599794634098209291T_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb),nth_VEBT_VEBT(Xs,Mb)) ) ) ).

% nth_enumerate_eq
tff(fact_1463_nth__enumerate__eq,axiom,
    ! [Mb: nat,Xs: list_int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_list_int_nat(size_size_list_int,Xs))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(Nb,Xs),Mb) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb),aa_nat_int(nth_int(Xs),Mb)) ) ) ).

% nth_enumerate_eq
tff(fact_1464_nth__enumerate__eq,axiom,
    ! [Mb: nat,Xs: list_nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( nth_Pr7617993195940197384at_nat(enumerate_nat(Nb,Xs),Mb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),aa_nat_nat(nth_nat(Xs),Mb)) ) ) ).

% nth_enumerate_eq
tff(fact_1465_nat__descend__induct,axiom,
    ! [Nb: nat,P2: fun_nat_o,Mb: nat] :
      ( ! [K: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
         => aa_nat_o(P2,K) )
     => ( ! [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
           => ( ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),I4)
                 => aa_nat_o(P2,I4) )
             => aa_nat_o(P2,K) ) )
       => aa_nat_o(P2,Mb) ) ) ).

% nat_descend_induct
tff(fact_1466_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D22)
       => ? [E: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E),D1)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1467_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D1)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D22)
       => ? [E: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E),D1)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1468_complete__interval,axiom,
    ! [A: real,B: real,P2: fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(P2,A)
       => ( ~ aa_real_o(P2,B)
         => ? [C: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
              & ! [X: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X),C) )
                 => aa_real_o(P2,X) )
              & ! [D3: real] :
                  ( ! [X3: real] :
                      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
                        & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),D3) )
                     => aa_real_o(P2,X3) )
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D3),C) ) ) ) ) ) ).

% complete_interval
tff(fact_1469_complete__interval,axiom,
    ! [A: nat,B: nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(P2,A)
       => ( ~ aa_nat_o(P2,B)
         => ? [C: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
              & ! [X: nat] :
                  ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),C) )
                 => aa_nat_o(P2,X) )
              & ! [D3: nat] :
                  ( ! [X3: nat] :
                      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X3)
                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),D3) )
                     => aa_nat_o(P2,X3) )
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D3),C) ) ) ) ) ) ).

% complete_interval
tff(fact_1470_complete__interval,axiom,
    ! [A: int,B: int,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(P2,A)
       => ( ~ aa_int_o(P2,B)
         => ? [C: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
              & ! [X: int] :
                  ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
                    & aa_int_o(aa_int_fun_int_o(ord_less_int,X),C) )
                 => aa_int_o(P2,X) )
              & ! [D3: int] :
                  ( ! [X3: int] :
                      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X3)
                        & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),D3) )
                     => aa_int_o(P2,X3) )
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D3),C) ) ) ) ) ) ).

% complete_interval
tff(fact_1471_pinf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ta) ) ).

% pinf(6)
tff(fact_1472_pinf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Ta) ) ).

% pinf(6)
tff(fact_1473_pinf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Ta) ) ).

% pinf(6)
tff(fact_1474_pinf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ta) ) ).

% pinf(6)
tff(fact_1475_pinf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta) ) ).

% pinf(6)
tff(fact_1476_pinf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X) ) ).

% pinf(8)
tff(fact_1477_pinf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X) ) ).

% pinf(8)
tff(fact_1478_pinf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X) ) ).

% pinf(8)
tff(fact_1479_pinf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X) ) ).

% pinf(8)
tff(fact_1480_pinf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X) ) ).

% pinf(8)
tff(fact_1481_minf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ta) ) ).

% minf(6)
tff(fact_1482_minf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Ta) ) ).

% minf(6)
tff(fact_1483_minf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Ta) ) ).

% minf(6)
tff(fact_1484_minf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ta) ) ).

% minf(6)
tff(fact_1485_minf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta) ) ).

% minf(6)
tff(fact_1486_minf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X) ) ).

% minf(8)
tff(fact_1487_minf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X) ) ).

% minf(8)
tff(fact_1488_minf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X) ) ).

% minf(8)
tff(fact_1489_minf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X) ) ).

% minf(8)
tff(fact_1490_minf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X) ) ).

% minf(8)
tff(fact_1491_empty__subsetI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(bot_bo2099793752762293965at_nat),A4) ).

% empty_subsetI
tff(fact_1492_empty__subsetI,axiom,
    ! [A4: set_real] : ord_less_eq_set_real(bot_bot_set_real,A4) ).

% empty_subsetI
tff(fact_1493_empty__subsetI,axiom,
    ! [A4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A4) ).

% empty_subsetI
tff(fact_1494_empty__subsetI,axiom,
    ! [A4: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A4) ).

% empty_subsetI
tff(fact_1495_subset__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A4),bot_bo2099793752762293965at_nat)
    <=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
tff(fact_1496_subset__empty,axiom,
    ! [A4: set_real] :
      ( ord_less_eq_set_real(A4,bot_bot_set_real)
    <=> ( A4 = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_1497_subset__empty,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),bot_bot_set_nat)
    <=> ( A4 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_1498_subset__empty,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),bot_bot_set_int)
    <=> ( A4 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_1499_length__enumerate,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(enumerate_VEBT_VEBT(Nb,Xs)) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) ).

% length_enumerate
tff(fact_1500_length__enumerate,axiom,
    ! [Nb: nat,Xs: list_int] : size_s2970893825323803983at_int(enumerate_int(Nb,Xs)) = aa_list_int_nat(size_size_list_int,Xs) ).

% length_enumerate
tff(fact_1501_length__enumerate,axiom,
    ! [Nb: nat,Xs: list_nat] : size_s5460976970255530739at_nat(enumerate_nat(Nb,Xs)) = aa_list_nat_nat(size_size_list_nat,Xs) ).

% length_enumerate
tff(fact_1502_bot_Oextremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(bot_bo2099793752762293965at_nat),A) ).

% bot.extremum
tff(fact_1503_bot_Oextremum,axiom,
    ! [A: set_real] : ord_less_eq_set_real(bot_bot_set_real,A) ).

% bot.extremum
tff(fact_1504_bot_Oextremum,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A) ).

% bot.extremum
tff(fact_1505_bot_Oextremum,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A) ).

% bot.extremum
tff(fact_1506_bot_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),A) ).

% bot.extremum
tff(fact_1507_bot_Oextremum__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A),bot_bo2099793752762293965at_nat)
    <=> ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
tff(fact_1508_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ord_less_eq_set_real(A,bot_bot_set_real)
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_1509_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_1510_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_1511_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_1512_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A),bot_bo2099793752762293965at_nat)
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1513_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ord_less_eq_set_real(A,bot_bot_set_real)
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_1514_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat)
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1515_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_1516_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1517_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(bot_bo2099793752762293965at_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1518_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> ord_less_set_real(bot_bot_set_real,A) ) ).

% bot.not_eq_extremum
tff(fact_1519_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,bot_bot_set_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1520_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> aa_set_int_o(ord_less_set_int(bot_bot_set_int),A) ) ).

% bot.not_eq_extremum
tff(fact_1521_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1522_bot_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ~ aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(A),bot_bo2099793752762293965at_nat) ).

% bot.extremum_strict
tff(fact_1523_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ ord_less_set_real(A,bot_bot_set_real) ).

% bot.extremum_strict
tff(fact_1524_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_1525_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ aa_set_int_o(ord_less_set_int(A),bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_1526_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),bot_bot_nat) ).

% bot.extremum_strict
tff(fact_1527_finite_OemptyI,axiom,
    aa_set_complex_o(finite3207457112153483333omplex,bot_bot_set_complex) ).

% finite.emptyI
tff(fact_1528_finite_OemptyI,axiom,
    aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,bot_bo2099793752762293965at_nat) ).

% finite.emptyI
tff(fact_1529_finite_OemptyI,axiom,
    finite_finite_real(bot_bot_set_real) ).

% finite.emptyI
tff(fact_1530_finite_OemptyI,axiom,
    aa_set_nat_o(finite_finite_nat,bot_bot_set_nat) ).

% finite.emptyI
tff(fact_1531_finite_OemptyI,axiom,
    aa_set_int_o(finite_finite_int,bot_bot_set_int) ).

% finite.emptyI
tff(fact_1532_infinite__imp__nonempty,axiom,
    ! [S2: set_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( S2 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_1533_infinite__imp__nonempty,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S2)
     => ( S2 != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
tff(fact_1534_infinite__imp__nonempty,axiom,
    ! [S2: set_real] :
      ( ~ finite_finite_real(S2)
     => ( S2 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
tff(fact_1535_infinite__imp__nonempty,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ( S2 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_1536_infinite__imp__nonempty,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
     => ( S2 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_1537_finite__transitivity__chain,axiom,
    ! [A4: set_set_nat,R: fun_se2170385732985038782_nat_o] :
      ( finite1152437895449049373et_nat(A4)
     => ( ! [X3: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X3),X3)
       => ( ! [X3: set_nat,Y: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X3),Y)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(R,Y),Z2)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X3),Z2) ) )
         => ( ! [X3: set_nat] :
                ( member_set_nat(X3,A4)
               => ? [Y3: set_nat] :
                    ( member_set_nat(Y3,A4)
                    & aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X3),Y3) ) )
           => ( A4 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1538_finite__transitivity__chain,axiom,
    ! [A4: set_complex,R: fun_co1148497455491383662plex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X3: complex] : ~ aa_complex_o(aa_com5808801942499186519plex_o(R,X3),X3)
       => ( ! [X3: complex,Y: complex,Z2: complex] :
              ( aa_complex_o(aa_com5808801942499186519plex_o(R,X3),Y)
             => ( aa_complex_o(aa_com5808801942499186519plex_o(R,Y),Z2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,X3),Z2) ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),A4)
               => ? [Y3: complex] :
                    ( aa_set_complex_o(member_complex(Y3),A4)
                    & aa_complex_o(aa_com5808801942499186519plex_o(R,X3),Y3) ) )
           => ( A4 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1539_finite__transitivity__chain,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,R: fun_Pr2028404973800837934_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( ! [X3: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X3),X3)
       => ( ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,Z2: product_prod_nat_nat] :
              ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X3),Y)
             => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,Y),Z2)
               => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X3),Z2) ) )
         => ( ! [X3: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4)
               => ? [Y3: product_prod_nat_nat] :
                    ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y3),A4)
                    & aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X3),Y3) ) )
           => ( A4 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1540_finite__transitivity__chain,axiom,
    ! [A4: set_real,R: fun_real_fun_real_o] :
      ( finite_finite_real(A4)
     => ( ! [X3: real] : ~ aa_real_o(aa_real_fun_real_o(R,X3),X3)
       => ( ! [X3: real,Y: real,Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(R,X3),Y)
             => ( aa_real_o(aa_real_fun_real_o(R,Y),Z2)
               => aa_real_o(aa_real_fun_real_o(R,X3),Z2) ) )
         => ( ! [X3: real] :
                ( member_real(X3,A4)
               => ? [Y3: real] :
                    ( member_real(Y3,A4)
                    & aa_real_o(aa_real_fun_real_o(R,X3),Y3) ) )
           => ( A4 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1541_finite__transitivity__chain,axiom,
    ! [A4: set_nat,R: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X3: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(R,X3),X3)
       => ( ! [X3: nat,Y: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X3),Y)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R,X3),Z2) ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),A4)
               => ? [Y3: nat] :
                    ( aa_set_nat_o(member_nat(Y3),A4)
                    & aa_nat_o(aa_nat_fun_nat_o(R,X3),Y3) ) )
           => ( A4 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1542_finite__transitivity__chain,axiom,
    ! [A4: set_int,R: fun_int_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X3: int] : ~ aa_int_o(aa_int_fun_int_o(R,X3),X3)
       => ( ! [X3: int,Y: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(R,X3),Y)
             => ( aa_int_o(aa_int_fun_int_o(R,Y),Z2)
               => aa_int_o(aa_int_fun_int_o(R,X3),Z2) ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),A4)
               => ? [Y3: int] :
                    ( aa_set_int_o(member_int(Y3),A4)
                    & aa_int_o(aa_int_fun_int_o(R,X3),Y3) ) )
           => ( A4 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1543_finite__has__maximal,axiom,
    ! [A4: set_real] :
      ( finite_finite_real(A4)
     => ( ( A4 != bot_bot_set_real )
       => ? [X3: real] :
            ( member_real(X3,A4)
            & ! [Xa: real] :
                ( member_real(Xa,A4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1544_finite__has__maximal,axiom,
    ! [A4: set_set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( ( A4 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( member_set_int(X3,A4)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A4)
               => ( aa_set_int_o(ord_less_eq_set_int(X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1545_finite__has__maximal,axiom,
    ! [A4: set_rat] :
      ( finite_finite_rat(A4)
     => ( ( A4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( member_rat(X3,A4)
            & ! [Xa: rat] :
                ( member_rat(Xa,A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1546_finite__has__maximal,axiom,
    ! [A4: set_num] :
      ( finite_finite_num(A4)
     => ( ( A4 != bot_bot_set_num )
       => ? [X3: num] :
            ( member_num(X3,A4)
            & ! [Xa: num] :
                ( member_num(Xa,A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1547_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1548_finite__has__maximal,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1549_finite__has__minimal,axiom,
    ! [A4: set_real] :
      ( finite_finite_real(A4)
     => ( ( A4 != bot_bot_set_real )
       => ? [X3: real] :
            ( member_real(X3,A4)
            & ! [Xa: real] :
                ( member_real(Xa,A4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1550_finite__has__minimal,axiom,
    ! [A4: set_set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( ( A4 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( member_set_int(X3,A4)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A4)
               => ( aa_set_int_o(ord_less_eq_set_int(Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1551_finite__has__minimal,axiom,
    ! [A4: set_rat] :
      ( finite_finite_rat(A4)
     => ( ( A4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( member_rat(X3,A4)
            & ! [Xa: rat] :
                ( member_rat(Xa,A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1552_finite__has__minimal,axiom,
    ! [A4: set_num] :
      ( finite_finite_num(A4)
     => ( ( A4 != bot_bot_set_num )
       => ? [X3: num] :
            ( member_num(X3,A4)
            & ! [Xa: num] :
                ( member_num(Xa,A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1553_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1554_finite__has__minimal,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1555_infinite__growing,axiom,
    ! [X5: set_real] :
      ( ( X5 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( member_real(X3,X5)
           => ? [Xa: real] :
                ( member_real(Xa,X5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Xa) ) )
       => ~ finite_finite_real(X5) ) ) ).

% infinite_growing
tff(fact_1556_infinite__growing,axiom,
    ! [X5: set_rat] :
      ( ( X5 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( member_rat(X3,X5)
           => ? [Xa: rat] :
                ( member_rat(Xa,X5)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Xa) ) )
       => ~ finite_finite_rat(X5) ) ) ).

% infinite_growing
tff(fact_1557_infinite__growing,axiom,
    ! [X5: set_num] :
      ( ( X5 != bot_bot_set_num )
     => ( ! [X3: num] :
            ( member_num(X3,X5)
           => ? [Xa: num] :
                ( member_num(Xa,X5)
                & aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Xa) ) )
       => ~ finite_finite_num(X5) ) ) ).

% infinite_growing
tff(fact_1558_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),X5)
           => ? [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),X5)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xa) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X5) ) ) ).

% infinite_growing
tff(fact_1559_infinite__growing,axiom,
    ! [X5: set_int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),X5)
           => ? [Xa: int] :
                ( aa_set_int_o(member_int(Xa),X5)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Xa) ) )
       => ~ aa_set_int_o(finite_finite_int,X5) ) ) ).

% infinite_growing
tff(fact_1560_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( finite_finite_real(S2)
     => ( ( S2 != bot_bot_set_real )
       => ? [X3: real] :
            ( member_real(X3,S2)
            & ~ ? [Xa: real] :
                  ( member_real(Xa,S2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1561_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( finite_finite_rat(S2)
     => ( ( S2 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( member_rat(X3,S2)
            & ~ ? [Xa: rat] :
                  ( member_rat(Xa,S2)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1562_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( finite_finite_num(S2)
     => ( ( S2 != bot_bot_set_num )
       => ? [X3: num] :
            ( member_num(X3,S2)
            & ~ ? [Xa: num] :
                  ( member_num(Xa,S2)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1563_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),S2)
            & ~ ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),S2)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1564_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),S2)
            & ~ ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),S2)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1565_divides__aux__eq,axiom,
    ! [Q3: nat,R2: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_1566_divides__aux__eq,axiom,
    ! [Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_1567_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),S2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X)),aa_complex_real(F,lattic8794016678065449205x_real(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1568_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: fun_real_real] :
      ( finite_finite_real(S2)
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X: real] :
              ( member_real(X,S2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,lattic8440615504127631091l_real(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1569_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),S2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X)),aa_nat_real(F,aa_set_nat_nat(lattic488527866317076247t_real(F),S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1570_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X: int] :
              ( aa_set_int_o(member_int(X),S2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(F,lattic2675449441010098035t_real(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1571_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),S2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X)),aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1572_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: fun_real_rat] :
      ( finite_finite_real(S2)
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X: real] :
              ( member_real(X,S2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,lattic4420706379359479199al_rat(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1573_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),S2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1574_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X: int] :
              ( aa_set_int_o(member_int(X),S2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(F,lattic7811156612396918303nt_rat(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1575_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),S2)
              & aa_num_o(aa_num_fun_num_o(ord_less_num,aa_complex_num(F,X)),aa_complex_num(F,lattic1922116423962787043ex_num(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1576_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: fun_real_num] :
      ( finite_finite_real(S2)
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X: real] :
              ( member_real(X,S2)
              & aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X)),aa_real_num(F,lattic1613168225601753569al_num(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1577_arg__min__least,axiom,
    ! [S2: set_complex,Yb: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ( aa_set_complex_o(member_complex(Yb),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S2))),aa_complex_rat(F,Yb)) ) ) ) ).

% arg_min_least
tff(fact_1578_arg__min__least,axiom,
    ! [S2: set_real,Yb: real,F: fun_real_rat] :
      ( finite_finite_real(S2)
     => ( ( S2 != bot_bot_set_real )
       => ( member_real(Yb,S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,lattic4420706379359479199al_rat(F,S2))),aa_real_rat(F,Yb)) ) ) ) ).

% arg_min_least
tff(fact_1579_arg__min__least,axiom,
    ! [S2: set_nat,Yb: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ( aa_set_nat_o(member_nat(Yb),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S2))),aa_nat_rat(F,Yb)) ) ) ) ).

% arg_min_least
tff(fact_1580_arg__min__least,axiom,
    ! [S2: set_int,Yb: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ( aa_set_int_o(member_int(Yb),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,lattic7811156612396918303nt_rat(F,S2))),aa_int_rat(F,Yb)) ) ) ) ).

% arg_min_least
tff(fact_1581_arg__min__least,axiom,
    ! [S2: set_complex,Yb: complex,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ( aa_set_complex_o(member_complex(Yb),S2)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(F,lattic1922116423962787043ex_num(F,S2))),aa_complex_num(F,Yb)) ) ) ) ).

% arg_min_least
tff(fact_1582_arg__min__least,axiom,
    ! [S2: set_real,Yb: real,F: fun_real_num] :
      ( finite_finite_real(S2)
     => ( ( S2 != bot_bot_set_real )
       => ( member_real(Yb,S2)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_real_num(F,lattic1613168225601753569al_num(F,S2))),aa_real_num(F,Yb)) ) ) ) ).

% arg_min_least
tff(fact_1583_arg__min__least,axiom,
    ! [S2: set_nat,Yb: nat,F: fun_nat_num] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ( aa_set_nat_o(member_nat(Yb),S2)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,aa_set_nat_nat(lattic4004264746738138117at_num(F),S2))),aa_nat_num(F,Yb)) ) ) ) ).

% arg_min_least
tff(fact_1584_arg__min__least,axiom,
    ! [S2: set_int,Yb: int,F: fun_int_num] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ( aa_set_int_o(member_int(Yb),S2)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,lattic5003618458639192673nt_num(F,S2))),aa_int_num(F,Yb)) ) ) ) ).

% arg_min_least
tff(fact_1585_arg__min__least,axiom,
    ! [S2: set_complex,Yb: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ( aa_set_complex_o(member_complex(Yb),S2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,lattic5364784637807008409ex_nat(F,S2))),aa_complex_nat(F,Yb)) ) ) ) ).

% arg_min_least
tff(fact_1586_arg__min__least,axiom,
    ! [S2: set_real,Yb: real,F: fun_real_nat] :
      ( finite_finite_real(S2)
     => ( ( S2 != bot_bot_set_real )
       => ( member_real(Yb,S2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,lattic5055836439445974935al_nat(F,S2))),aa_real_nat(F,Yb)) ) ) ) ).

% arg_min_least
tff(fact_1587_subset__emptyI,axiom,
    ! [A4: set_complex] :
      ( ! [X3: complex] : ~ aa_set_complex_o(member_complex(X3),A4)
     => aa_set_complex_o(ord_le211207098394363844omplex(A4),bot_bot_set_complex) ) ).

% subset_emptyI
tff(fact_1588_subset__emptyI,axiom,
    ! [A4: set_set_nat] :
      ( ! [X3: set_nat] : ~ member_set_nat(X3,A4)
     => ord_le6893508408891458716et_nat(A4,bot_bot_set_set_nat) ) ).

% subset_emptyI
tff(fact_1589_subset__emptyI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4)
     => aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A4),bot_bo2099793752762293965at_nat) ) ).

% subset_emptyI
tff(fact_1590_subset__emptyI,axiom,
    ! [A4: set_real] :
      ( ! [X3: real] : ~ member_real(X3,A4)
     => ord_less_eq_set_real(A4,bot_bot_set_real) ) ).

% subset_emptyI
tff(fact_1591_subset__emptyI,axiom,
    ! [A4: set_nat] :
      ( ! [X3: nat] : ~ aa_set_nat_o(member_nat(X3),A4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),bot_bot_set_nat) ) ).

% subset_emptyI
tff(fact_1592_subset__emptyI,axiom,
    ! [A4: set_int] :
      ( ! [X3: int] : ~ aa_set_int_o(member_int(X3),A4)
     => aa_set_int_o(ord_less_eq_set_int(A4),bot_bot_set_int) ) ).

% subset_emptyI
tff(fact_1593_find__Some__iff2,axiom,
    ! [Xb: product_prod_nat_nat,P2: fun_Pr4796943543331071514_nat_o,Xs: list_P6011104703257516679at_nat] :
      ( ( some_P7363390416028606310at_nat(Xb) = find_P8199882355184865565at_nat(P2,Xs) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s5460976970255530739at_nat(Xs))
          & aa_Pro6150637604445964947_nat_o(P2,nth_Pr7617993195940197384at_nat(Xs,I3))
          & ( Xb = nth_Pr7617993195940197384at_nat(Xs,I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_Pro6150637604445964947_nat_o(P2,nth_Pr7617993195940197384at_nat(Xs,J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1594_find__Some__iff2,axiom,
    ! [Xb: num,P2: fun_num_o,Xs: list_num] :
      ( ( some_num(Xb) = find_num(P2,Xs) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_num(Xs))
          & aa_num_o(P2,aa_nat_num(nth_num(Xs),I3))
          & ( Xb = aa_nat_num(nth_num(Xs),I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_num_o(P2,aa_nat_num(nth_num(Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1595_find__Some__iff2,axiom,
    ! [Xb: vEBT_VEBT,P2: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ( some_VEBT_VEBT(Xb) = find_VEBT_VEBT(P2,Xs) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
          & aa_VEBT_VEBT_o(P2,nth_VEBT_VEBT(Xs,I3))
          & ( Xb = nth_VEBT_VEBT(Xs,I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_VEBT_VEBT_o(P2,nth_VEBT_VEBT(Xs,J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1596_find__Some__iff2,axiom,
    ! [Xb: int,P2: fun_int_o,Xs: list_int] :
      ( ( some_int(Xb) = find_int(P2,Xs) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,Xs))
          & aa_int_o(P2,aa_nat_int(nth_int(Xs),I3))
          & ( Xb = aa_nat_int(nth_int(Xs),I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_int_o(P2,aa_nat_int(nth_int(Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1597_find__Some__iff2,axiom,
    ! [Xb: nat,P2: fun_nat_o,Xs: list_nat] :
      ( ( some_nat(Xb) = find_nat(P2,Xs) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs))
          & aa_nat_o(P2,aa_nat_nat(nth_nat(Xs),I3))
          & ( Xb = aa_nat_nat(nth_nat(Xs),I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_nat_o(P2,aa_nat_nat(nth_nat(Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1598_find__Some__iff,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,Xs: list_P6011104703257516679at_nat,Xb: product_prod_nat_nat] :
      ( ( find_P8199882355184865565at_nat(P2,Xs) = some_P7363390416028606310at_nat(Xb) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s5460976970255530739at_nat(Xs))
          & aa_Pro6150637604445964947_nat_o(P2,nth_Pr7617993195940197384at_nat(Xs,I3))
          & ( Xb = nth_Pr7617993195940197384at_nat(Xs,I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_Pro6150637604445964947_nat_o(P2,nth_Pr7617993195940197384at_nat(Xs,J3)) ) ) ) ).

% find_Some_iff
tff(fact_1599_find__Some__iff,axiom,
    ! [P2: fun_num_o,Xs: list_num,Xb: num] :
      ( ( find_num(P2,Xs) = some_num(Xb) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_num(Xs))
          & aa_num_o(P2,aa_nat_num(nth_num(Xs),I3))
          & ( Xb = aa_nat_num(nth_num(Xs),I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_num_o(P2,aa_nat_num(nth_num(Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_1600_find__Some__iff,axiom,
    ! [P2: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ( find_VEBT_VEBT(P2,Xs) = some_VEBT_VEBT(Xb) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
          & aa_VEBT_VEBT_o(P2,nth_VEBT_VEBT(Xs,I3))
          & ( Xb = nth_VEBT_VEBT(Xs,I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_VEBT_VEBT_o(P2,nth_VEBT_VEBT(Xs,J3)) ) ) ) ).

% find_Some_iff
tff(fact_1601_find__Some__iff,axiom,
    ! [P2: fun_int_o,Xs: list_int,Xb: int] :
      ( ( find_int(P2,Xs) = some_int(Xb) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,Xs))
          & aa_int_o(P2,aa_nat_int(nth_int(Xs),I3))
          & ( Xb = aa_nat_int(nth_int(Xs),I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_int_o(P2,aa_nat_int(nth_int(Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_1602_find__Some__iff,axiom,
    ! [P2: fun_nat_o,Xs: list_nat,Xb: nat] :
      ( ( find_nat(P2,Xs) = some_nat(Xb) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs))
          & aa_nat_o(P2,aa_nat_nat(nth_nat(Xs),I3))
          & ( Xb = aa_nat_nat(nth_nat(Xs),I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_nat_o(P2,aa_nat_nat(nth_nat(Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_1603_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2))
       => ( nth_Pr4953567300277697838T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys2),I) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,I),nth_VEBT_VEBT(Ys2,I)) ) ) ) ).

% nth_zip
tff(fact_1604_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Ys2))
       => ( nth_Pr6837108013167703752BT_int(zip_VEBT_VEBT_int(Xs,Ys2),I) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,I),aa_nat_int(nth_int(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_1605_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Ys2))
       => ( nth_Pr1791586995822124652BT_nat(zip_VEBT_VEBT_nat(Xs,Ys2),I) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,I),aa_nat_nat(nth_nat(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_1606_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2))
       => ( nth_Pr3474266648193625910T_VEBT(zip_int_VEBT_VEBT(Xs,Ys2),I) = produc3329399203697025711T_VEBT(aa_nat_int(nth_int(Xs),I),nth_VEBT_VEBT(Ys2,I)) ) ) ) ).

% nth_zip
tff(fact_1607_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Ys2))
       => ( nth_Pr4439495888332055232nt_int(zip_int_int(Xs,Ys2),I) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_1608_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Ys2))
       => ( nth_Pr8617346907841251940nt_nat(zip_int_nat(Xs,Ys2),I) = product_Pair_int_nat(aa_nat_int(nth_int(Xs),I),aa_nat_nat(nth_nat(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_1609_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys2: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2))
       => ( nth_Pr744662078594809490T_VEBT(zip_nat_VEBT_VEBT(Xs,Ys2),I) = produc599794634098209291T_VEBT(aa_nat_nat(nth_nat(Xs),I),nth_VEBT_VEBT(Ys2,I)) ) ) ) ).

% nth_zip
tff(fact_1610_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Ys2))
       => ( nth_Pr3440142176431000676at_int(zip_nat_int(Xs,Ys2),I) = product_Pair_nat_int(aa_nat_nat(nth_nat(Xs),I),aa_nat_int(nth_int(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_1611_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Ys2))
       => ( nth_Pr7617993195940197384at_nat(zip_nat_nat(Xs,Ys2),I) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_1612_nth__zip,axiom,
    ! [I: nat,Xs: list_f8851413139013959296e_term,Ys2: list_P5578671422887162913nteger] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s502031211236231956e_term(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6527587076508218509nteger(Ys2))
       => ( nth_Pr2854691652959048409nteger(zip_fu7084642285130276018nteger(Xs,Ys2),I) = produc7264323995540650962nteger(nth_fu2298125310669257513e_term(Xs,I),nth_Pr2304437835452373666nteger(Ys2,I)) ) ) ) ).

% nth_zip
tff(fact_1613_rotate1__length01,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),one_one_nat)
     => ( rotate1_VEBT_VEBT(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_1614_rotate1__length01,axiom,
    ! [Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),one_one_nat)
     => ( rotate1_int(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_1615_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat)
     => ( rotate1_nat(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_1616_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_1617_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_1618_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_1619_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_1620_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_1621_set__rotate1,axiom,
    ! [Xs: list_VEBT_VEBT] : set_VEBT_VEBT2(rotate1_VEBT_VEBT(Xs)) = set_VEBT_VEBT2(Xs) ).

% set_rotate1
tff(fact_1622_set__rotate1,axiom,
    ! [Xs: list_int] : set_int2(rotate1_int(Xs)) = set_int2(Xs) ).

% set_rotate1
tff(fact_1623_set__rotate1,axiom,
    ! [Xs: list_nat] : set_nat2(rotate1_nat(Xs)) = set_nat2(Xs) ).

% set_rotate1
tff(fact_1624_length__rotate1,axiom,
    ! [Xs: list_VEBT_VEBT] : aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,rotate1_VEBT_VEBT(Xs)) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) ).

% length_rotate1
tff(fact_1625_length__rotate1,axiom,
    ! [Xs: list_int] : aa_list_int_nat(size_size_list_int,rotate1_int(Xs)) = aa_list_int_nat(size_size_list_int,Xs) ).

% length_rotate1
tff(fact_1626_length__rotate1,axiom,
    ! [Xs: list_nat] : aa_list_nat_nat(size_size_list_nat,rotate1_nat(Xs)) = aa_list_nat_nat(size_size_list_nat,Xs) ).

% length_rotate1
tff(fact_1627_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_1628_find__cong,axiom,
    ! [Xs: list_complex,Ys2: list_complex,P2: fun_complex_o,Q: fun_complex_o] :
      ( ( Xs = Ys2 )
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),set_complex2(Ys2))
           => ( aa_complex_o(P2,X3)
            <=> aa_complex_o(Q,X3) ) )
       => ( find_complex(P2,Xs) = find_complex(Q,Ys2) ) ) ) ).

% find_cong
tff(fact_1629_find__cong,axiom,
    ! [Xs: list_real,Ys2: list_real,P2: fun_real_o,Q: fun_real_o] :
      ( ( Xs = Ys2 )
     => ( ! [X3: real] :
            ( member_real(X3,set_real2(Ys2))
           => ( aa_real_o(P2,X3)
            <=> aa_real_o(Q,X3) ) )
       => ( find_real(P2,Xs) = find_real(Q,Ys2) ) ) ) ).

% find_cong
tff(fact_1630_find__cong,axiom,
    ! [Xs: list_set_nat,Ys2: list_set_nat,P2: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ( Xs = Ys2 )
     => ( ! [X3: set_nat] :
            ( member_set_nat(X3,set_set_nat2(Ys2))
           => ( aa_set_nat_o(P2,X3)
            <=> aa_set_nat_o(Q,X3) ) )
       => ( find_set_nat(P2,Xs) = find_set_nat(Q,Ys2) ) ) ) ).

% find_cong
tff(fact_1631_find__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,P2: fun_VEBT_VEBT_o,Q: fun_VEBT_VEBT_o] :
      ( ( Xs = Ys2 )
     => ( ! [X3: vEBT_VEBT] :
            ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Ys2))
           => ( aa_VEBT_VEBT_o(P2,X3)
            <=> aa_VEBT_VEBT_o(Q,X3) ) )
       => ( find_VEBT_VEBT(P2,Xs) = find_VEBT_VEBT(Q,Ys2) ) ) ) ).

% find_cong
tff(fact_1632_find__cong,axiom,
    ! [Xs: list_int,Ys2: list_int,P2: fun_int_o,Q: fun_int_o] :
      ( ( Xs = Ys2 )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),set_int2(Ys2))
           => ( aa_int_o(P2,X3)
            <=> aa_int_o(Q,X3) ) )
       => ( find_int(P2,Xs) = find_int(Q,Ys2) ) ) ) ).

% find_cong
tff(fact_1633_find__cong,axiom,
    ! [Xs: list_nat,Ys2: list_nat,P2: fun_nat_o,Q: fun_nat_o] :
      ( ( Xs = Ys2 )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),set_nat2(Ys2))
           => ( aa_nat_o(P2,X3)
            <=> aa_nat_o(Q,X3) ) )
       => ( find_nat(P2,Xs) = find_nat(Q,Ys2) ) ) ) ).

% find_cong
tff(fact_1634_set__zip__rightD,axiom,
    ! [Xb: int,Yb: int,Xs: list_int,Ys2: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Yb),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys2)))
     => aa_set_int_o(member_int(Yb),set_int2(Ys2)) ) ).

% set_zip_rightD
tff(fact_1635_set__zip__rightD,axiom,
    ! [Xb: fun_Co3460408610320954864e_term,Yb: produc8923325533196201883nteger,Xs: list_f8851413139013959296e_term,Ys2: list_P5578671422887162913nteger] :
      ( member6901806533512420791nteger(produc7264323995540650962nteger(Xb,Yb),set_Pr6295439789841872971nteger(zip_fu7084642285130276018nteger(Xs,Ys2)))
     => member157494554546826820nteger(Yb,set_Pr920681315882439344nteger(Ys2)) ) ).

% set_zip_rightD
tff(fact_1636_set__zip__rightD,axiom,
    ! [Xb: fun_Pr5098669352780829690e_term,Yb: produc8923325533196201883nteger,Xs: list_f4306079564415238026e_term,Ys2: list_P5578671422887162913nteger] :
      ( member8978266922497061677nteger(produc565448451787809096nteger(Xb,Yb),set_Pr5524914931742404033nteger(zip_fu4451626411769283112nteger(Xs,Ys2)))
     => member157494554546826820nteger(Yb,set_Pr920681315882439344nteger(Ys2)) ) ).

% set_zip_rightD
tff(fact_1637_set__zip__rightD,axiom,
    ! [Xb: fun_Pr7763680713840462443e_term,Yb: product_prod_int_int,Xs: list_f280343417242313083e_term,Ys2: list_P5707943133018811711nt_int] :
      ( member753276645541486746nt_int(produc7320969046608631221nt_int(Xb,Yb),set_Pr8602918816529533742nt_int(zip_fu5048005673116055189nt_int(Xs,Ys2)))
     => member5262025264175285858nt_int(Yb,set_Pr2470121279949933262nt_int(Ys2)) ) ).

% set_zip_rightD
tff(fact_1638_set__zip__rightD,axiom,
    ! [Xb: fun_in5730379987311621695e_term,Yb: product_prod_int_int,Xs: list_f628289730585369423e_term,Ys2: list_P5707943133018811711nt_int] :
      ( member5890090881338784710nt_int(produc4729913460834754785nt_int(Xb,Yb),set_Pr4881580080894854746nt_int(zip_fu2910578966527623617nt_int(Xs,Ys2)))
     => member5262025264175285858nt_int(Yb,set_Pr2470121279949933262nt_int(Ys2)) ) ).

% set_zip_rightD
tff(fact_1639_set__zip__leftD,axiom,
    ! [Xb: int,Yb: int,Xs: list_int,Ys2: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Yb),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys2)))
     => aa_set_int_o(member_int(Xb),set_int2(Xs)) ) ).

% set_zip_leftD
tff(fact_1640_set__zip__leftD,axiom,
    ! [Xb: fun_Co3460408610320954864e_term,Yb: produc8923325533196201883nteger,Xs: list_f8851413139013959296e_term,Ys2: list_P5578671422887162913nteger] :
      ( member6901806533512420791nteger(produc7264323995540650962nteger(Xb,Yb),set_Pr6295439789841872971nteger(zip_fu7084642285130276018nteger(Xs,Ys2)))
     => member7516538326899768071e_term(Xb,set_fu2578164854786925467e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1641_set__zip__leftD,axiom,
    ! [Xb: fun_Pr5098669352780829690e_term,Yb: produc8923325533196201883nteger,Xs: list_f4306079564415238026e_term,Ys2: list_P5578671422887162913nteger] :
      ( member8978266922497061677nteger(produc565448451787809096nteger(Xb,Yb),set_Pr5524914931742404033nteger(zip_fu4451626411769283112nteger(Xs,Ys2)))
     => member8464590939221061905e_term(Xb,set_fu3771136004723407269e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1642_set__zip__leftD,axiom,
    ! [Xb: fun_Pr7763680713840462443e_term,Yb: product_prod_int_int,Xs: list_f280343417242313083e_term,Ys2: list_P5707943133018811711nt_int] :
      ( member753276645541486746nt_int(produc7320969046608631221nt_int(Xb,Yb),set_Pr8602918816529533742nt_int(zip_fu5048005673116055189nt_int(Xs,Ys2)))
     => member1495369047980753794e_term(Xb,set_fu3262800575041149974e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1643_set__zip__leftD,axiom,
    ! [Xb: fun_in5730379987311621695e_term,Yb: product_prod_int_int,Xs: list_f628289730585369423e_term,Ys2: list_P5707943133018811711nt_int] :
      ( member5890090881338784710nt_int(produc4729913460834754785nt_int(Xb,Yb),set_Pr4881580080894854746nt_int(zip_fu2910578966527623617nt_int(Xs,Ys2)))
     => member4871225538671630166e_term(Xb,set_fu1748814359475392490e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1644_in__set__zipE,axiom,
    ! [Xb: complex,Yb: complex,Xs: list_complex,Ys2: list_complex] :
      ( member5793383173714906214omplex(produc101793102246108661omplex(Xb,Yb),set_Pr8199049879907524818omplex(zip_complex_complex(Xs,Ys2)))
     => ~ ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
         => ~ aa_set_complex_o(member_complex(Yb),set_complex2(Ys2)) ) ) ).

% in_set_zipE
tff(fact_1645_in__set__zipE,axiom,
    ! [Xb: complex,Yb: real,Xs: list_complex,Ys2: list_real] :
      ( member47443559803733732x_real(produc1746590499379883635x_real(Xb,Yb),set_Pr1225976482156248400x_real(zip_complex_real(Xs,Ys2)))
     => ~ ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
         => ~ member_real(Yb,set_real2(Ys2)) ) ) ).

% in_set_zipE
tff(fact_1646_in__set__zipE,axiom,
    ! [Xb: real,Yb: complex,Xs: list_real,Ys2: list_complex] :
      ( member7358116576843751780omplex(produc1693001998875562995omplex(Xb,Yb),set_Pr8536649499196266448omplex(zip_real_complex(Xs,Ys2)))
     => ~ ( member_real(Xb,set_real2(Xs))
         => ~ aa_set_complex_o(member_complex(Yb),set_complex2(Ys2)) ) ) ).

% in_set_zipE
tff(fact_1647_in__set__zipE,axiom,
    ! [Xb: real,Yb: real,Xs: list_real,Ys2: list_real] :
      ( member7849222048561428706l_real(produc4511245868158468465l_real(Xb,Yb),set_Pr5999470521830281550l_real(zip_real_real(Xs,Ys2)))
     => ~ ( member_real(Xb,set_real2(Xs))
         => ~ member_real(Yb,set_real2(Ys2)) ) ) ).

% in_set_zipE
tff(fact_1648_in__set__zipE,axiom,
    ! [Xb: complex,Yb: vEBT_VEBT,Xs: list_complex,Ys2: list_VEBT_VEBT] :
      ( member1978952105866562066T_VEBT(produc2757191886755552429T_VEBT(Xb,Yb),set_Pr5158653123227461798T_VEBT(zip_co9157518722488180109T_VEBT(Xs,Ys2)))
     => ~ ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
         => ~ member_VEBT_VEBT(Yb,set_VEBT_VEBT2(Ys2)) ) ) ).

% in_set_zipE
tff(fact_1649_in__set__zipE,axiom,
    ! [Xb: real,Yb: vEBT_VEBT,Xs: list_real,Ys2: list_VEBT_VEBT] :
      ( member7262085504369356948T_VEBT(produc6931449550656315951T_VEBT(Xb,Yb),set_Pr8897343066327330088T_VEBT(zip_real_VEBT_VEBT(Xs,Ys2)))
     => ~ ( member_real(Xb,set_real2(Xs))
         => ~ member_VEBT_VEBT(Yb,set_VEBT_VEBT2(Ys2)) ) ) ).

% in_set_zipE
tff(fact_1650_in__set__zipE,axiom,
    ! [Xb: complex,Yb: int,Xs: list_complex,Ys2: list_int] :
      ( member595073364599660772ex_int(produc1367138851071493491ex_int(Xb,Yb),set_Pr4995810437751016784ex_int(zip_complex_int(Xs,Ys2)))
     => ~ ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
         => ~ aa_set_int_o(member_int(Yb),set_int2(Ys2)) ) ) ).

% in_set_zipE
tff(fact_1651_in__set__zipE,axiom,
    ! [Xb: real,Yb: int,Xs: list_real,Ys2: list_int] :
      ( member1627681773268152802al_int(produc3179012173361985393al_int(Xb,Yb),set_Pr8219819362198175822al_int(zip_real_int(Xs,Ys2)))
     => ~ ( member_real(Xb,set_real2(Xs))
         => ~ aa_set_int_o(member_int(Yb),set_int2(Ys2)) ) ) ).

% in_set_zipE
tff(fact_1652_in__set__zipE,axiom,
    ! [Xb: complex,Yb: nat,Xs: list_complex,Ys2: list_nat] :
      ( member4772924384108857480ex_nat(produc1369629321580543767ex_nat(Xb,Yb),set_Pr9173661457260213492ex_nat(zip_complex_nat(Xs,Ys2)))
     => ~ ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
         => ~ aa_set_nat_o(member_nat(Yb),set_nat2(Ys2)) ) ) ).

% in_set_zipE
tff(fact_1653_in__set__zipE,axiom,
    ! [Xb: real,Yb: nat,Xs: list_real,Ys2: list_nat] :
      ( member5805532792777349510al_nat(produc3181502643871035669al_nat(Xb,Yb),set_Pr3174298344852596722al_nat(zip_real_nat(Xs,Ys2)))
     => ~ ( member_real(Xb,set_real2(Xs))
         => ~ aa_set_nat_o(member_nat(Yb),set_nat2(Ys2)) ) ) ).

% in_set_zipE
tff(fact_1654_zip__same,axiom,
    ! [A: complex,B: complex,Xs: list_complex] :
      ( member5793383173714906214omplex(produc101793102246108661omplex(A,B),set_Pr8199049879907524818omplex(zip_complex_complex(Xs,Xs)))
    <=> ( aa_set_complex_o(member_complex(A),set_complex2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1655_zip__same,axiom,
    ! [A: real,B: real,Xs: list_real] :
      ( member7849222048561428706l_real(produc4511245868158468465l_real(A,B),set_Pr5999470521830281550l_real(zip_real_real(Xs,Xs)))
    <=> ( member_real(A,set_real2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1656_zip__same,axiom,
    ! [A: set_nat,B: set_nat,Xs: list_set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(A,B),set_Pr9040384385603167362et_nat(zip_set_nat_set_nat(Xs,Xs)))
    <=> ( member_set_nat(A,set_set_nat2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1657_zip__same,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(A,B),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Xs)))
    <=> ( member_VEBT_VEBT(A,set_VEBT_VEBT2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1658_zip__same,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Xs)))
    <=> ( aa_set_nat_o(member_nat(A),set_nat2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1659_zip__same,axiom,
    ! [A: int,B: int,Xs: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Xs)))
    <=> ( aa_set_int_o(member_int(A),set_int2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1660_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_complex,Yb: complex] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = size_s3451745648224563538omplex(Ys2) )
     => ( aa_set_complex_o(member_complex(Yb),set_complex2(Ys2))
       => ~ ! [X3: vEBT_VEBT] : ~ member3207599676835851048omplex(produc5617778602380981643omplex(X3,Yb),set_Pr6387300694196750780omplex(zip_VE2794733401258833515omplex(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1661_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_real,Yb: real] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = size_size_list_real(Ys2) )
     => ( member_real(Yb,set_real2(Ys2))
       => ~ ! [X3: vEBT_VEBT] : ~ member8675245146396747942T_real(produc8117437818029410057T_real(X3,Yb),set_Pr1087130671499945274T_real(zip_VEBT_VEBT_real(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1662_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,Yb: vEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2) )
     => ( member_VEBT_VEBT(Yb,set_VEBT_VEBT2(Ys2))
       => ~ ! [X3: vEBT_VEBT] : ~ member568628332442017744T_VEBT(produc537772716801021591T_VEBT(X3,Yb),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1663_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_int,Yb: int] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_list_int_nat(size_size_list_int,Ys2) )
     => ( aa_set_int_o(member_int(Yb),set_int2(Ys2))
       => ~ ! [X3: vEBT_VEBT] : ~ member5419026705395827622BT_int(produc736041933913180425BT_int(X3,Yb),set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1664_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_nat,Yb: nat] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_list_nat_nat(size_size_list_nat,Ys2) )
     => ( aa_set_nat_o(member_nat(Yb),set_nat2(Ys2))
       => ~ ! [X3: vEBT_VEBT] : ~ member373505688050248522BT_nat(produc738532404422230701BT_nat(X3,Yb),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1665_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys2: list_complex,Yb: complex] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = size_s3451745648224563538omplex(Ys2) )
     => ( aa_set_complex_o(member_complex(Yb),set_complex2(Ys2))
       => ~ ! [X3: int] : ~ member8811922270175639012omplex(produc7948753499206759283omplex(X3,Yb),set_Pr3989287306472219216omplex(zip_int_complex(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1666_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys2: list_real,Yb: real] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = size_size_list_real(Ys2) )
     => ( member_real(Yb,set_real2(Ys2))
       => ~ ! [X3: int] : ~ member2744130022092475746t_real(produc801115645435158769t_real(X3,Yb),set_Pr112895574167722958t_real(zip_int_real(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1667_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys2: list_VEBT_VEBT,Yb: vEBT_VEBT] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2) )
     => ( member_VEBT_VEBT(Yb,set_VEBT_VEBT2(Ys2))
       => ~ ! [X3: int] : ~ member2056185340421749780T_VEBT(produc3329399203697025711T_VEBT(X3,Yb),set_Pr8714266321650254504T_VEBT(zip_int_VEBT_VEBT(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1668_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys2: list_int,Yb: int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys2) )
     => ( aa_set_int_o(member_int(Yb),set_int2(Ys2))
       => ~ ! [X3: int] : ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Yb),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1669_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys2: list_nat,Yb: nat] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_nat_nat(size_size_list_nat,Ys2) )
     => ( aa_set_nat_o(member_nat(Yb),set_nat2(Ys2))
       => ~ ! [X3: int] : ~ member216504246829706758nt_nat(product_Pair_int_nat(X3,Yb),set_Pr6647972299459129970nt_nat(zip_int_nat(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1670_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_complex,Ys2: list_VEBT_VEBT,Xb: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2) )
     => ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
       => ~ ! [Y: vEBT_VEBT] : ~ member1978952105866562066T_VEBT(produc2757191886755552429T_VEBT(Xb,Y),set_Pr5158653123227461798T_VEBT(zip_co9157518722488180109T_VEBT(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1671_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys2: list_VEBT_VEBT,Xb: real] :
      ( ( size_size_list_real(Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2) )
     => ( member_real(Xb,set_real2(Xs))
       => ~ ! [Y: vEBT_VEBT] : ~ member7262085504369356948T_VEBT(produc6931449550656315951T_VEBT(Xb,Y),set_Pr8897343066327330088T_VEBT(zip_real_VEBT_VEBT(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1672_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_complex,Ys2: list_int,Xb: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = aa_list_int_nat(size_size_list_int,Ys2) )
     => ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
       => ~ ! [Y: int] : ~ member595073364599660772ex_int(produc1367138851071493491ex_int(Xb,Y),set_Pr4995810437751016784ex_int(zip_complex_int(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1673_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys2: list_int,Xb: real] :
      ( ( size_size_list_real(Xs) = aa_list_int_nat(size_size_list_int,Ys2) )
     => ( member_real(Xb,set_real2(Xs))
       => ~ ! [Y: int] : ~ member1627681773268152802al_int(produc3179012173361985393al_int(Xb,Y),set_Pr8219819362198175822al_int(zip_real_int(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1674_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_complex,Ys2: list_nat,Xb: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = aa_list_nat_nat(size_size_list_nat,Ys2) )
     => ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
       => ~ ! [Y: nat] : ~ member4772924384108857480ex_nat(produc1369629321580543767ex_nat(Xb,Y),set_Pr9173661457260213492ex_nat(zip_complex_nat(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1675_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys2: list_nat,Xb: real] :
      ( ( size_size_list_real(Xs) = aa_list_nat_nat(size_size_list_nat,Ys2) )
     => ( member_real(Xb,set_real2(Xs))
       => ~ ! [Y: nat] : ~ member5805532792777349510al_nat(produc3181502643871035669al_nat(Xb,Y),set_Pr3174298344852596722al_nat(zip_real_nat(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1676_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2) )
     => ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
       => ~ ! [Y: vEBT_VEBT] : ~ member568628332442017744T_VEBT(produc537772716801021591T_VEBT(Xb,Y),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1677_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_int,Xb: vEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_list_int_nat(size_size_list_int,Ys2) )
     => ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
       => ~ ! [Y: int] : ~ member5419026705395827622BT_int(produc736041933913180425BT_int(Xb,Y),set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1678_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_nat,Xb: vEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_list_nat_nat(size_size_list_nat,Ys2) )
     => ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
       => ~ ! [Y: nat] : ~ member373505688050248522BT_nat(produc738532404422230701BT_nat(Xb,Y),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1679_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_int,Ys2: list_VEBT_VEBT,Xb: int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2) )
     => ( aa_set_int_o(member_int(Xb),set_int2(Xs))
       => ~ ! [Y: vEBT_VEBT] : ~ member2056185340421749780T_VEBT(produc3329399203697025711T_VEBT(Xb,Y),set_Pr8714266321650254504T_VEBT(zip_int_VEBT_VEBT(Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1680_ssubst__Pair__rhs,axiom,
    ! [R2: int,S3: int,R: set_Pr958786334691620121nt_int,S4: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S3),R)
     => ( ( S4 = S3 )
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1681_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Co3460408610320954864e_term,S3: produc8923325533196201883nteger,R: set_Pr1200682188532975830nteger,S4: produc8923325533196201883nteger] :
      ( member6901806533512420791nteger(produc7264323995540650962nteger(R2,S3),R)
     => ( ( S4 = S3 )
       => member6901806533512420791nteger(produc7264323995540650962nteger(R2,S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1682_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr5098669352780829690e_term,S3: produc8923325533196201883nteger,R: set_Pr3486312109254157132nteger,S4: produc8923325533196201883nteger] :
      ( member8978266922497061677nteger(produc565448451787809096nteger(R2,S3),R)
     => ( ( S4 = S3 )
       => member8978266922497061677nteger(produc565448451787809096nteger(R2,S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1683_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr7763680713840462443e_term,S3: product_prod_int_int,R: set_Pr6700895561065273913nt_int,S4: product_prod_int_int] :
      ( member753276645541486746nt_int(produc7320969046608631221nt_int(R2,S3),R)
     => ( ( S4 = S3 )
       => member753276645541486746nt_int(produc7320969046608631221nt_int(R2,S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1684_ssubst__Pair__rhs,axiom,
    ! [R2: fun_in5730379987311621695e_term,S3: product_prod_int_int,R: set_Pr4166680992137330533nt_int,S4: product_prod_int_int] :
      ( member5890090881338784710nt_int(produc4729913460834754785nt_int(R2,S3),R)
     => ( ( S4 = S3 )
       => member5890090881338784710nt_int(produc4729913460834754785nt_int(R2,S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1685_find__None__iff,axiom,
    ! [P2: fun_complex_o,Xs: list_complex] :
      ( ( find_complex(P2,Xs) = none_complex )
    <=> ~ ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),set_complex2(Xs))
            & aa_complex_o(P2,X4) ) ) ).

% find_None_iff
tff(fact_1686_find__None__iff,axiom,
    ! [P2: fun_real_o,Xs: list_real] :
      ( ( find_real(P2,Xs) = none_real )
    <=> ~ ? [X4: real] :
            ( member_real(X4,set_real2(Xs))
            & aa_real_o(P2,X4) ) ) ).

% find_None_iff
tff(fact_1687_find__None__iff,axiom,
    ! [P2: fun_set_nat_o,Xs: list_set_nat] :
      ( ( find_set_nat(P2,Xs) = none_set_nat )
    <=> ~ ? [X4: set_nat] :
            ( member_set_nat(X4,set_set_nat2(Xs))
            & aa_set_nat_o(P2,X4) ) ) ).

% find_None_iff
tff(fact_1688_find__None__iff,axiom,
    ! [P2: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ( find_VEBT_VEBT(P2,Xs) = none_VEBT_VEBT )
    <=> ~ ? [X4: vEBT_VEBT] :
            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
            & aa_VEBT_VEBT_o(P2,X4) ) ) ).

% find_None_iff
tff(fact_1689_find__None__iff,axiom,
    ! [P2: fun_int_o,Xs: list_int] :
      ( ( find_int(P2,Xs) = none_int )
    <=> ~ ? [X4: int] :
            ( aa_set_int_o(member_int(X4),set_int2(Xs))
            & aa_int_o(P2,X4) ) ) ).

% find_None_iff
tff(fact_1690_find__None__iff,axiom,
    ! [P2: fun_nat_o,Xs: list_nat] :
      ( ( find_nat(P2,Xs) = none_nat )
    <=> ~ ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
            & aa_nat_o(P2,X4) ) ) ).

% find_None_iff
tff(fact_1691_find__None__iff,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,Xs: list_P6011104703257516679at_nat] :
      ( ( find_P8199882355184865565at_nat(P2,Xs) = none_P5556105721700978146at_nat )
    <=> ~ ? [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),set_Pr5648618587558075414at_nat(Xs))
            & aa_Pro6150637604445964947_nat_o(P2,X4) ) ) ).

% find_None_iff
tff(fact_1692_find__None__iff,axiom,
    ! [P2: fun_num_o,Xs: list_num] :
      ( ( find_num(P2,Xs) = none_num )
    <=> ~ ? [X4: num] :
            ( member_num(X4,set_num2(Xs))
            & aa_num_o(P2,X4) ) ) ).

% find_None_iff
tff(fact_1693_find__None__iff2,axiom,
    ! [P2: fun_complex_o,Xs: list_complex] :
      ( ( none_complex = find_complex(P2,Xs) )
    <=> ~ ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),set_complex2(Xs))
            & aa_complex_o(P2,X4) ) ) ).

% find_None_iff2
tff(fact_1694_find__None__iff2,axiom,
    ! [P2: fun_real_o,Xs: list_real] :
      ( ( none_real = find_real(P2,Xs) )
    <=> ~ ? [X4: real] :
            ( member_real(X4,set_real2(Xs))
            & aa_real_o(P2,X4) ) ) ).

% find_None_iff2
tff(fact_1695_find__None__iff2,axiom,
    ! [P2: fun_set_nat_o,Xs: list_set_nat] :
      ( ( none_set_nat = find_set_nat(P2,Xs) )
    <=> ~ ? [X4: set_nat] :
            ( member_set_nat(X4,set_set_nat2(Xs))
            & aa_set_nat_o(P2,X4) ) ) ).

% find_None_iff2
tff(fact_1696_find__None__iff2,axiom,
    ! [P2: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ( none_VEBT_VEBT = find_VEBT_VEBT(P2,Xs) )
    <=> ~ ? [X4: vEBT_VEBT] :
            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
            & aa_VEBT_VEBT_o(P2,X4) ) ) ).

% find_None_iff2
tff(fact_1697_find__None__iff2,axiom,
    ! [P2: fun_int_o,Xs: list_int] :
      ( ( none_int = find_int(P2,Xs) )
    <=> ~ ? [X4: int] :
            ( aa_set_int_o(member_int(X4),set_int2(Xs))
            & aa_int_o(P2,X4) ) ) ).

% find_None_iff2
tff(fact_1698_find__None__iff2,axiom,
    ! [P2: fun_nat_o,Xs: list_nat] :
      ( ( none_nat = find_nat(P2,Xs) )
    <=> ~ ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
            & aa_nat_o(P2,X4) ) ) ).

% find_None_iff2
tff(fact_1699_find__None__iff2,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,Xs: list_P6011104703257516679at_nat] :
      ( ( none_P5556105721700978146at_nat = find_P8199882355184865565at_nat(P2,Xs) )
    <=> ~ ? [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),set_Pr5648618587558075414at_nat(Xs))
            & aa_Pro6150637604445964947_nat_o(P2,X4) ) ) ).

% find_None_iff2
tff(fact_1700_find__None__iff2,axiom,
    ! [P2: fun_num_o,Xs: list_num] :
      ( ( none_num = find_num(P2,Xs) )
    <=> ~ ? [X4: num] :
            ( member_num(X4,set_num2(Xs))
            & aa_num_o(P2,X4) ) ) ).

% find_None_iff2
tff(fact_1701_dbl__inc__def,axiom,
    ! [Xb: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,Xb) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(Xb),Xb)),one_one_complex) ).

% dbl_inc_def
tff(fact_1702_dbl__inc__def,axiom,
    ! [Xb: real] : aa_real_real(neg_nu8295874005876285629c_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb)),one_one_real) ).

% dbl_inc_def
tff(fact_1703_dbl__inc__def,axiom,
    ! [Xb: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,Xb) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(Xb),Xb)),one_one_rat) ).

% dbl_inc_def
tff(fact_1704_dbl__inc__def,axiom,
    ! [Xb: int] : aa_int_int(neg_nu5851722552734809277nc_int,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb)),one_one_int) ).

% dbl_inc_def
tff(fact_1705__C5_Ohyps_C_I9_J,axiom,
    ( ( mi != ma )
   => ! [I4: nat] :
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I4 )
           => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na)) )
          & ! [X: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),na) = I4 )
                & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),na)) )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),X)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),ma) ) ) ) ) ) ).

% "5.hyps"(9)
tff(fact_1706_pair__lessI2,axiom,
    ! [A: nat,B: nat,S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S3),Ta)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Ta)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_1707_intind,axiom,
    ! [I: nat,Nb: nat,P2: fun_int_o,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_int_o(P2,Xb)
       => aa_int_o(P2,aa_nat_int(nth_int(replicate_int(Nb,Xb)),I)) ) ) ).

% intind
tff(fact_1708_intind,axiom,
    ! [I: nat,Nb: nat,P2: fun_nat_o,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_nat_o(P2,Xb)
       => aa_nat_o(P2,aa_nat_nat(nth_nat(replicate_nat(Nb,Xb)),I)) ) ) ).

% intind
tff(fact_1709_intind,axiom,
    ! [I: nat,Nb: nat,P2: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_VEBT_VEBT_o(P2,Xb)
       => aa_VEBT_VEBT_o(P2,nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xb),I)) ) ) ).

% intind
tff(fact_1710_pair__less__iff1,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Yb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Z)),fun_pair_less)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Z) ) ).

% pair_less_iff1
tff(fact_1711_gen__length__def,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] : aa_lis7044684121217598542BT_nat(gen_length_VEBT_VEBT(Nb),Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)) ).

% gen_length_def
tff(fact_1712_gen__length__def,axiom,
    ! [Nb: nat,Xs: list_int] : aa_list_int_nat(gen_length_int(Nb),Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_list_int_nat(size_size_list_int,Xs)) ).

% gen_length_def
tff(fact_1713_gen__length__def,axiom,
    ! [Nb: nat,Xs: list_nat] : aa_list_nat_nat(gen_length_nat(Nb),Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_list_nat_nat(size_size_list_nat,Xs)) ).

% gen_length_def
tff(fact_1714_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_1715_nth__rotate1,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( nth_VEBT_VEBT(rotate1_VEBT_VEBT(Xs),Nb) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Nb)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))) ) ) ).

% nth_rotate1
tff(fact_1716_nth__rotate1,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_list_int_nat(size_size_list_int,Xs))
     => ( aa_nat_int(nth_int(rotate1_int(Xs)),Nb) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Nb)),aa_list_int_nat(size_size_list_int,Xs))) ) ) ).

% nth_rotate1
tff(fact_1717_nth__rotate1,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_nat(nth_nat(rotate1_nat(Xs)),Nb) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Nb)),aa_list_nat_nat(size_size_list_nat,Xs))) ) ) ).

% nth_rotate1
tff(fact_1718_length__code,axiom,
    size_s6755466524823107622T_VEBT = gen_length_VEBT_VEBT(zero_zero_nat) ).

% length_code
tff(fact_1719_length__code,axiom,
    size_size_list_int = gen_length_int(zero_zero_nat) ).

% length_code
tff(fact_1720_length__code,axiom,
    size_size_list_nat = gen_length_nat(zero_zero_nat) ).

% length_code
tff(fact_1721_nth__equal__first__eq,axiom,
    ! [Xb: complex,Xs: list_complex,Nb: nat] :
      ( ~ aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),size_s3451745648224563538omplex(Xs))
       => ( ( aa_nat_complex(nth_complex(cons_complex(Xb,Xs)),Nb) = Xb )
        <=> ( Nb = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_1722_nth__equal__first__eq,axiom,
    ! [Xb: real,Xs: list_real,Nb: nat] :
      ( ~ member_real(Xb,set_real2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),size_size_list_real(Xs))
       => ( ( aa_nat_real(nth_real(cons_real(Xb,Xs)),Nb) = Xb )
        <=> ( Nb = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_1723_nth__equal__first__eq,axiom,
    ! [Xb: set_nat,Xs: list_set_nat,Nb: nat] :
      ( ~ member_set_nat(Xb,set_set_nat2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),size_s3254054031482475050et_nat(Xs))
       => ( ( nth_set_nat(cons_set_nat(Xb,Xs),Nb) = Xb )
        <=> ( Nb = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_1724_nth__equal__first__eq,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,Nb: nat] :
      ( ~ member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
       => ( ( nth_VEBT_VEBT(cons_VEBT_VEBT(Xb,Xs),Nb) = Xb )
        <=> ( Nb = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_1725_nth__equal__first__eq,axiom,
    ! [Xb: int,Xs: list_int,Nb: nat] :
      ( ~ aa_set_int_o(member_int(Xb),set_int2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_list_int_nat(size_size_list_int,Xs))
       => ( ( aa_nat_int(nth_int(cons_int(Xb,Xs)),Nb) = Xb )
        <=> ( Nb = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_1726_nth__equal__first__eq,axiom,
    ! [Xb: nat,Xs: list_nat,Nb: nat] :
      ( ~ aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( ( aa_nat_nat(nth_nat(cons_nat(Xb,Xs)),Nb) = Xb )
        <=> ( Nb = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_1727__C5_Ohyps_C_I2_J,axiom,
    aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) ).

% "5.hyps"(2)
tff(fact_1728__C5_Ohyps_C_I8_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% "5.hyps"(8)
tff(fact_1729__C5_Oprems_C_I1_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% "5.prems"(1)
tff(fact_1730_list_Oinject,axiom,
    ! [X21: int,X222: list_int,Y21: int,Y22: list_int] :
      ( ( cons_int(X21,X222) = cons_int(Y21,Y22) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
tff(fact_1731_list_Oinject,axiom,
    ! [X21: nat,X222: list_nat,Y21: nat,Y22: list_nat] :
      ( ( cons_nat(X21,X222) = cons_nat(Y21,Y22) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
tff(fact_1732__C5_Oprems_C_I2_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% "5.prems"(2)
tff(fact_1733_low__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% low_def
tff(fact_1734__C5_OIH_C_I1_J,axiom,
    ! [X: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList))
     => ( aa_nat_o(vEBT_invar_vebt(X),na)
        & ! [Xa: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))
           => ! [Xb2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))
               => ( aa_nat_o(vEBT_V8194947554948674370ptions(X),Xa)
                 => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(X,Xb2)),Xa) ) ) ) ) ) ).

% "5.IH"(1)
tff(fact_1735__C5_Ohyps_C_I5_J,axiom,
    ! [I4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
     => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),X_13)
      <=> aa_nat_o(vEBT_V8194947554948674370ptions(summary),I4) ) ) ).

% "5.hyps"(5)
tff(fact_1736_high__bound__aux,axiom,
    ! [Maa: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ).

% high_bound_aux
tff(fact_1737_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,Nb: nat] :
      ( aa_nat_o(vEBT_vebt_member(Tree),Xb)
     => ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% member_bound
tff(fact_1738_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_1739_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_1740_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_1741_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_1742_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_1743_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_1744_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_1745_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_1746_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_1747_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W2)),Z)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W2))),Z) ).

% add_numeral_left
tff(fact_1748_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,W2)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W2))),Z) ).

% add_numeral_left
tff(fact_1749_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: rat] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,W2)),Z)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W2))),Z) ).

% add_numeral_left
tff(fact_1750_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W2)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),W2))),Z) ).

% add_numeral_left
tff(fact_1751_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W2)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W2))),Z) ).

% add_numeral_left
tff(fact_1752_mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% mod_0
tff(fact_1753_mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_1754_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_1755_mod__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),zero_zero_nat) = A ).

% mod_by_0
tff(fact_1756_mod__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),zero_zero_int) = A ).

% mod_by_0
tff(fact_1757_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_1758_mod__self,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),A) = zero_zero_nat ).

% mod_self
tff(fact_1759_mod__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),A) = zero_zero_int ).

% mod_self
tff(fact_1760_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_1761_set__n__deg__not__0,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Mb: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_1762_insert__simp__mima,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        | ( Xb = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_1763_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xb)),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_1764_replicate__eq__replicate,axiom,
    ! [Mb: nat,Xb: vEBT_VEBT,Nb: nat,Yb: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(Mb,Xb) = replicate_VEBT_VEBT(Nb,Yb) )
    <=> ( ( Mb = Nb )
        & ( ( Mb != zero_zero_nat )
         => ( Xb = Yb ) ) ) ) ).

% replicate_eq_replicate
tff(fact_1765_length__replicate,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] : aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,replicate_VEBT_VEBT(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_1766_length__replicate,axiom,
    ! [Nb: nat,Xb: int] : aa_list_int_nat(size_size_list_int,replicate_int(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_1767_length__replicate,axiom,
    ! [Nb: nat,Xb: nat] : aa_list_nat_nat(size_size_list_nat,replicate_nat(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_1768_mi__ma__2__deg,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)) ) ) ).

% mi_ma_2_deg
tff(fact_1769__C5_OIH_C_I2_J,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(summary),Xb)
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(summary,Yb)),Xb) ) ) ) ).

% "5.IH"(2)
tff(fact_1770_mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_1771_mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_1772_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_1773_Suc__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Nb),one)) ).

% Suc_numeral
tff(fact_1774_nth__Cons__Suc,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,Nb: nat] : nth_VEBT_VEBT(cons_VEBT_VEBT(Xb,Xs),aa_nat_nat(suc,Nb)) = nth_VEBT_VEBT(Xs,Nb) ).

% nth_Cons_Suc
tff(fact_1775_nth__Cons__Suc,axiom,
    ! [Xb: int,Xs: list_int,Nb: nat] : aa_nat_int(nth_int(cons_int(Xb,Xs)),aa_nat_nat(suc,Nb)) = aa_nat_int(nth_int(Xs),Nb) ).

% nth_Cons_Suc
tff(fact_1776_nth__Cons__Suc,axiom,
    ! [Xb: nat,Xs: list_nat,Nb: nat] : aa_nat_nat(nth_nat(cons_nat(Xb,Xs)),aa_nat_nat(suc,Nb)) = aa_nat_nat(nth_nat(Xs),Nb) ).

% nth_Cons_Suc
tff(fact_1777_nth__Cons__0,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] : nth_VEBT_VEBT(cons_VEBT_VEBT(Xb,Xs),zero_zero_nat) = Xb ).

% nth_Cons_0
tff(fact_1778_nth__Cons__0,axiom,
    ! [Xb: int,Xs: list_int] : aa_nat_int(nth_int(cons_int(Xb,Xs)),zero_zero_nat) = Xb ).

% nth_Cons_0
tff(fact_1779_nth__Cons__0,axiom,
    ! [Xb: nat,Xs: list_nat] : aa_nat_nat(nth_nat(cons_nat(Xb,Xs)),zero_zero_nat) = Xb ).

% nth_Cons_0
tff(fact_1780_zip__Cons__Cons,axiom,
    ! [Xb: int,Xs: list_int,Yb: nat,Ys2: list_nat] : zip_int_nat(cons_int(Xb,Xs),cons_nat(Yb,Ys2)) = cons_P7512249878480867347nt_nat(product_Pair_int_nat(Xb,Yb),zip_int_nat(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_1781_zip__Cons__Cons,axiom,
    ! [Xb: nat,Xs: list_nat,Yb: int,Ys2: list_int] : zip_nat_int(cons_nat(Xb,Xs),cons_int(Yb,Ys2)) = cons_P2335045147070616083at_int(product_Pair_nat_int(Xb,Yb),zip_nat_int(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_1782_zip__Cons__Cons,axiom,
    ! [Xb: nat,Xs: list_nat,Yb: nat,Ys2: list_nat] : zip_nat_nat(cons_nat(Xb,Xs),cons_nat(Yb,Ys2)) = cons_P6512896166579812791at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Yb),zip_nat_nat(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_1783_zip__Cons__Cons,axiom,
    ! [Xb: int,Xs: list_int,Yb: int,Ys2: list_int] : zip_int_int(cons_int(Xb,Xs),cons_int(Yb,Ys2)) = cons_P3334398858971670639nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Yb),zip_int_int(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_1784_zip__Cons__Cons,axiom,
    ! [Xb: fun_Co3460408610320954864e_term,Xs: list_f8851413139013959296e_term,Yb: produc8923325533196201883nteger,Ys2: list_P5578671422887162913nteger] : zip_fu7084642285130276018nteger(cons_f4064889718614483898e_term(Xb,Xs),cons_P9044669534377732177nteger(Yb,Ys2)) = cons_P6847588673944434154nteger(produc7264323995540650962nteger(Xb,Yb),zip_fu7084642285130276018nteger(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_1785_zip__Cons__Cons,axiom,
    ! [Xb: fun_Pr5098669352780829690e_term,Xs: list_f4306079564415238026e_term,Yb: produc8923325533196201883nteger,Ys2: list_P5578671422887162913nteger] : zip_fu4451626411769283112nteger(cons_f7541796142091620804e_term(Xb,Xs),cons_P9044669534377732177nteger(Yb,Ys2)) = cons_P8226300992106120032nteger(produc565448451787809096nteger(Xb,Yb),zip_fu4451626411769283112nteger(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_1786_zip__Cons__Cons,axiom,
    ! [Xb: fun_Pr7763680713840462443e_term,Xs: list_f280343417242313083e_term,Yb: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] : zip_fu5048005673116055189nt_int(cons_f7531370473056858165e_term(Xb,Xs),cons_P3334398858971670639nt_int(Yb,Ys2)) = cons_P8208993393157215437nt_int(produc7320969046608631221nt_int(Xb,Yb),zip_fu5048005673116055189nt_int(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_1787_zip__Cons__Cons,axiom,
    ! [Xb: fun_in5730379987311621695e_term,Xs: list_f628289730585369423e_term,Yb: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] : zip_fu2910578966527623617nt_int(cons_f6442722764679334921e_term(Xb,Xs),cons_P3334398858971670639nt_int(Yb,Ys2)) = cons_P6394518999824463865nt_int(produc4729913460834754785nt_int(Xb,Yb),zip_fu2910578966527623617nt_int(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_1788__092_060open_062low_Ax_An_A_060_A2_A_094_An_A_092_060and_062_Alow_Ay_An_A_060_A2_A_094_An_092_060close_062,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ya),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)) ) ).

% \<open>low x n < 2 ^ n \<and> low y n < 2 ^ n\<close>
tff(fact_1789_Ball__set__replicate,axiom,
    ! [Nb: nat,A: int,P2: fun_int_o] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(replicate_int(Nb,A)))
         => aa_int_o(P2,X4) )
    <=> ( aa_int_o(P2,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1790_Ball__set__replicate,axiom,
    ! [Nb: nat,A: nat,P2: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(replicate_nat(Nb,A)))
         => aa_nat_o(P2,X4) )
    <=> ( aa_nat_o(P2,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1791_Ball__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P2: fun_VEBT_VEBT_o] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
         => aa_VEBT_VEBT_o(P2,X4) )
    <=> ( aa_VEBT_VEBT_o(P2,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1792_Bex__set__replicate,axiom,
    ! [Nb: nat,A: int,P2: fun_int_o] :
      ( ? [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(replicate_int(Nb,A)))
          & aa_int_o(P2,X4) )
    <=> ( aa_int_o(P2,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1793_Bex__set__replicate,axiom,
    ! [Nb: nat,A: nat,P2: fun_nat_o] :
      ( ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(replicate_nat(Nb,A)))
          & aa_nat_o(P2,X4) )
    <=> ( aa_nat_o(P2,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1794_Bex__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P2: fun_VEBT_VEBT_o] :
      ( ? [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
          & aa_VEBT_VEBT_o(P2,X4) )
    <=> ( aa_VEBT_VEBT_o(P2,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1795_in__set__replicate,axiom,
    ! [Xb: complex,Nb: nat,Yb: complex] :
      ( aa_set_complex_o(member_complex(Xb),set_complex2(replicate_complex(Nb,Yb)))
    <=> ( ( Xb = Yb )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1796_in__set__replicate,axiom,
    ! [Xb: real,Nb: nat,Yb: real] :
      ( member_real(Xb,set_real2(replicate_real(Nb,Yb)))
    <=> ( ( Xb = Yb )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1797_in__set__replicate,axiom,
    ! [Xb: set_nat,Nb: nat,Yb: set_nat] :
      ( member_set_nat(Xb,set_set_nat2(replicate_set_nat(Nb,Yb)))
    <=> ( ( Xb = Yb )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1798_in__set__replicate,axiom,
    ! [Xb: int,Nb: nat,Yb: int] :
      ( aa_set_int_o(member_int(Xb),set_int2(replicate_int(Nb,Yb)))
    <=> ( ( Xb = Yb )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1799_in__set__replicate,axiom,
    ! [Xb: nat,Nb: nat,Yb: nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(replicate_nat(Nb,Yb)))
    <=> ( ( Xb = Yb )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1800_in__set__replicate,axiom,
    ! [Xb: vEBT_VEBT,Nb: nat,Yb: vEBT_VEBT] :
      ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Yb)))
    <=> ( ( Xb = Yb )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1801_nth__replicate,axiom,
    ! [I: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_nat_int(nth_int(replicate_int(Nb,Xb)),I) = Xb ) ) ).

% nth_replicate
tff(fact_1802_nth__replicate,axiom,
    ! [I: nat,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_nat_nat(nth_nat(replicate_nat(Nb,Xb)),I) = Xb ) ) ).

% nth_replicate
tff(fact_1803_nth__replicate,axiom,
    ! [I: nat,Nb: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xb),I) = Xb ) ) ).

% nth_replicate
tff(fact_1804_xyprop,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ya),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)) ) ).

% xyprop
tff(fact_1805_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_1806_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_1807_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_1808_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_1809_one__plus__numeral,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_1810_one__plus__numeral,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_1811_one__plus__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_1812_one__plus__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_1813_one__plus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_1814_numeral__plus__one,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_1815_numeral__plus__one,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_1816_numeral__plus__one,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_1817_numeral__plus__one,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_1818_numeral__plus__one,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_1819_enumerate__simps_I2_J,axiom,
    ! [Nb: nat,Xb: int,Xs: list_int] : enumerate_int(Nb,cons_int(Xb,Xs)) = cons_P2335045147070616083at_int(product_Pair_nat_int(Nb,Xb),enumerate_int(aa_nat_nat(suc,Nb),Xs)) ).

% enumerate_simps(2)
tff(fact_1820_enumerate__simps_I2_J,axiom,
    ! [Nb: nat,Xb: nat,Xs: list_nat] : enumerate_nat(Nb,cons_nat(Xb,Xs)) = cons_P6512896166579812791at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Nb),Xb),enumerate_nat(aa_nat_nat(suc,Nb),Xs)) ).

% enumerate_simps(2)
tff(fact_1821_one__add__one,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1822_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1823_one__add__one,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1824_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1825_one__add__one,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1826_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc'
tff(fact_1827_add__2__eq__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc
tff(fact_1828_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_1829_numeral__Bit0,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb)) ).

% numeral_Bit0
tff(fact_1830_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_Bit0
tff(fact_1831_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_Bit0
tff(fact_1832_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_Bit0
tff(fact_1833_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_Bit0
tff(fact_1834_replicate__Suc,axiom,
    ! [Nb: nat,Xb: int] : replicate_int(aa_nat_nat(suc,Nb),Xb) = cons_int(Xb,replicate_int(Nb,Xb)) ).

% replicate_Suc
tff(fact_1835_replicate__Suc,axiom,
    ! [Nb: nat,Xb: nat] : replicate_nat(aa_nat_nat(suc,Nb),Xb) = cons_nat(Xb,replicate_nat(Nb,Xb)) ).

% replicate_Suc
tff(fact_1836_replicate__Suc,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] : replicate_VEBT_VEBT(aa_nat_nat(suc,Nb),Xb) = cons_VEBT_VEBT(Xb,replicate_VEBT_VEBT(Nb,Xb)) ).

% replicate_Suc
tff(fact_1837_not__Cons__self2,axiom,
    ! [Xb: int,Xs: list_int] : cons_int(Xb,Xs) != Xs ).

% not_Cons_self2
tff(fact_1838_not__Cons__self2,axiom,
    ! [Xb: nat,Xs: list_nat] : cons_nat(Xb,Xs) != Xs ).

% not_Cons_self2
tff(fact_1839_add__One__commute,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),Nb) = aa_num_num(plus_plus_num(Nb),one) ).

% add_One_commute
tff(fact_1840_le__num__One__iff,axiom,
    ! [Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),one)
    <=> ( Xb = one ) ) ).

% le_num_One_iff
tff(fact_1841_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_1842_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_1843_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_1844_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1845_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q3: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1846_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q3: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1847_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_1848_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% nat_1_add_1
tff(fact_1849_gen__length__code_I2_J,axiom,
    ! [Nb: nat,Xb: int,Xs: list_int] : aa_list_int_nat(gen_length_int(Nb),cons_int(Xb,Xs)) = aa_list_int_nat(gen_length_int(aa_nat_nat(suc,Nb)),Xs) ).

% gen_length_code(2)
tff(fact_1850_gen__length__code_I2_J,axiom,
    ! [Nb: nat,Xb: nat,Xs: list_nat] : aa_list_nat_nat(gen_length_nat(Nb),cons_nat(Xb,Xs)) = aa_list_nat_nat(gen_length_nat(aa_nat_nat(suc,Nb)),Xs) ).

% gen_length_code(2)
tff(fact_1851_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_1852_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_1853_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_1854_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_1855_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys2: list_nat,Xy: product_prod_int_nat,Xys: list_P8198026277950538467nt_nat] :
      ( ( zip_int_nat(Xs,Ys2) = cons_P7512249878480867347nt_nat(Xy,Xys) )
     => ~ ! [X3: int,Xs4: list_int] :
            ( ( Xs = cons_int(X3,Xs4) )
           => ! [Y: nat,Ys3: list_nat] :
                ( ( Ys2 = cons_nat(Y,Ys3) )
               => ( ( Xy = product_Pair_int_nat(X3,Y) )
                 => ( Xys != zip_int_nat(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1856_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys2: list_int,Xy: product_prod_nat_int,Xys: list_P3521021558325789923at_int] :
      ( ( zip_nat_int(Xs,Ys2) = cons_P2335045147070616083at_int(Xy,Xys) )
     => ~ ! [X3: nat,Xs4: list_nat] :
            ( ( Xs = cons_nat(X3,Xs4) )
           => ! [Y: int,Ys3: list_int] :
                ( ( Ys2 = cons_int(Y,Ys3) )
               => ( ( Xy = product_Pair_nat_int(X3,Y) )
                 => ( Xys != zip_nat_int(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1857_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( zip_nat_nat(Xs,Ys2) = cons_P6512896166579812791at_nat(Xy,Xys) )
     => ~ ! [X3: nat,Xs4: list_nat] :
            ( ( Xs = cons_nat(X3,Xs4) )
           => ! [Y: nat,Ys3: list_nat] :
                ( ( Ys2 = cons_nat(Y,Ys3) )
               => ( ( Xy = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y) )
                 => ( Xys != zip_nat_nat(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1858_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys2: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
      ( ( zip_int_int(Xs,Ys2) = cons_P3334398858971670639nt_int(Xy,Xys) )
     => ~ ! [X3: int,Xs4: list_int] :
            ( ( Xs = cons_int(X3,Xs4) )
           => ! [Y: int,Ys3: list_int] :
                ( ( Ys2 = cons_int(Y,Ys3) )
               => ( ( Xy = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) )
                 => ( Xys != zip_int_int(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1859_zip__eq__ConsE,axiom,
    ! [Xs: list_f8851413139013959296e_term,Ys2: list_P5578671422887162913nteger,Xy: produc7509114972854644512nteger,Xys: list_P1054633106646461488nteger] :
      ( ( zip_fu7084642285130276018nteger(Xs,Ys2) = cons_P6847588673944434154nteger(Xy,Xys) )
     => ~ ! [X3: fun_Co3460408610320954864e_term,Xs4: list_f8851413139013959296e_term] :
            ( ( Xs = cons_f4064889718614483898e_term(X3,Xs4) )
           => ! [Y: produc8923325533196201883nteger,Ys3: list_P5578671422887162913nteger] :
                ( ( Ys2 = cons_P9044669534377732177nteger(Y,Ys3) )
               => ( ( Xy = produc7264323995540650962nteger(X3,Y) )
                 => ( Xys != zip_fu7084642285130276018nteger(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1860_zip__eq__ConsE,axiom,
    ! [Xs: list_f4306079564415238026e_term,Ys2: list_P5578671422887162913nteger,Xy: produc2453742332425399446nteger,Xys: list_P2959130015091922598nteger] :
      ( ( zip_fu4451626411769283112nteger(Xs,Ys2) = cons_P8226300992106120032nteger(Xy,Xys) )
     => ~ ! [X3: fun_Pr5098669352780829690e_term,Xs4: list_f4306079564415238026e_term] :
            ( ( Xs = cons_f7541796142091620804e_term(X3,Xs4) )
           => ! [Y: produc8923325533196201883nteger,Ys3: list_P5578671422887162913nteger] :
                ( ( Ys2 = cons_P9044669534377732177nteger(Y,Ys3) )
               => ( ( Xy = produc565448451787809096nteger(X3,Y) )
                 => ( Xys != zip_fu4451626411769283112nteger(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1861_zip__eq__ConsE,axiom,
    ! [Xs: list_f280343417242313083e_term,Ys2: list_P5707943133018811711nt_int,Xy: produc3043004577601930755nt_int,Xys: list_P2843741347166045587nt_int] :
      ( ( zip_fu5048005673116055189nt_int(Xs,Ys2) = cons_P8208993393157215437nt_int(Xy,Xys) )
     => ~ ! [X3: fun_Pr7763680713840462443e_term,Xs4: list_f280343417242313083e_term] :
            ( ( Xs = cons_f7531370473056858165e_term(X3,Xs4) )
           => ! [Y: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
                ( ( Ys2 = cons_P3334398858971670639nt_int(Y,Ys3) )
               => ( ( Xy = produc7320969046608631221nt_int(X3,Y) )
                 => ( Xys != zip_fu5048005673116055189nt_int(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1862_zip__eq__ConsE,axiom,
    ! [Xs: list_f628289730585369423e_term,Ys2: list_P5707943133018811711nt_int,Xy: produc4788131091073775919nt_int,Xys: list_P932735248015874751nt_int] :
      ( ( zip_fu2910578966527623617nt_int(Xs,Ys2) = cons_P6394518999824463865nt_int(Xy,Xys) )
     => ~ ! [X3: fun_in5730379987311621695e_term,Xs4: list_f628289730585369423e_term] :
            ( ( Xs = cons_f6442722764679334921e_term(X3,Xs4) )
           => ! [Y: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
                ( ( Ys2 = cons_P3334398858971670639nt_int(Y,Ys3) )
               => ( ( Xy = produc4729913460834754785nt_int(X3,Y) )
                 => ( Xys != zip_fu2910578966527623617nt_int(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1863_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_1864_Suc__nat__number__of__add,axiom,
    ! [V: num,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),one))),Nb) ).

% Suc_nat_number_of_add
tff(fact_1865_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != numera6690914467698888265omplex(Nb) ).

% zero_neq_numeral
tff(fact_1866_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_num_real(numeral_numeral_real,Nb) ).

% zero_neq_numeral
tff(fact_1867_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,Nb) ).

% zero_neq_numeral
tff(fact_1868_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Nb) ).

% zero_neq_numeral
tff(fact_1869_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Nb) ).

% zero_neq_numeral
tff(fact_1870_list_Oset__intros_I2_J,axiom,
    ! [Yb: complex,X222: list_complex,X21: complex] :
      ( aa_set_complex_o(member_complex(Yb),set_complex2(X222))
     => aa_set_complex_o(member_complex(Yb),set_complex2(cons_complex(X21,X222))) ) ).

% list.set_intros(2)
tff(fact_1871_list_Oset__intros_I2_J,axiom,
    ! [Yb: real,X222: list_real,X21: real] :
      ( member_real(Yb,set_real2(X222))
     => member_real(Yb,set_real2(cons_real(X21,X222))) ) ).

% list.set_intros(2)
tff(fact_1872_list_Oset__intros_I2_J,axiom,
    ! [Yb: set_nat,X222: list_set_nat,X21: set_nat] :
      ( member_set_nat(Yb,set_set_nat2(X222))
     => member_set_nat(Yb,set_set_nat2(cons_set_nat(X21,X222))) ) ).

% list.set_intros(2)
tff(fact_1873_list_Oset__intros_I2_J,axiom,
    ! [Yb: vEBT_VEBT,X222: list_VEBT_VEBT,X21: vEBT_VEBT] :
      ( member_VEBT_VEBT(Yb,set_VEBT_VEBT2(X222))
     => member_VEBT_VEBT(Yb,set_VEBT_VEBT2(cons_VEBT_VEBT(X21,X222))) ) ).

% list.set_intros(2)
tff(fact_1874_list_Oset__intros_I2_J,axiom,
    ! [Yb: int,X222: list_int,X21: int] :
      ( aa_set_int_o(member_int(Yb),set_int2(X222))
     => aa_set_int_o(member_int(Yb),set_int2(cons_int(X21,X222))) ) ).

% list.set_intros(2)
tff(fact_1875_list_Oset__intros_I2_J,axiom,
    ! [Yb: nat,X222: list_nat,X21: nat] :
      ( aa_set_nat_o(member_nat(Yb),set_nat2(X222))
     => aa_set_nat_o(member_nat(Yb),set_nat2(cons_nat(X21,X222))) ) ).

% list.set_intros(2)
tff(fact_1876_list_Oset__intros_I1_J,axiom,
    ! [X21: complex,X222: list_complex] : aa_set_complex_o(member_complex(X21),set_complex2(cons_complex(X21,X222))) ).

% list.set_intros(1)
tff(fact_1877_list_Oset__intros_I1_J,axiom,
    ! [X21: real,X222: list_real] : member_real(X21,set_real2(cons_real(X21,X222))) ).

% list.set_intros(1)
tff(fact_1878_list_Oset__intros_I1_J,axiom,
    ! [X21: set_nat,X222: list_set_nat] : member_set_nat(X21,set_set_nat2(cons_set_nat(X21,X222))) ).

% list.set_intros(1)
tff(fact_1879_list_Oset__intros_I1_J,axiom,
    ! [X21: vEBT_VEBT,X222: list_VEBT_VEBT] : member_VEBT_VEBT(X21,set_VEBT_VEBT2(cons_VEBT_VEBT(X21,X222))) ).

% list.set_intros(1)
tff(fact_1880_list_Oset__intros_I1_J,axiom,
    ! [X21: int,X222: list_int] : aa_set_int_o(member_int(X21),set_int2(cons_int(X21,X222))) ).

% list.set_intros(1)
tff(fact_1881_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X222: list_nat] : aa_set_nat_o(member_nat(X21),set_nat2(cons_nat(X21,X222))) ).

% list.set_intros(1)
tff(fact_1882_list_Oset__cases,axiom,
    ! [E2: complex,A: list_complex] :
      ( aa_set_complex_o(member_complex(E2),set_complex2(A))
     => ( ! [Z22: list_complex] : A != cons_complex(E2,Z22)
       => ~ ! [Z1: complex,Z22: list_complex] :
              ( ( A = cons_complex(Z1,Z22) )
             => ~ aa_set_complex_o(member_complex(E2),set_complex2(Z22)) ) ) ) ).

% list.set_cases
tff(fact_1883_list_Oset__cases,axiom,
    ! [E2: real,A: list_real] :
      ( member_real(E2,set_real2(A))
     => ( ! [Z22: list_real] : A != cons_real(E2,Z22)
       => ~ ! [Z1: real,Z22: list_real] :
              ( ( A = cons_real(Z1,Z22) )
             => ~ member_real(E2,set_real2(Z22)) ) ) ) ).

% list.set_cases
tff(fact_1884_list_Oset__cases,axiom,
    ! [E2: set_nat,A: list_set_nat] :
      ( member_set_nat(E2,set_set_nat2(A))
     => ( ! [Z22: list_set_nat] : A != cons_set_nat(E2,Z22)
       => ~ ! [Z1: set_nat,Z22: list_set_nat] :
              ( ( A = cons_set_nat(Z1,Z22) )
             => ~ member_set_nat(E2,set_set_nat2(Z22)) ) ) ) ).

% list.set_cases
tff(fact_1885_list_Oset__cases,axiom,
    ! [E2: vEBT_VEBT,A: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(E2,set_VEBT_VEBT2(A))
     => ( ! [Z22: list_VEBT_VEBT] : A != cons_VEBT_VEBT(E2,Z22)
       => ~ ! [Z1: vEBT_VEBT,Z22: list_VEBT_VEBT] :
              ( ( A = cons_VEBT_VEBT(Z1,Z22) )
             => ~ member_VEBT_VEBT(E2,set_VEBT_VEBT2(Z22)) ) ) ) ).

% list.set_cases
tff(fact_1886_list_Oset__cases,axiom,
    ! [E2: int,A: list_int] :
      ( aa_set_int_o(member_int(E2),set_int2(A))
     => ( ! [Z22: list_int] : A != cons_int(E2,Z22)
       => ~ ! [Z1: int,Z22: list_int] :
              ( ( A = cons_int(Z1,Z22) )
             => ~ aa_set_int_o(member_int(E2),set_int2(Z22)) ) ) ) ).

% list.set_cases
tff(fact_1887_list_Oset__cases,axiom,
    ! [E2: nat,A: list_nat] :
      ( aa_set_nat_o(member_nat(E2),set_nat2(A))
     => ( ! [Z22: list_nat] : A != cons_nat(E2,Z22)
       => ~ ! [Z1: nat,Z22: list_nat] :
              ( ( A = cons_nat(Z1,Z22) )
             => ~ aa_set_nat_o(member_nat(E2),set_nat2(Z22)) ) ) ) ).

% list.set_cases
tff(fact_1888_set__ConsD,axiom,
    ! [Yb: complex,Xb: complex,Xs: list_complex] :
      ( aa_set_complex_o(member_complex(Yb),set_complex2(cons_complex(Xb,Xs)))
     => ( ( Yb = Xb )
        | aa_set_complex_o(member_complex(Yb),set_complex2(Xs)) ) ) ).

% set_ConsD
tff(fact_1889_set__ConsD,axiom,
    ! [Yb: real,Xb: real,Xs: list_real] :
      ( member_real(Yb,set_real2(cons_real(Xb,Xs)))
     => ( ( Yb = Xb )
        | member_real(Yb,set_real2(Xs)) ) ) ).

% set_ConsD
tff(fact_1890_set__ConsD,axiom,
    ! [Yb: set_nat,Xb: set_nat,Xs: list_set_nat] :
      ( member_set_nat(Yb,set_set_nat2(cons_set_nat(Xb,Xs)))
     => ( ( Yb = Xb )
        | member_set_nat(Yb,set_set_nat2(Xs)) ) ) ).

% set_ConsD
tff(fact_1891_set__ConsD,axiom,
    ! [Yb: vEBT_VEBT,Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Yb,set_VEBT_VEBT2(cons_VEBT_VEBT(Xb,Xs)))
     => ( ( Yb = Xb )
        | member_VEBT_VEBT(Yb,set_VEBT_VEBT2(Xs)) ) ) ).

% set_ConsD
tff(fact_1892_set__ConsD,axiom,
    ! [Yb: int,Xb: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Yb),set_int2(cons_int(Xb,Xs)))
     => ( ( Yb = Xb )
        | aa_set_int_o(member_int(Yb),set_int2(Xs)) ) ) ).

% set_ConsD
tff(fact_1893_set__ConsD,axiom,
    ! [Yb: nat,Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Yb),set_nat2(cons_nat(Xb,Xs)))
     => ( ( Yb = Xb )
        | aa_set_nat_o(member_nat(Yb),set_nat2(Xs)) ) ) ).

% set_ConsD
tff(fact_1894_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_1895_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_12)
               => ( ! [X3: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_1896_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_12)
               => ( ! [X3: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_1897_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,modulo364778990260209775nteger(A,B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1898_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1899_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1900_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1901_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1902_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1903_gcd__nat__induct,axiom,
    ! [P2: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
      ( ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,M),zero_zero_nat)
     => ( ! [M: nat,N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( aa_nat_o(aa_nat_fun_nat_o(P2,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2))
             => aa_nat_o(aa_nat_fun_nat_o(P2,M),N2) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P2,Mb),Nb) ) ) ).

% gcd_nat_induct
tff(fact_1904_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_1905_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_1906_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_1907_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_1908_zero__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_le_numeral
tff(fact_1909_zero__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_le_numeral
tff(fact_1910_zero__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_le_numeral
tff(fact_1911_zero__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_le_numeral
tff(fact_1912_zero__less__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_less_numeral
tff(fact_1913_zero__less__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_less_numeral
tff(fact_1914_zero__less__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_less_numeral
tff(fact_1915_zero__less__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_less_numeral
tff(fact_1916_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_1917_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_1918_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_1919_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_1920_one__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,Nb)) ).

% one_le_numeral
tff(fact_1921_one__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_le_numeral
tff(fact_1922_one__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% one_le_numeral
tff(fact_1923_one__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) ).

% one_le_numeral
tff(fact_1924_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Xb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xb)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_1925_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Xb)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_1926_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Xb)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_1927_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_1928_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Xb)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_1929_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => ( X3 = Xb ) )
     => ( replicate_VEBT_VEBT(aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_1930_replicate__length__same,axiom,
    ! [Xs: list_int,Xb: int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),set_int2(Xs))
         => ( X3 = Xb ) )
     => ( replicate_int(aa_list_int_nat(size_size_list_int,Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_1931_replicate__length__same,axiom,
    ! [Xs: list_nat,Xb: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => ( X3 = Xb ) )
     => ( replicate_nat(aa_list_nat_nat(size_size_list_nat,Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_1932_replicate__eqI,axiom,
    ! [Xs: list_complex,Nb: nat,Xb: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = Nb )
     => ( ! [Y: complex] :
            ( aa_set_complex_o(member_complex(Y),set_complex2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_complex(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_1933_replicate__eqI,axiom,
    ! [Xs: list_real,Nb: nat,Xb: real] :
      ( ( size_size_list_real(Xs) = Nb )
     => ( ! [Y: real] :
            ( member_real(Y,set_real2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_real(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_1934_replicate__eqI,axiom,
    ! [Xs: list_set_nat,Nb: nat,Xb: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = Nb )
     => ( ! [Y: set_nat] :
            ( member_set_nat(Y,set_set_nat2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_set_nat(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_1935_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,Nb: nat,Xb: vEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = Nb )
     => ( ! [Y: vEBT_VEBT] :
            ( member_VEBT_VEBT(Y,set_VEBT_VEBT2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_VEBT_VEBT(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_1936_replicate__eqI,axiom,
    ! [Xs: list_int,Nb: nat,Xb: int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = Nb )
     => ( ! [Y: int] :
            ( aa_set_int_o(member_int(Y),set_int2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_int(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_1937_replicate__eqI,axiom,
    ! [Xs: list_nat,Nb: nat,Xb: nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = Nb )
     => ( ! [Y: nat] :
            ( aa_set_nat_o(member_nat(Y),set_nat2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_nat(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_1938_length__Suc__conv,axiom,
    ! [Xs: list_VEBT_VEBT,Nb: nat] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_nat_nat(suc,Nb) )
    <=> ? [Y4: vEBT_VEBT,Ys4: list_VEBT_VEBT] :
          ( ( Xs = cons_VEBT_VEBT(Y4,Ys4) )
          & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys4) = Nb ) ) ) ).

% length_Suc_conv
tff(fact_1939_length__Suc__conv,axiom,
    ! [Xs: list_int,Nb: nat] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_nat_nat(suc,Nb) )
    <=> ? [Y4: int,Ys4: list_int] :
          ( ( Xs = cons_int(Y4,Ys4) )
          & ( aa_list_int_nat(size_size_list_int,Ys4) = Nb ) ) ) ).

% length_Suc_conv
tff(fact_1940_length__Suc__conv,axiom,
    ! [Xs: list_nat,Nb: nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_nat_nat(suc,Nb) )
    <=> ? [Y4: nat,Ys4: list_nat] :
          ( ( Xs = cons_nat(Y4,Ys4) )
          & ( aa_list_nat_nat(size_size_list_nat,Ys4) = Nb ) ) ) ).

% length_Suc_conv
tff(fact_1941_Suc__length__conv,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( ( aa_nat_nat(suc,Nb) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) )
    <=> ? [Y4: vEBT_VEBT,Ys4: list_VEBT_VEBT] :
          ( ( Xs = cons_VEBT_VEBT(Y4,Ys4) )
          & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys4) = Nb ) ) ) ).

% Suc_length_conv
tff(fact_1942_Suc__length__conv,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( ( aa_nat_nat(suc,Nb) = aa_list_int_nat(size_size_list_int,Xs) )
    <=> ? [Y4: int,Ys4: list_int] :
          ( ( Xs = cons_int(Y4,Ys4) )
          & ( aa_list_int_nat(size_size_list_int,Ys4) = Nb ) ) ) ).

% Suc_length_conv
tff(fact_1943_Suc__length__conv,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( ( aa_nat_nat(suc,Nb) = aa_list_nat_nat(size_size_list_nat,Xs) )
    <=> ? [Y4: nat,Ys4: list_nat] :
          ( ( Xs = cons_nat(Y4,Ys4) )
          & ( aa_list_nat_nat(size_size_list_nat,Ys4) = Nb ) ) ) ).

% Suc_length_conv
tff(fact_1944_set__subset__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),set_VEBT_VEBT2(cons_VEBT_VEBT(Xb,Xs))) ).

% set_subset_Cons
tff(fact_1945_set__subset__Cons,axiom,
    ! [Xs: list_nat,Xb: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Xs)),set_nat2(cons_nat(Xb,Xs))) ).

% set_subset_Cons
tff(fact_1946_set__subset__Cons,axiom,
    ! [Xs: list_int,Xb: int] : aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),set_int2(cons_int(Xb,Xs))) ).

% set_subset_Cons
tff(fact_1947_impossible__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2))
     => ( Xs != cons_VEBT_VEBT(Xb,Ys2) ) ) ).

% impossible_Cons
tff(fact_1948_impossible__Cons,axiom,
    ! [Xs: list_int,Ys2: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),aa_list_int_nat(size_size_list_int,Ys2))
     => ( Xs != cons_int(Xb,Ys2) ) ) ).

% impossible_Cons
tff(fact_1949_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_nat_nat(size_size_list_nat,Ys2))
     => ( Xs != cons_nat(Xb,Ys2) ) ) ).

% impossible_Cons
tff(fact_1950_set__encode__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_nat(nat_set_encode,A4) = aa_set_nat_nat(nat_set_encode,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% set_encode_eq
tff(fact_1951_find_Osimps_I2_J,axiom,
    ! [P2: fun_int_o,Xb: int,Xs: list_int] :
      find_int(P2,cons_int(Xb,Xs)) = $ite(aa_int_o(P2,Xb),some_int(Xb),find_int(P2,Xs)) ).

% find.simps(2)
tff(fact_1952_find_Osimps_I2_J,axiom,
    ! [P2: fun_nat_o,Xb: nat,Xs: list_nat] :
      find_nat(P2,cons_nat(Xb,Xs)) = $ite(aa_nat_o(P2,Xb),some_nat(Xb),find_nat(P2,Xs)) ).

% find.simps(2)
tff(fact_1953_find_Osimps_I2_J,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,Xb: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      find_P8199882355184865565at_nat(P2,cons_P6512896166579812791at_nat(Xb,Xs)) = $ite(aa_Pro6150637604445964947_nat_o(P2,Xb),some_P7363390416028606310at_nat(Xb),find_P8199882355184865565at_nat(P2,Xs)) ).

% find.simps(2)
tff(fact_1954_find_Osimps_I2_J,axiom,
    ! [P2: fun_num_o,Xb: num,Xs: list_num] :
      find_num(P2,cons_num(Xb,Xs)) = $ite(aa_num_o(P2,Xb),some_num(Xb),find_num(P2,Xs)) ).

% find.simps(2)
tff(fact_1955_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A,B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1956_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1957_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1958_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( modulo364778990260209775nteger(A,B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1959_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1960_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1961_Suc__le__length__iff,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
    <=> ? [X4: vEBT_VEBT,Ys4: list_VEBT_VEBT] :
          ( ( Xs = cons_VEBT_VEBT(X4,Ys4) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys4)) ) ) ).

% Suc_le_length_iff
tff(fact_1962_Suc__le__length__iff,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_list_int_nat(size_size_list_int,Xs))
    <=> ? [X4: int,Ys4: list_int] :
          ( ( Xs = cons_int(X4,Ys4) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_list_int_nat(size_size_list_int,Ys4)) ) ) ).

% Suc_le_length_iff
tff(fact_1963_Suc__le__length__iff,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_list_nat_nat(size_size_list_nat,Xs))
    <=> ? [X4: nat,Ys4: list_nat] :
          ( ( Xs = cons_nat(X4,Ys4) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_list_nat_nat(size_size_list_nat,Ys4)) ) ) ).

% Suc_le_length_iff
tff(fact_1964_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                   => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),X_13)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mia = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I2 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),Nb) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_1965_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_1966_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                   => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),X_13)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mia = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I2 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),Nb) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_1967_count__list_Osimps_I2_J,axiom,
    ! [Xb: int,Xs: list_int,Yb: int] :
      aa_int_nat(count_list_int(cons_int(Xb,Xs)),Yb) = $ite(Xb = Yb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(count_list_int(Xs),Yb)),one_one_nat),aa_int_nat(count_list_int(Xs),Yb)) ).

% count_list.simps(2)
tff(fact_1968_count__list_Osimps_I2_J,axiom,
    ! [Xb: nat,Xs: list_nat,Yb: nat] :
      aa_nat_nat(count_list_nat(cons_nat(Xb,Xs)),Yb) = $ite(Xb = Yb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(count_list_nat(Xs),Yb)),one_one_nat),aa_nat_nat(count_list_nat(Xs),Yb)) ).

% count_list.simps(2)
tff(fact_1969_set__encode__inf,axiom,
    ! [A4: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(nat_set_encode,A4) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_1970_pair__lessI1,axiom,
    ! [A: nat,B: nat,S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Ta)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_1971_list_Osize_I4_J,axiom,
    ! [X21: vEBT_VEBT,X222: list_VEBT_VEBT] : aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,cons_VEBT_VEBT(X21,X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_1972_list_Osize_I4_J,axiom,
    ! [X21: int,X222: list_int] : aa_list_int_nat(size_size_list_int,cons_int(X21,X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_int_nat(size_size_list_int,X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_1973_list_Osize_I4_J,axiom,
    ! [X21: nat,X222: list_nat] : aa_list_nat_nat(size_size_list_nat,cons_nat(X21,X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_nat_nat(size_size_list_nat,X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_1974_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
    <=> ( ( ? [A5: $o,B5: $o] : A1 = vEBT_Leaf((A5),(B5))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X4),N3) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N3)
            & ( aa_lis7044684121217598542BT_nat(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))),N3) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
        | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X4),N3) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N3))
            & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
        | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X4),N3) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N3)
            & ( aa_lis7044684121217598542BT_nat(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))),N3) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
               => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),X_13)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I3) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I3 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N3)) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N3) = I3 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N3)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X4)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma2) ) ) ) ) ) )
        | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X4),N3) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N3))
            & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3)))
               => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),X_13)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I3) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I3 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N3)) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N3) = I3 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N3)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X4)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_1975_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
     => ( ( ? [A3: $o,B3: $o] : A1 = vEBT_Leaf((A3),(B3))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                     => aa_nat_o(vEBT_invar_vebt(X),N2) )
                 => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
                   => ( ( aa_lis7044684121217598542BT_nat(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 = N2 )
                       => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M) )
                         => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_1)
                           => ~ ! [X: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                       => aa_nat_o(vEBT_invar_vebt(X),N2) )
                   => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
                     => ( ( aa_lis7044684121217598542BT_nat(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,N2) )
                         => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M) )
                           => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_1)
                             => ~ ! [X: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_invar_vebt(X),N2) )
                     => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
                       => ( ( aa_lis7044684121217598542BT_nat(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 = N2 )
                           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M) )
                             => ( ! [I4: nat] :
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                                   => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),X_13)
                                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I4) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X: vEBT_VEBT] :
                                        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                     => ~ ( ( Mi != Ma )
                                         => ! [I4: nat] :
                                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N2) = I4 )
                                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N2)) )
                                                & ! [X: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N2) = I4 )
                                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N2)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_invar_vebt(X),N2) )
                       => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
                         => ( ( aa_lis7044684121217598542BT_nat(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,N2) )
                             => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M) )
                               => ( ! [I4: nat] :
                                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                                     => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),X_13)
                                      <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I4) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X: vEBT_VEBT] :
                                          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                         => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                       => ~ ( ( Mi != Ma )
                                           => ! [I4: nat] :
                                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N2) = I4 )
                                                   => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N2)) )
                                                  & ! [X: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N2) = I4 )
                                                        & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N2)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_1976_mod2__gr__0,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_1977_add__self__mod__2,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_1978_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_1979_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1980_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1981_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1982_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1983_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1984_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1985_sum__power2__eq__zero__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Yb = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1986_sum__power2__eq__zero__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Yb = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1987_sum__power2__eq__zero__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Yb = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1988_zero__less__power2,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_1989_zero__less__power2,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_1990_zero__less__power2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_1991_power2__eq__iff__nonneg,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Yb ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1992_power2__eq__iff__nonneg,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
       => ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Yb ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1993_power2__eq__iff__nonneg,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Yb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Yb ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1994_power2__eq__iff__nonneg,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Yb ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1995_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_1996_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_1997_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_1998_power__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_1999_power__decreasing__iff,axiom,
    ! [B: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Mb)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2000_power__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2001_power__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),Mb)),aa_nat_int(power_power_int(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2002_mod__add__self2,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self2
tff(fact_2003_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self2
tff(fact_2004_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self2
tff(fact_2005_mod__add__self1,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self1
tff(fact_2006_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self1
tff(fact_2007_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B,A),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self1
tff(fact_2008_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Nb)) = zero_zero_rat ).

% power_0_Suc
tff(fact_2009_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = zero_zero_nat ).

% power_0_Suc
tff(fact_2010_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,Nb)) = zero_zero_real ).

% power_0_Suc
tff(fact_2011_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,Nb)) = zero_zero_int ).

% power_0_Suc
tff(fact_2012_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,Nb)) = zero_zero_complex ).

% power_0_Suc
tff(fact_2013_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_2014_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_2015_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_real ).

% power_zero_numeral
tff(fact_2016_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_int ).

% power_zero_numeral
tff(fact_2017_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_2018_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_2019_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2020_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_2021_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_2022_power__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_2023_nat__power__eq__Suc__0__iff,axiom,
    ! [Xb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Mb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = zero_zero_nat )
        | ( Xb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2024_nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_2025_mod__by__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_2026_power__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2027_power__eq__0__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2028_power__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2029_power__eq__0__iff,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_nat_int(power_power_int(A),Nb) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2030_power__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(power_power_complex(A),Nb) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2031_power__strict__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2032_power__strict__decreasing__iff,axiom,
    ! [B: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Mb)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2033_power__strict__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2034_power__strict__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),Mb)),aa_nat_int(power_power_int(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2035_power__increasing__iff,axiom,
    ! [B: real,Xb: nat,Yb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Yb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ) ).

% power_increasing_iff
tff(fact_2036_power__increasing__iff,axiom,
    ! [B: rat,Xb: nat,Yb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Xb)),aa_nat_rat(power_power_rat(B),Yb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ) ).

% power_increasing_iff
tff(fact_2037_power__increasing__iff,axiom,
    ! [B: nat,Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Yb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ) ).

% power_increasing_iff
tff(fact_2038_power__increasing__iff,axiom,
    ! [B: int,Xb: nat,Yb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),Xb)),aa_nat_int(power_power_int(B),Yb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ) ).

% power_increasing_iff
tff(fact_2039_power__mono__iff,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2040_power__mono__iff,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2041_power__mono__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2042_power__mono__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2043_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_2044_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_2045_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_2046_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_2047_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_2048_mod2__Suc__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_2049_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% two_realpow_ge_one
tff(fact_2050_power__not__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),Nb) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_2051_power__not__zero,axiom,
    ! [A: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_2052_power__not__zero,axiom,
    ! [A: real,Nb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_2053_power__not__zero,axiom,
    ! [A: int,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),Nb) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_2054_power__not__zero,axiom,
    ! [A: complex,Nb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),Nb) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_2055_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) ).

% mod_add_right_eq
tff(fact_2056_mod__add__right__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2))),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ).

% mod_add_right_eq
tff(fact_2057_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(B,C2)),C2) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C2) ).

% mod_add_right_eq
tff(fact_2058_mod__add__left__eq,axiom,
    ! [A: nat,C2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C2)),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) ).

% mod_add_left_eq
tff(fact_2059_mod__add__left__eq,axiom,
    ! [A: int,C2: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C2)),B)),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ).

% mod_add_left_eq
tff(fact_2060_mod__add__left__eq,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C2),B),C2) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C2) ).

% mod_add_left_eq
tff(fact_2061_mod__add__cong,axiom,
    ! [A: nat,C2: nat,A2: nat,B: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) ) ) ) ).

% mod_add_cong
tff(fact_2062_mod__add__cong,axiom,
    ! [A: int,C2: int,A2: int,B: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) ) ) ) ).

% mod_add_cong
tff(fact_2063_mod__add__cong,axiom,
    ! [A: code_integer,C2: code_integer,A2: code_integer,B: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger(A,C2) = modulo364778990260209775nteger(A2,C2) )
     => ( ( modulo364778990260209775nteger(B,C2) = modulo364778990260209775nteger(B2,C2) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C2) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A2,B2),C2) ) ) ) ).

% mod_add_cong
tff(fact_2064_mod__add__eq,axiom,
    ! [A: nat,C2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) ).

% mod_add_eq
tff(fact_2065_mod__add__eq,axiom,
    ! [A: int,C2: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2))),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ).

% mod_add_eq
tff(fact_2066_mod__add__eq,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C2),modulo364778990260209775nteger(B,C2)),C2) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C2) ).

% mod_add_eq
tff(fact_2067_mod__Suc__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),Nb) ).

% mod_Suc_Suc_eq
tff(fact_2068_mod__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% mod_Suc_eq
tff(fact_2069_mod__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Mb) ).

% mod_less_eq_dividend
tff(fact_2070_power__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono
tff(fact_2071_power__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono
tff(fact_2072_power__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% power_mono
tff(fact_2073_power__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono
tff(fact_2074_zero__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_le_power
tff(fact_2075_zero__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_le_power
tff(fact_2076_zero__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% zero_le_power
tff(fact_2077_zero__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% zero_le_power
tff(fact_2078_zero__less__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_less_power
tff(fact_2079_zero__less__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_less_power
tff(fact_2080_zero__less__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% zero_less_power
tff(fact_2081_zero__less__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% zero_less_power
tff(fact_2082_one__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% one_le_power
tff(fact_2083_one__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% one_le_power
tff(fact_2084_one__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% one_le_power
tff(fact_2085_one__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% one_le_power
tff(fact_2086_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_2087_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_2088_power__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_2089_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2090_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2091_nat__power__less__imp__less,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_2092_mod__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = Nb,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))) ).

% mod_Suc
tff(fact_2093_mod__induct,axiom,
    ! [P2: fun_nat_o,Nb: nat,P: nat,Mb: nat] :
      ( aa_nat_o(P2,Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),P)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),P)
         => ( ! [N2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),P)
               => ( aa_nat_o(P2,N2)
                 => aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),P)) ) )
           => aa_nat_o(P2,Mb) ) ) ) ) ).

% mod_induct
tff(fact_2094_mod__less__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_2095_mod__Suc__le__divisor,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_2096_power__less__imp__less__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2097_power__less__imp__less__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2098_power__less__imp__less__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2099_power__less__imp__less__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2100_power__le__one,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),one_one_real) ) ) ).

% power_le_one
tff(fact_2101_power__le__one,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),one_one_rat) ) ) ).

% power_le_one
tff(fact_2102_power__le__one,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat) ) ) ).

% power_le_one
tff(fact_2103_power__le__one,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),one_one_int) ) ) ).

% power_le_one
tff(fact_2104_power__inject__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2105_power__inject__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2106_power__inject__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2107_power__inject__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2108_power__le__imp__le__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2109_power__le__imp__le__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2110_power__le__imp__le__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2111_power__le__imp__le__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2112_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_2113_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_2114_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_2115_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(power_power_int(zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_2116_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_2117_power__gt1,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_2118_power__gt1,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_2119_power__gt1,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_2120_power__gt1,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_2121_power__increasing,axiom,
    ! [Nb: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)) ) ) ).

% power_increasing
tff(fact_2122_power__increasing,axiom,
    ! [Nb: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(A),N4)) ) ) ).

% power_increasing
tff(fact_2123_power__increasing,axiom,
    ! [Nb: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).

% power_increasing
tff(fact_2124_power__increasing,axiom,
    ! [Nb: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(A),N4)) ) ) ).

% power_increasing
tff(fact_2125_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = zero_zero_rat ) ) ).

% zero_power
tff(fact_2126_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = zero_zero_nat ) ) ).

% zero_power
tff(fact_2127_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = zero_zero_real ) ) ).

% zero_power
tff(fact_2128_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(power_power_int(zero_zero_int),Nb) = zero_zero_int ) ) ).

% zero_power
tff(fact_2129_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = zero_zero_complex ) ) ).

% zero_power
tff(fact_2130_power__gt__expt,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),K2)) ) ).

% power_gt_expt
tff(fact_2131_nat__one__le__power,axiom,
    ! [I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb)) ) ).

% nat_one_le_power
tff(fact_2132_mod__le__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_2133_power__Suc__le__self,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2134_power__Suc__le__self,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2135_power__Suc__le__self,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2136_power__Suc__le__self,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2137_power__Suc__less__one,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_2138_power__Suc__less__one,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_2139_power__Suc__less__one,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_2140_power__Suc__less__one,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_2141_power__strict__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2142_power__strict__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2143_power__strict__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2144_power__strict__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2145_power__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2146_power__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2147_power__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2148_power__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2149_power__le__imp__le__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2150_power__le__imp__le__exp,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2151_power__le__imp__le__exp,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2152_power__le__imp__le__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2153_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2154_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2155_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2156_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2157_power__eq__imp__eq__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2158_power__eq__imp__eq__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2159_power__eq__imp__eq__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2160_power__eq__imp__eq__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2161_zero__power2,axiom,
    aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat ).

% zero_power2
tff(fact_2162_zero__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% zero_power2
tff(fact_2163_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_2164_zero__power2,axiom,
    aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int ).

% zero_power2
tff(fact_2165_zero__power2,axiom,
    aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex ).

% zero_power2
tff(fact_2166_self__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% self_le_power
tff(fact_2167_self__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% self_le_power
tff(fact_2168_self__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% self_le_power
tff(fact_2169_self__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% self_le_power
tff(fact_2170_one__less__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% one_less_power
tff(fact_2171_one__less__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% one_less_power
tff(fact_2172_one__less__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% one_less_power
tff(fact_2173_one__less__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% one_less_power
tff(fact_2174_power2__nat__le__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_2175_power2__nat__le__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_2176_self__le__ge2__pow,axiom,
    ! [K2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Mb)) ) ).

% self_le_ge2_pow
tff(fact_2177_zero__le__power2,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_2178_zero__le__power2,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_2179_zero__le__power2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_2180_power2__eq__imp__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
         => ( Xb = Yb ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2181_power2__eq__imp__eq,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
         => ( Xb = Yb ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2182_power2__eq__imp__eq,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Yb)
         => ( Xb = Yb ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2183_power2__eq__imp__eq,axiom,
    ! [Xb: int,Yb: int] :
      ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
         => ( Xb = Yb ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2184_power2__le__imp__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ).

% power2_le_imp_le
tff(fact_2185_power2__le__imp__le,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ) ).

% power2_le_imp_le
tff(fact_2186_power2__le__imp__le,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Yb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ) ).

% power2_le_imp_le
tff(fact_2187_power2__le__imp__le,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ) ).

% power2_le_imp_le
tff(fact_2188_power__strict__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2189_power__strict__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2190_power__strict__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2191_power__strict__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2192_power2__less__0,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_2193_power2__less__0,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat) ).

% power2_less_0
tff(fact_2194_power2__less__0,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int) ).

% power2_less_0
tff(fact_2195_nat__induct2,axiom,
    ! [P2: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P2,zero_zero_nat)
     => ( aa_nat_o(P2,one_one_nat)
       => ( ! [N2: nat] :
              ( aa_nat_o(P2,N2)
             => aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(P2,Nb) ) ) ) ).

% nat_induct2
tff(fact_2196_power2__less__imp__less,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ).

% power2_less_imp_less
tff(fact_2197_power2__less__imp__less,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb) ) ) ).

% power2_less_imp_less
tff(fact_2198_power2__less__imp__less,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Yb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb) ) ) ).

% power2_less_imp_less
tff(fact_2199_power2__less__imp__less,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb) ) ) ).

% power2_less_imp_less
tff(fact_2200_sum__power2__le__zero__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Yb = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2201_sum__power2__le__zero__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Yb = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2202_sum__power2__le__zero__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Yb = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2203_sum__power2__ge__zero,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_2204_sum__power2__ge__zero,axiom,
    ! [Xb: rat,Yb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_2205_sum__power2__ge__zero,axiom,
    ! [Xb: int,Yb: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_2206_not__sum__power2__lt__zero,axiom,
    ! [Xb: real,Yb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real) ).

% not_sum_power2_lt_zero
tff(fact_2207_not__sum__power2__lt__zero,axiom,
    ! [Xb: rat,Yb: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_2208_not__sum__power2__lt__zero,axiom,
    ! [Xb: int,Yb: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_2209_sum__power2__gt__zero__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_real )
        | ( Yb != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2210_sum__power2__gt__zero__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_rat )
        | ( Yb != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2211_sum__power2__gt__zero__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_int )
        | ( Yb != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2212_ex__power__ivl2,axiom,
    ! [B: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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_2213_ex__power__ivl1,axiom,
    ! [B: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K2)
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),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_2214__C00_C,axiom,
    ( ( deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) )
    & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
    & ( aa_nat_nat(suc,na) = m )
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),na)
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)
    & ( na = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% "00"
tff(fact_2215_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_2216_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
        & ( ( Xb = Mia )
          | ( Xb = Maa )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista))
            & aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).

% member_inv
tff(fact_2217_both__member__options__from__complete__tree__to__child,axiom,
    ! [Dega: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( Xb = Mia )
          | ( Xb = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_2218_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(Info,Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_2219_semiring__norm_I69_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),one) ).

% semiring_norm(69)
tff(fact_2220_semiring__norm_I2_J,axiom,
    aa_num_num(plus_plus_num(one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_2221_bits__mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_2222_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_2223_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_2224_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2225_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2226_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2227_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2228_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_2229_high__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% high_def
tff(fact_2230_bits__div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_div_0
tff(fact_2231_bits__div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_2232_bits__div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_2233_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_2234_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_2235_div__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_2236_div__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_2237_div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% div_0
tff(fact_2238_div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_2239_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_2240_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_2241_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_2242_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_2243_div__by__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_2244_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_2245_div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_2246_div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_2247_divide__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C2),A) = aa_complex_complex(divide1717551699836669952omplex(C2),B) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2248_divide__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2249_divide__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C2),A) = aa_rat_rat(divide_divide_rat(C2),B) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2250_divide__cancel__right,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C2) = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2251_divide__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2252_divide__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C2) = aa_rat_rat(divide_divide_rat(B),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2253_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_2254_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_2255_division__ring__divide__zero,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_2256_bits__mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_mod_0
tff(fact_2257_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_2258_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_2259_semiring__norm_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(Mb),Nb)) ).

% semiring_norm(6)
tff(fact_2260_semiring__norm_I71_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(71)
tff(fact_2261_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),Nb) ).

% semiring_norm(68)
tff(fact_2262_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2263_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2264_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2265_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_2266_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% div_self
tff(fact_2267_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% div_self
tff(fact_2268_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),A) = one_one_nat ) ) ).

% div_self
tff(fact_2269_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).

% div_self
tff(fact_2270_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(A),B) )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2271_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2272_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(A),B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2273_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_2274_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% divide_self
tff(fact_2275_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_2276_divide__self__if,axiom,
    ! [A: complex] :
      aa_complex_complex(divide1717551699836669952omplex(A),A) = $ite(A = zero_zero_complex,zero_zero_complex,one_one_complex) ).

% divide_self_if
tff(fact_2277_divide__self__if,axiom,
    ! [A: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_2278_divide__self__if,axiom,
    ! [A: rat] :
      aa_rat_rat(divide_divide_rat(A),A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_2279_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2280_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2281_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2282_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(B),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2283_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_2284_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_2285_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_2286_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_2287_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_2288_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_2289_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_2290_mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% mod_div_trivial
tff(fact_2291_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_2292_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_2293_div__by__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = Mb ).

% div_by_Suc_0
tff(fact_2294_div__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat ) ) ).

% div_less
tff(fact_2295_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_divide_1_iff
tff(fact_2296_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_divide_1_iff
tff(fact_2297_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_2298_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_2299_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% divide_less_0_1_iff
tff(fact_2300_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_2301_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_2302_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_2303_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_2304_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_2305_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_2306_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_2307_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_2308_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_2309_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_divide_1_iff
tff(fact_2310_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_divide_1_iff
tff(fact_2311_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2312_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2313_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_2314_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_2315_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_2316_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_2317_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2318_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2319_div2__Suc__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_2320_add__self__div__2,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Mb ).

% add_self_div_2
tff(fact_2321_bits__1__div__2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_2322_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_2323_one__div__two__eq__zero,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_2324_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_2325_real__arch__pow__inv,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N2)),Yb) ) ) ).

% real_arch_pow_inv
tff(fact_2326_less__eq__real__def,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% less_eq_real_def
tff(fact_2327_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X: real] : member_real(X,S2)
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( member_real(X3,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Z3) )
       => ? [Y: real] :
            ( ! [X: real] :
                ( member_real(X,S2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) )
            & ! [Z3: real] :
                ( ! [X3: real] :
                    ( member_real(X3,S2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Z3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z3) ) ) ) ) ).

% complete_real
tff(fact_2328_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),B)),C2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C2)),aa_complex_complex(divide1717551699836669952omplex(B),C2)) ).

% add_divide_distrib
tff(fact_2329_add__divide__distrib,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)) ).

% add_divide_distrib
tff(fact_2330_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),C2)),aa_rat_rat(divide_divide_rat(B),C2)) ).

% add_divide_distrib
tff(fact_2331_div__le__mono,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),K2)) ) ).

% div_le_mono
tff(fact_2332_div__le__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ).

% div_le_dividend
tff(fact_2333_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)) ) ) ).

% divide_right_mono_neg
tff(fact_2334_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C2)),aa_rat_rat(divide_divide_rat(A),C2)) ) ) ).

% divide_right_mono_neg
tff(fact_2335_divide__nonpos__nonpos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2336_divide__nonpos__nonpos,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2337_divide__nonpos__nonneg,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_2338_divide__nonpos__nonneg,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_2339_divide__nonneg__nonpos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_2340_divide__nonneg__nonpos,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_2341_divide__nonneg__nonneg,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2342_divide__nonneg__nonneg,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2343_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_2344_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_2345_divide__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)) ) ) ).

% divide_right_mono
tff(fact_2346_divide__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C2)),aa_rat_rat(divide_divide_rat(B),C2)) ) ) ).

% divide_right_mono
tff(fact_2347_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% divide_le_0_iff
tff(fact_2348_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% divide_le_0_iff
tff(fact_2349_divide__neg__neg,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)) ) ) ).

% divide_neg_neg
tff(fact_2350_divide__neg__neg,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% divide_neg_neg
tff(fact_2351_divide__neg__pos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_2352_divide__neg__pos,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_2353_divide__pos__neg,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_2354_divide__pos__neg,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_2355_divide__pos__pos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)) ) ) ).

% divide_pos_pos
tff(fact_2356_divide__pos__pos,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% divide_pos_pos
tff(fact_2357_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% divide_less_0_iff
tff(fact_2358_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% divide_less_0_iff
tff(fact_2359_divide__less__cancel,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        & ( C2 != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_2360_divide__less__cancel,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C2)),aa_rat_rat(divide_divide_rat(B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( C2 != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_2361_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_2362_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_2363_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)) ) ) ).

% divide_strict_right_mono
tff(fact_2364_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C2)),aa_rat_rat(divide_divide_rat(B),C2)) ) ) ).

% divide_strict_right_mono
tff(fact_2365_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2366_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C2)),aa_rat_rat(divide_divide_rat(B),C2)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2367_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2368_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2369_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2370_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2371_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2372_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = A )
    <=> ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2373_div__add1__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2))),C2)) ).

% div_add1_eq
tff(fact_2374_div__add1__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2))),C2)) ).

% div_add1_eq
tff(fact_2375_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] : divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C2) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C2),divide6298287555418463151nteger(B,C2)),divide6298287555418463151nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C2),modulo364778990260209775nteger(B,C2)),C2)) ).

% div_add1_eq
tff(fact_2376_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_2377_Suc__div__le__mono,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Mb)),Nb)) ).

% Suc_div_le_mono
tff(fact_2378_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_positive
tff(fact_2379_div__positive,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_positive
tff(fact_2380_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2381_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2382_divide__nonpos__pos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_2383_divide__nonpos__pos,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_2384_divide__nonpos__neg,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)) ) ) ).

% divide_nonpos_neg
tff(fact_2385_divide__nonpos__neg,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% divide_nonpos_neg
tff(fact_2386_divide__nonneg__pos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)) ) ) ).

% divide_nonneg_pos
tff(fact_2387_divide__nonneg__pos,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% divide_nonneg_pos
tff(fact_2388_divide__nonneg__neg,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_2389_divide__nonneg__neg,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_2390_divide__le__cancel,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_2391_divide__le__cancel,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C2)),aa_rat_rat(divide_divide_rat(B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_2392_frac__less2,axiom,
    ! [Xb: real,Yb: real,W2: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Yb),W2)) ) ) ) ) ).

% frac_less2
tff(fact_2393_frac__less2,axiom,
    ! [Xb: rat,Yb: rat,W2: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Yb),W2)) ) ) ) ) ).

% frac_less2
tff(fact_2394_frac__less,axiom,
    ! [Xb: real,Yb: real,W2: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W2),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Yb),W2)) ) ) ) ) ).

% frac_less
tff(fact_2395_frac__less,axiom,
    ! [Xb: rat,Yb: rat,W2: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Yb),W2)) ) ) ) ) ).

% frac_less
tff(fact_2396_frac__le,axiom,
    ! [Yb: real,Xb: real,W2: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W2),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Yb),W2)) ) ) ) ) ).

% frac_le
tff(fact_2397_frac__le,axiom,
    ! [Yb: rat,Xb: rat,W2: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Yb),W2)) ) ) ) ) ).

% frac_le
tff(fact_2398_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_2399_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_2400_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_2401_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_2402_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_2403_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != 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,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_2404_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_2405_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != 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),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_2406_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B) ) ).

% gt_half_sum
tff(fact_2407_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_2408_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_2409_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_2410_div__greater__zero__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_2411_div__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K2),Mb)) ) ) ).

% div_le_mono2
tff(fact_2412_div__less__dividend,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ) ) ).

% div_less_dividend
tff(fact_2413_div__eq__dividend__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Mb )
      <=> ( Nb = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_2414_div__less__mono,axiom,
    ! [A4: nat,B4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),Nb) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),Nb) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_2415_div__exp__eq,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ).

% div_exp_eq
tff(fact_2416_div__exp__eq,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ).

% div_exp_eq
tff(fact_2417_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_2418_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_2419_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_2420_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_2421_field__sum__of__halves,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_2422_field__sum__of__halves,axiom,
    ! [Xb: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(Xb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_2423_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_2424_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_2425_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_2426_div__exp__mod__exp__eq,axiom,
    ! [A: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_2427_div__exp__mod__exp__eq,axiom,
    ! [A: int,Nb: nat,Mb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_2428_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,Nb: nat,Mb: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_2429_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% half_gt_zero_iff
tff(fact_2430_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% half_gt_zero_iff
tff(fact_2431_half__gt__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_2432_half__gt__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_2433_field__less__half__sum,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_2434_field__less__half__sum,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),Yb)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_2435_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2436_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_2437_verit__le__mono__div,axiom,
    ! [A4: nat,B4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),Nb)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),Nb) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_2438_not__exp__less__eq__0__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_2439_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_2440_low__inv,axiom,
    ! [Xb: nat,Nb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Yb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xb)),Nb) = Xb ) ) ).

% low_inv
tff(fact_2441_high__inv,axiom,
    ! [Xb: nat,Nb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Yb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xb)),Nb) = Yb ) ) ).

% high_inv
tff(fact_2442_enat__ord__number_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Mb),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(1)
tff(fact_2443_pos2,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% pos2
tff(fact_2444_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xc: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,Xaa),Xba),Xc) = Yb )
     => ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(Xb,produc487386426758144856at_nat(Xaa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xba),Xc))))
       => ~ ( ( Yb = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xba),Xaa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xba),aa_nat_nat(aa_nat_fun_nat_nat(Xb,Xaa),Xc))) )
           => ~ accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(Xb,produc487386426758144856at_nat(Xaa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xba),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_2445_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc: nat] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(F,produc487386426758144856at_nat(A,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Acc))))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_2446_insert__simp__excp,axiom,
    ! [Mia: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Xb: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mia),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_2447_insert__simp__norm,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_2448_realpow__pos__nth,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),Nb) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_2449_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% zle_add1_eq_le
tff(fact_2450_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT,Yb: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),I,Yb) = list_u1324408373059187874T_VEBT(Xs,I,Yb) ).

% list_update_overwrite
tff(fact_2451_mult__cancel__right,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C2) = aa_complex_complex(times_times_complex(B),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2452_mult__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2453_mult__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C2) = aa_rat_rat(times_times_rat(B),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2454_mult__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2455_mult__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2456_mult__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(C2),A) = aa_complex_complex(times_times_complex(C2),B) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2457_mult__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2458_mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(C2),A) = aa_rat_rat(times_times_rat(C2),B) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2459_mult__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2460_mult__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) )
    <=> ( ( C2 = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2461_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_2462_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_2463_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_2464_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_2465_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_2466_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_2467_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_2468_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_2469_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_2470_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_2471_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_2472_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_2473_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_2474_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_2475_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_2476_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_2477_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_2478_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_2479_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_2480_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_2481_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_2482_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_2483_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_2484_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_2485_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_2486_mult__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_2487_mult__0__right,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_2488_mult__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb) )
    <=> ( ( Mb = Nb )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_2489_mult__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2) )
    <=> ( ( Mb = Nb )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_2490_mod__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K2)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = K2 ) ) ) ).

% mod_neg_neg_trivial
tff(fact_2491_mod__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = K2 ) ) ) ).

% mod_pos_pos_trivial
tff(fact_2492_div__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_2493_div__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_2494_max__bot,axiom,
    ! [Xb: set_Pr1261947904930325089at_nat] : ord_ma7524802468073614006at_nat(bot_bo2099793752762293965at_nat,Xb) = Xb ).

% max_bot
tff(fact_2495_max__bot,axiom,
    ! [Xb: set_real] : ord_max_set_real(bot_bot_set_real,Xb) = Xb ).

% max_bot
tff(fact_2496_max__bot,axiom,
    ! [Xb: set_nat] : ord_max_set_nat(bot_bot_set_nat,Xb) = Xb ).

% max_bot
tff(fact_2497_max__bot,axiom,
    ! [Xb: set_int] : ord_max_set_int(bot_bot_set_int,Xb) = Xb ).

% max_bot
tff(fact_2498_max__bot,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),Xb) = Xb ).

% max_bot
tff(fact_2499_max__bot,axiom,
    ! [Xb: extended_enat] : ord_ma741700101516333627d_enat(bot_bo4199563552545308370d_enat,Xb) = Xb ).

% max_bot
tff(fact_2500_max__bot2,axiom,
    ! [Xb: set_Pr1261947904930325089at_nat] : ord_ma7524802468073614006at_nat(Xb,bot_bo2099793752762293965at_nat) = Xb ).

% max_bot2
tff(fact_2501_max__bot2,axiom,
    ! [Xb: set_real] : ord_max_set_real(Xb,bot_bot_set_real) = Xb ).

% max_bot2
tff(fact_2502_max__bot2,axiom,
    ! [Xb: set_nat] : ord_max_set_nat(Xb,bot_bot_set_nat) = Xb ).

% max_bot2
tff(fact_2503_max__bot2,axiom,
    ! [Xb: set_int] : ord_max_set_int(Xb,bot_bot_set_int) = Xb ).

% max_bot2
tff(fact_2504_max__bot2,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),bot_bot_nat) = Xb ).

% max_bot2
tff(fact_2505_max__bot2,axiom,
    ! [Xb: extended_enat] : ord_ma741700101516333627d_enat(Xb,bot_bo4199563552545308370d_enat) = Xb ).

% max_bot2
tff(fact_2506_nat__1__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_2507_nat__mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = one_one_nat )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_2508_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT] : aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,list_u1324408373059187874T_VEBT(Xs,I,Xb)) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) ).

% length_list_update
tff(fact_2509_length__list__update,axiom,
    ! [Xs: list_int,I: nat,Xb: int] : aa_list_int_nat(size_size_list_int,list_update_int(Xs,I,Xb)) = aa_list_int_nat(size_size_list_int,Xs) ).

% length_list_update
tff(fact_2510_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,Xb: nat] : aa_list_nat_nat(size_size_list_nat,list_update_nat(Xs,I,Xb)) = aa_list_nat_nat(size_size_list_nat,Xs) ).

% length_list_update
tff(fact_2511_max__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)) ).

% max_Suc_Suc
tff(fact_2512_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_2513_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_2514_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_2515_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_2516_max__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Nb) = Nb ).

% max_0L
tff(fact_2517_max__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),zero_zero_nat) = Nb ).

% max_0R
tff(fact_2518_list__update__id,axiom,
    ! [Xs: list_int,I: nat] : list_update_int(Xs,I,aa_nat_int(nth_int(Xs),I)) = Xs ).

% list_update_id
tff(fact_2519_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),I)) = Xs ).

% list_update_id
tff(fact_2520_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] : list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,I)) = Xs ).

% list_update_id
tff(fact_2521_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,Xb: int] :
      ( ( I != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_2522_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Xb: nat] :
      ( ( I != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_2523_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ( I != J )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),J) = nth_VEBT_VEBT(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_2524_mult__cancel__right2,axiom,
    ! [A: complex,C2: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C2) = C2 )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_2525_mult__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = C2 )
    <=> ( ( C2 = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_2526_mult__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C2) = C2 )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_2527_mult__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_2528_mult__cancel__right1,axiom,
    ! [C2: complex,B: complex] :
      ( ( C2 = aa_complex_complex(times_times_complex(B),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_2529_mult__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( ( C2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_2530_mult__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( ( C2 = aa_rat_rat(times_times_rat(B),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_2531_mult__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( C2 = aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_2532_mult__cancel__left2,axiom,
    ! [C2: complex,A: complex] :
      ( ( aa_complex_complex(times_times_complex(C2),A) = C2 )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_2533_mult__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = C2 )
    <=> ( ( C2 = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_2534_mult__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( ( aa_rat_rat(times_times_rat(C2),A) = C2 )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_2535_mult__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_2536_mult__cancel__left1,axiom,
    ! [C2: complex,B: complex] :
      ( ( C2 = aa_complex_complex(times_times_complex(C2),B) )
    <=> ( ( C2 = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_2537_mult__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( ( C2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) )
    <=> ( ( C2 = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_2538_mult__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( ( C2 = aa_rat_rat(times_times_rat(C2),B) )
    <=> ( ( C2 = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_2539_mult__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( C2 = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) )
    <=> ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_2540_sum__squares__eq__zero__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Yb)) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Yb = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2541_sum__squares__eq__zero__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Yb),Yb)) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Yb = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2542_sum__squares__eq__zero__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Yb)) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Yb = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2543_div__mult__mult1__if,axiom,
    ! [C2: nat,A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) = $ite(C2 = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% div_mult_mult1_if
tff(fact_2544_div__mult__mult1__if,axiom,
    ! [C2: int,A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) = $ite(C2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% div_mult_mult1_if
tff(fact_2545_div__mult__mult2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_2546_div__mult__mult2,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult2
tff(fact_2547_div__mult__mult1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_2548_div__mult__mult1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != 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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult1
tff(fact_2549_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(C2),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2550_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2551_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(C2),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2552_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2553_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2554_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2555_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2556_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2557_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(B),C2)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2558_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2559_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2560_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(B),C2)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2561_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2562_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(B),C2)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2563_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2564_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2565_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2566_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2567_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2568_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(C2),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2569_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2570_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2571_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: complex,A: complex,B: complex] :
      aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(C2),B)) = $ite(C2 = zero_zero_complex,zero_zero_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2572_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: real,A: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) = $ite(C2 = zero_zero_real,zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2573_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: rat,A: rat,B: rat] :
      aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)) = $ite(C2 = zero_zero_rat,zero_zero_rat,aa_rat_rat(divide_divide_rat(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2574_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_2575_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V))) ).

% distrib_right_numeral
tff(fact_2576_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_2577_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_2578_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_2579_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C2: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(B),C2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C2)) ).

% distrib_left_numeral
tff(fact_2580_distrib__left__numeral,axiom,
    ! [V: num,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C2)) ).

% distrib_left_numeral
tff(fact_2581_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(B),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C2)) ).

% distrib_left_numeral
tff(fact_2582_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C2)) ).

% distrib_left_numeral
tff(fact_2583_distrib__left__numeral,axiom,
    ! [V: num,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C2)) ).

% distrib_left_numeral
tff(fact_2584_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_2585_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_2586_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_2587_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),B) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_2588_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_2589_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_2590_mod__mult__self4,axiom,
    ! [B: nat,C2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_2591_mod__mult__self4,axiom,
    ! [B: int,C2: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self4
tff(fact_2592_mod__mult__self4,axiom,
    ! [B: code_integer,C2: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self4
tff(fact_2593_mod__mult__self3,axiom,
    ! [C2: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_2594_mod__mult__self3,axiom,
    ! [C2: int,B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self3
tff(fact_2595_mod__mult__self3,axiom,
    ! [C2: code_integer,B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self3
tff(fact_2596_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_2597_mod__mult__self2,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self2
tff(fact_2598_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self2
tff(fact_2599_mod__mult__self1,axiom,
    ! [A: nat,C2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_2600_mod__mult__self1,axiom,
    ! [A: int,C2: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self1
tff(fact_2601_mod__mult__self1,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self1
tff(fact_2602_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V)) = $ite(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V)),numera1916890842035813515d_enat(V),numera1916890842035813515d_enat(U)) ).

% max_number_of(1)
tff(fact_2603_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(numera6620942414471956472nteger(U),numera6620942414471956472nteger(V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)),numera6620942414471956472nteger(V),numera6620942414471956472nteger(U)) ).

% max_number_of(1)
tff(fact_2604_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,V),aa_num_real(numeral_numeral_real,U)) ).

% max_number_of(1)
tff(fact_2605_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(1)
tff(fact_2606_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V),aa_num_nat(numeral_numeral_nat,U)) ).

% max_number_of(1)
tff(fact_2607_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(1)
tff(fact_2608_max__0__1_I3_J,axiom,
    ! [Xb: num] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(3)
tff(fact_2609_max__0__1_I3_J,axiom,
    ! [Xb: num] : ord_max_Code_integer(zero_z3403309356797280102nteger,numera6620942414471956472nteger(Xb)) = numera6620942414471956472nteger(Xb) ).

% max_0_1(3)
tff(fact_2610_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),aa_num_real(numeral_numeral_real,Xb)) = aa_num_real(numeral_numeral_real,Xb) ).

% max_0_1(3)
tff(fact_2611_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_rat_rat(ord_max_rat(zero_zero_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_num_rat(numeral_numeral_rat,Xb) ).

% max_0_1(3)
tff(fact_2612_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(3)
tff(fact_2613_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),aa_num_int(numeral_numeral_int,Xb)) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(3)
tff(fact_2614_max__0__1_I4_J,axiom,
    ! [Xb: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(Xb),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(4)
tff(fact_2615_max__0__1_I4_J,axiom,
    ! [Xb: num] : ord_max_Code_integer(numera6620942414471956472nteger(Xb),zero_z3403309356797280102nteger) = numera6620942414471956472nteger(Xb) ).

% max_0_1(4)
tff(fact_2616_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,Xb)),zero_zero_real) = aa_num_real(numeral_numeral_real,Xb) ).

% max_0_1(4)
tff(fact_2617_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,Xb)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,Xb) ).

% max_0_1(4)
tff(fact_2618_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xb)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(4)
tff(fact_2619_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,Xb)),zero_zero_int) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(4)
tff(fact_2620_max__0__1_I1_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_2621_max__0__1_I1_J,axiom,
    aa_rat_rat(ord_max_rat(zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_2622_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_2623_max__0__1_I1_J,axiom,
    ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_2624_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_2625_max__0__1_I1_J,axiom,
    ord_max_Code_integer(zero_z3403309356797280102nteger,one_one_Code_integer) = one_one_Code_integer ).

% max_0_1(1)
tff(fact_2626_max__0__1_I2_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_2627_max__0__1_I2_J,axiom,
    aa_rat_rat(ord_max_rat(one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_2628_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_2629_max__0__1_I2_J,axiom,
    ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat,zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_2630_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_2631_max__0__1_I2_J,axiom,
    ord_max_Code_integer(one_one_Code_integer,zero_z3403309356797280102nteger) = one_one_Code_integer ).

% max_0_1(2)
tff(fact_2632_mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_2633_one__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_2634_mult__less__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% mult_less_cancel2
tff(fact_2635_nat__0__less__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_2636_nat__mult__less__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_2637_mult__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc_right
tff(fact_2638_nat__mult__div__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) = $ite(K2 = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_2639_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),I)
     => ( list_u1324408373059187874T_VEBT(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_2640_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),I)
     => ( list_update_int(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_2641_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),I)
     => ( list_update_nat(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_2642_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2643_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2644_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2645_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2646_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W2)) = A )
    <=> $ite(numera6690914467698888265omplex(W2) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2)),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2647_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2)) = A )
    <=> $ite(aa_num_real(numeral_numeral_real,W2) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2648_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2)) = A )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2649_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W2)) )
    <=> $ite(numera6690914467698888265omplex(W2) != zero_zero_complex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2)) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2650_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2)) )
    <=> $ite(aa_num_real(numeral_numeral_real,W2) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2651_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2)) )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2652_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2653_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2654_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2655_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2656_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2657_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2658_div__mult__self4,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_2659_div__mult__self4,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_2660_div__mult__self3,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_2661_div__mult__self3,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_2662_div__mult__self2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_2663_div__mult__self2,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( B != 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,B),C2))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_2664_div__mult__self1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_2665_div__mult__self1,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( B != 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,C2),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_2666_half__negative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_2667_one__le__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ) ) ).

% one_le_mult_iff
tff(fact_2668_mult__le__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% mult_le_cancel2
tff(fact_2669_nat__mult__le__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2670_div__mult__self1__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Nb) = Mb ) ) ).

% div_mult_self1_is_m
tff(fact_2671_div__mult__self__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Nb) = Mb ) ) ).

% div_mult_self_is_m
tff(fact_2672_power__add__numeral2,axiom,
    ! [A: complex,Mb: num,Nb: num,B: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_2673_power__add__numeral2,axiom,
    ! [A: real,Mb: num,Nb: num,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_2674_power__add__numeral2,axiom,
    ! [A: rat,Mb: num,Nb: num,B: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_2675_power__add__numeral2,axiom,
    ! [A: nat,Mb: num,Nb: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_2676_power__add__numeral2,axiom,
    ! [A: int,Mb: num,Nb: num,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_2677_power__add__numeral,axiom,
    ! [A: complex,Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_2678_power__add__numeral,axiom,
    ! [A: real,Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_2679_power__add__numeral,axiom,
    ! [A: rat,Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_2680_power__add__numeral,axiom,
    ! [A: nat,Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_2681_power__add__numeral,axiom,
    ! [A: int,Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_2682_Suc__mod__mult__self1,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self1
tff(fact_2683_Suc__mod__mult__self2,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self2
tff(fact_2684_Suc__mod__mult__self3,axiom,
    ! [K2: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self3
tff(fact_2685_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K2: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self4
tff(fact_2686_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_2687_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_2688_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_2689_half__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% half_nonnegative_int_iff
tff(fact_2690_Suc__times__numeral__mod__eq,axiom,
    ! [K2: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K2) != 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,K2)),Nb))),aa_num_nat(numeral_numeral_nat,K2)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_2691_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_2692_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_int_nat(size_size_list_int,Xs))
       => ( set_int2(list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_2693_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_2694_zmod__le__nonneg__dividend,axiom,
    ! [Mb: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Mb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),K2)),Mb) ) ).

% zmod_le_nonneg_dividend
tff(fact_2695_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% neg_mod_conj
tff(fact_2696_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ) ).

% pos_mod_conj
tff(fact_2697_zmod__trivial__iff,axiom,
    ! [I: int,K2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I),K2) = I )
    <=> ( ( K2 = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_2698_verit__le__mono__div__int,axiom,
    ! [A4: int,B4: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),Nb)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),Nb) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_2699_neg__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_2700_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_2701_mod__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_2702_zdiv__mono__strict,axiom,
    ! [A4: int,B4: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),Nb) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),Nb) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_2703_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_2704_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_2705_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_2706_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_2707_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_2708_pos__imp__zdiv__pos__iff,axiom,
    ! [K2: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_2709_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_2710_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_2711_div__positive__int,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L)) ) ) ).

% div_positive_int
tff(fact_2712_div__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L))
    <=> ( ( K2 = zero_zero_int )
        | ( L = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).

% div_int_pos_iff
tff(fact_2713_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z) ) ).

% zless_imp_add1_zle
tff(fact_2714_zdiv__mono2__neg,axiom,
    ! [A: int,B2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_2715_zdiv__mono1__neg,axiom,
    ! [A: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% zdiv_mono1_neg
tff(fact_2716_zdiv__eq__0__iff,axiom,
    ! [I: int,K2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K2) = zero_zero_int )
    <=> ( ( K2 = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_2717_zdiv__mono2,axiom,
    ! [A: int,B2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ) ).

% zdiv_mono2
tff(fact_2718_zdiv__mono1,axiom,
    ! [A: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B)) ) ) ).

% zdiv_mono1
tff(fact_2719_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ).

% le_imp_0_less
tff(fact_2720_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% add1_zle_eq
tff(fact_2721_int__ge__induct,axiom,
    ! [K2: int,I: int,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),I)
     => ( aa_int_o(P2,K2)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),I2)
             => ( aa_int_o(P2,I2)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P2,I) ) ) ) ).

% int_ge_induct
tff(fact_2722_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_2723_conj__le__cong,axiom,
    ! [Xb: int,X6: int,P2: $o,P3: $o] :
      ( ( Xb = X6 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
         => ( (P2)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
            & (P2) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
            & (P3) ) ) ) ) ).

% conj_le_cong
tff(fact_2724_imp__le__cong,axiom,
    ! [Xb: int,X6: int,P2: $o,P3: $o] :
      ( ( Xb = X6 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
         => ( (P2)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
           => (P2) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
           => (P3) ) ) ) ) ).

% imp_le_cong
tff(fact_2725_verit__la__generic,axiom,
    ! [A: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xb)
      | ( A = Xb )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),A) ) ).

% verit_la_generic
tff(fact_2726_less__eq__int__code_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% less_eq_int_code(1)
tff(fact_2727_i0__lb,axiom,
    ! [Nb: extended_enat] : ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat,Nb) ).

% i0_lb
tff(fact_2728_ile0__eq,axiom,
    ! [Nb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Nb,zero_z5237406670263579293d_enat)
    <=> ( Nb = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_2729_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_2730_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_2731_int__div__less__self,axiom,
    ! [Xb: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),K2)),Xb) ) ) ).

% int_div_less_self
tff(fact_2732_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_neg_pos_less0
tff(fact_2733_less__int__code_I1_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_int_code(1)
tff(fact_2734_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2735_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C2) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2736_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2737_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2738_mult_Oassoc,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% mult.assoc
tff(fact_2739_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C2) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C2)) ).

% mult.assoc
tff(fact_2740_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% mult.assoc
tff(fact_2741_mult_Oassoc,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% mult.assoc
tff(fact_2742_mult_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) ).

% mult.commute
tff(fact_2743_mult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),B) = aa_rat_rat(times_times_rat(B),A) ).

% mult.commute
tff(fact_2744_mult_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) ).

% mult.commute
tff(fact_2745_mult_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) ).

% mult.commute
tff(fact_2746_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% mult.left_commute
tff(fact_2747_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C2: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(times_times_rat(A),C2)) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C2)) ).

% mult.left_commute
tff(fact_2748_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% mult.left_commute
tff(fact_2749_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% mult.left_commute
tff(fact_2750_nat__mult__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ).

% nat_mult_max_left
tff(fact_2751_nat__mult__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)) ).

% nat_mult_max_right
tff(fact_2752_list__update__swap,axiom,
    ! [I: nat,I5: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,X6: vEBT_VEBT] :
      ( ( I != I5 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),I5,X6) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I5,X6),I,Xb) ) ) ).

% list_update_swap
tff(fact_2753_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_2754_plus__int__code_I1_J,axiom,
    ! [K2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),zero_zero_int) = K2 ).

% plus_int_code(1)
tff(fact_2755_neg__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)) ) ).

% neg_mod_bound
tff(fact_2756_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_2757_zip__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT,Ys2: list_VEBT_VEBT,Yb: vEBT_VEBT] : zip_VE537291747668921783T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),list_u1324408373059187874T_VEBT(Ys2,I,Yb)) = list_u6961636818849549845T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys2),I,produc537772716801021591T_VEBT(Xb,Yb)) ).

% zip_update
tff(fact_2758_zip__update,axiom,
    ! [Xs: list_int,I: nat,Xb: int,Ys2: list_int,Yb: int] : zip_int_int(list_update_int(Xs,I,Xb),list_update_int(Ys2,I,Yb)) = list_u3002344382305578791nt_int(zip_int_int(Xs,Ys2),I,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Yb)) ).

% zip_update
tff(fact_2759_zip__update,axiom,
    ! [Xs: list_f8851413139013959296e_term,I: nat,Xb: fun_Co3460408610320954864e_term,Ys2: list_P5578671422887162913nteger,Yb: produc8923325533196201883nteger] : zip_fu7084642285130276018nteger(list_u2080314064485400322e_term(Xs,I,Xb),list_u2254550707601501961nteger(Ys2,I,Yb)) = list_u5986614341521392946nteger(zip_fu7084642285130276018nteger(Xs,Ys2),I,produc7264323995540650962nteger(Xb,Yb)) ).

% zip_update
tff(fact_2760_zip__update,axiom,
    ! [Xs: list_f4306079564415238026e_term,I: nat,Xb: fun_Pr5098669352780829690e_term,Ys2: list_P5578671422887162913nteger,Yb: produc8923325533196201883nteger] : zip_fu4451626411769283112nteger(list_u3429793794183195916e_term(Xs,I,Xb),list_u2254550707601501961nteger(Ys2,I,Yb)) = list_u5815421275070861992nteger(zip_fu4451626411769283112nteger(Xs,Ys2),I,produc565448451787809096nteger(Xb,Yb)) ).

% zip_update
tff(fact_2761_zip__update,axiom,
    ! [Xs: list_f280343417242313083e_term,I: nat,Xb: fun_Pr7763680713840462443e_term,Ys2: list_P5707943133018811711nt_int,Yb: product_prod_int_int] : zip_fu5048005673116055189nt_int(list_u7283322402012212093e_term(Xs,I,Xb),list_u3002344382305578791nt_int(Ys2,I,Yb)) = list_u7912136156437333013nt_int(zip_fu5048005673116055189nt_int(Xs,Ys2),I,produc7320969046608631221nt_int(Xb,Yb)) ).

% zip_update
tff(fact_2762_zip__update,axiom,
    ! [Xs: list_f628289730585369423e_term,I: nat,Xb: fun_in5730379987311621695e_term,Ys2: list_P5707943133018811711nt_int,Yb: product_prod_int_int] : zip_fu2910578966527623617nt_int(list_u528859344425011025e_term(Xs,I,Xb),list_u3002344382305578791nt_int(Ys2,I,Yb)) = list_u4453756226229724993nt_int(zip_fu2910578966527623617nt_int(Xs,Ys2),I,produc4729913460834754785nt_int(Xb,Yb)) ).

% zip_update
tff(fact_2763_max__absorb2,axiom,
    ! [Xb: extended_enat,Yb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Xb,Yb)
     => ( ord_ma741700101516333627d_enat(Xb,Yb) = Yb ) ) ).

% max_absorb2
tff(fact_2764_max__absorb2,axiom,
    ! [Xb: code_integer,Yb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),Yb)
     => ( ord_max_Code_integer(Xb,Yb) = Yb ) ) ).

% max_absorb2
tff(fact_2765_max__absorb2,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Yb)
     => ( ord_max_set_int(Xb,Yb) = Yb ) ) ).

% max_absorb2
tff(fact_2766_max__absorb2,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => ( aa_rat_rat(ord_max_rat(Xb),Yb) = Yb ) ) ).

% max_absorb2
tff(fact_2767_max__absorb2,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
     => ( aa_num_num(ord_max_num(Xb),Yb) = Yb ) ) ).

% max_absorb2
tff(fact_2768_max__absorb2,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb) = Yb ) ) ).

% max_absorb2
tff(fact_2769_max__absorb2,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb) = Yb ) ) ).

% max_absorb2
tff(fact_2770_max__absorb1,axiom,
    ! [Yb: extended_enat,Xb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Yb,Xb)
     => ( ord_ma741700101516333627d_enat(Xb,Yb) = Xb ) ) ).

% max_absorb1
tff(fact_2771_max__absorb1,axiom,
    ! [Yb: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Yb),Xb)
     => ( ord_max_Code_integer(Xb,Yb) = Xb ) ) ).

% max_absorb1
tff(fact_2772_max__absorb1,axiom,
    ! [Yb: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Yb),Xb)
     => ( ord_max_set_int(Xb,Yb) = Xb ) ) ).

% max_absorb1
tff(fact_2773_max__absorb1,axiom,
    ! [Yb: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb)
     => ( aa_rat_rat(ord_max_rat(Xb),Yb) = Xb ) ) ).

% max_absorb1
tff(fact_2774_max__absorb1,axiom,
    ! [Yb: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb)
     => ( aa_num_num(ord_max_num(Xb),Yb) = Xb ) ) ).

% max_absorb1
tff(fact_2775_max__absorb1,axiom,
    ! [Yb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb) = Xb ) ) ).

% max_absorb1
tff(fact_2776_max__absorb1,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb) = Xb ) ) ).

% max_absorb1
tff(fact_2777_max__def,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ord_ma741700101516333627d_enat(A,B) = $ite(ord_le2932123472753598470d_enat(A,B),B,A) ).

% max_def
tff(fact_2778_max__def,axiom,
    ! [A: code_integer,B: code_integer] :
      ord_max_Code_integer(A,B) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B),B,A) ).

% max_def
tff(fact_2779_max__def,axiom,
    ! [A: set_int,B: set_int] :
      ord_max_set_int(A,B) = $ite(aa_set_int_o(ord_less_eq_set_int(A),B),B,A) ).

% max_def
tff(fact_2780_max__def,axiom,
    ! [A: rat,B: rat] :
      aa_rat_rat(ord_max_rat(A),B) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B),B,A) ).

% max_def
tff(fact_2781_max__def,axiom,
    ! [A: num,B: num] :
      aa_num_num(ord_max_num(A),B) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B),B,A) ).

% max_def
tff(fact_2782_max__def,axiom,
    ! [A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),B,A) ).

% max_def
tff(fact_2783_max__def,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B),B,A) ).

% max_def
tff(fact_2784_max__add__distrib__right,axiom,
    ! [Xb: real,Yb: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Yb),Z)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Z)) ).

% max_add_distrib_right
tff(fact_2785_max__add__distrib__right,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] : aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(ord_max_rat(Yb),Z)) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xb),Yb)),aa_rat_rat(plus_plus_rat(Xb),Z)) ).

% max_add_distrib_right
tff(fact_2786_max__add__distrib__right,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Yb),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Yb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Z)) ).

% max_add_distrib_right
tff(fact_2787_max__add__distrib__right,axiom,
    ! [Xb: int,Yb: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(ord_max_int,Yb),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Yb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Z)) ).

% max_add_distrib_right
tff(fact_2788_max__add__distrib__right,axiom,
    ! [Xb: code_integer,Yb: code_integer,Z: code_integer] : plus_p5714425477246183910nteger(Xb,ord_max_Code_integer(Yb,Z)) = ord_max_Code_integer(plus_p5714425477246183910nteger(Xb,Yb),plus_p5714425477246183910nteger(Xb,Z)) ).

% max_add_distrib_right
tff(fact_2789_max__add__distrib__left,axiom,
    ! [Xb: real,Yb: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Yb)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Yb),Z)) ).

% max_add_distrib_left
tff(fact_2790_max__add__distrib__left,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(ord_max_rat(Xb),Yb)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xb),Z)),aa_rat_rat(plus_plus_rat(Yb),Z)) ).

% max_add_distrib_left
tff(fact_2791_max__add__distrib__left,axiom,
    ! [Xb: nat,Yb: 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,Xb),Yb)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Yb),Z)) ).

% max_add_distrib_left
tff(fact_2792_max__add__distrib__left,axiom,
    ! [Xb: int,Yb: 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,Xb),Yb)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Yb),Z)) ).

% max_add_distrib_left
tff(fact_2793_max__add__distrib__left,axiom,
    ! [Xb: code_integer,Yb: code_integer,Z: code_integer] : plus_p5714425477246183910nteger(ord_max_Code_integer(Xb,Yb),Z) = ord_max_Code_integer(plus_p5714425477246183910nteger(Xb,Z),plus_p5714425477246183910nteger(Yb,Z)) ).

% max_add_distrib_left
tff(fact_2794_nat__add__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Q3)) ).

% nat_add_max_left
tff(fact_2795_nat__add__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q3)) ).

% nat_add_max_right
tff(fact_2796_mult__right__cancel,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C2) = aa_complex_complex(times_times_complex(B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2797_mult__right__cancel,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2798_mult__right__cancel,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C2) = aa_rat_rat(times_times_rat(B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2799_mult__right__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2800_mult__right__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2801_mult__left__cancel,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C2),A) = aa_complex_complex(times_times_complex(C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2802_mult__left__cancel,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2803_mult__left__cancel,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C2),A) = aa_rat_rat(times_times_rat(C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2804_mult__left__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2805_mult__left__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2806_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_2807_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_2808_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_2809_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_2810_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_2811_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_2812_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_2813_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_2814_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_2815_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_2816_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_2817_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_2818_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_2819_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_2820_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_2821_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2822_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_2823_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2824_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_2825_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_2826_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_2827_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_2828_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_2829_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_2830_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_2831_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C2: 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)),C2)) = 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)),C2) ).

% combine_common_factor
tff(fact_2832_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),E2)),C2) ).

% combine_common_factor
tff(fact_2833_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C2: 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)),C2)) = 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)),C2) ).

% combine_common_factor
tff(fact_2834_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C2: 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)),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E2)),C2) ).

% combine_common_factor
tff(fact_2835_distrib__right,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% distrib_right
tff(fact_2836_distrib__right,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)) ).

% distrib_right
tff(fact_2837_distrib__right,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% distrib_right
tff(fact_2838_distrib__right,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% distrib_right
tff(fact_2839_distrib__left,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) = 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),C2)) ).

% distrib_left
tff(fact_2840_distrib__left,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C2)) ).

% distrib_left
tff(fact_2841_distrib__left,axiom,
    ! [A: nat,B: nat,C2: 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),C2)) = 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),C2)) ).

% distrib_left
tff(fact_2842_distrib__left,axiom,
    ! [A: int,B: int,C2: 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),C2)) = 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),C2)) ).

% distrib_left
tff(fact_2843_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_2844_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)) ).

% comm_semiring_class.distrib
tff(fact_2845_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_2846_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_2847_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) = 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),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_2848_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_2849_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C2: 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),C2)) = 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),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_2850_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_2851_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_2852_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_2853_crossproduct__noteq,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ( A != B )
        & ( C2 != D ) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2854_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ( A != B )
        & ( C2 != D ) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D)) != aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),D)),aa_rat_rat(times_times_rat(B),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2855_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ( A != B )
        & ( C2 != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) != 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),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2856_crossproduct__noteq,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( A != B )
        & ( C2 != D ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) != 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),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2857_crossproduct__eq,axiom,
    ! [W2: real,Yb: real,Xb: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Yb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),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,Xb),Yb)) )
    <=> ( ( W2 = Xb )
        | ( Yb = Z ) ) ) ).

% crossproduct_eq
tff(fact_2858_crossproduct__eq,axiom,
    ! [W2: rat,Yb: rat,Xb: rat,Z: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W2),Yb)),aa_rat_rat(times_times_rat(Xb),Z)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W2),Z)),aa_rat_rat(times_times_rat(Xb),Yb)) )
    <=> ( ( W2 = Xb )
        | ( Yb = Z ) ) ) ).

% crossproduct_eq
tff(fact_2859_crossproduct__eq,axiom,
    ! [W2: nat,Yb: nat,Xb: 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),Yb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),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,Xb),Yb)) )
    <=> ( ( W2 = Xb )
        | ( Yb = Z ) ) ) ).

% crossproduct_eq
tff(fact_2860_crossproduct__eq,axiom,
    ! [W2: int,Yb: int,Xb: 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),Yb)),aa_int_int(aa_int_fun_int_int(times_times_int,Xb),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,Xb),Yb)) )
    <=> ( ( W2 = Xb )
        | ( Yb = Z ) ) ) ).

% crossproduct_eq
tff(fact_2861_Suc__mult__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Nb) )
    <=> ( Mb = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_2862_mult__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% mult_0
tff(fact_2863_nat__mult__eq__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb) )
    <=> ( ( K2 = zero_zero_nat )
        | ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_2864_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),J)) ) ).

% mult_le_mono2
tff(fact_2865_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K2)) ) ).

% mult_le_mono1
tff(fact_2866_mult__le__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_2867_le__square,axiom,
    ! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb)) ).

% le_square
tff(fact_2868_le__cube,axiom,
    ! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb))) ).

% le_cube
tff(fact_2869_add__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)) ).

% add_mult_distrib
tff(fact_2870_add__mult__distrib2,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) ).

% add_mult_distrib2
tff(fact_2871_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K2: 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)),K2)) = 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)),K2) ).

% left_add_mult_distrib
tff(fact_2872_nat__mult__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Nb) = Nb ).

% nat_mult_1
tff(fact_2873_nat__mult__1__right,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),one_one_nat) = Nb ).

% nat_mult_1_right
tff(fact_2874_mult__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2875_mult__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2876_mult__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2877_mult__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2878_mult__mono_H,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2879_mult__mono_H,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2880_mult__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2881_mult__mono_H,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2882_zero__le__square,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)) ).

% zero_le_square
tff(fact_2883_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),A)) ).

% zero_le_square
tff(fact_2884_zero__le__square,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A)) ).

% zero_le_square
tff(fact_2885_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ).

% split_mult_pos_le
tff(fact_2886_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ).

% split_mult_pos_le
tff(fact_2887_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% split_mult_pos_le
tff(fact_2888_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2889_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2890_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2891_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2892_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2893_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2894_mult__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) ) ) ).

% mult_left_mono
tff(fact_2895_mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)) ) ) ).

% mult_left_mono
tff(fact_2896_mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% mult_left_mono
tff(fact_2897_mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_left_mono
tff(fact_2898_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_2899_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_2900_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_2901_mult__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ) ) ).

% mult_right_mono
tff(fact_2902_mult__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)) ) ) ).

% mult_right_mono
tff(fact_2903_mult__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ) ).

% mult_right_mono
tff(fact_2904_mult__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_right_mono
tff(fact_2905_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2906_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2907_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2908_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_2909_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_2910_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_2911_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_2912_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2913_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2914_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2915_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2916_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_2917_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2918_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2919_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_2920_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_2921_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2922_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2923_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_2924_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2925_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2926_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2927_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2928_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_2929_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_2930_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_2931_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2932_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2933_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2934_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2935_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2936_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2937_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2938_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2939_mult__less__cancel__right__disj,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2940_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2941_mult__less__cancel__right__disj,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2942_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_2943_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_2944_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_2945_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_2946_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2947_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2948_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2949_mult__less__cancel__left__disj,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2950_mult__less__cancel__left__disj,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2951_mult__less__cancel__left__disj,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2952_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2953_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2954_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2955_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2956_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2957_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2958_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2959_mult__less__cancel__left__pos,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2960_mult__less__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2961_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2962_mult__less__cancel__left__neg,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2963_mult__less__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2964_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2965_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2966_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B),A))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2967_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2968_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2969_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos
tff(fact_2970_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_2971_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos
tff(fact_2972_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos
tff(fact_2973_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_2974_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_2975_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_2976_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_2977_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_2978_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_2979_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_2980_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2981_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_pos_pos
tff(fact_2982_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2983_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2984_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_2985_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_2986_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_2987_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_2988_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_2989_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_2990_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_2991_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_2992_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% mult_less_0_iff
tff(fact_2993_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_2994_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_2995_not__square__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real) ).

% not_square_less_zero
tff(fact_2996_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_2997_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int) ).

% not_square_less_zero
tff(fact_2998_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_neg_neg
tff(fact_2999_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_neg_neg
tff(fact_3000_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_neg_neg
tff(fact_3001_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C2: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R2),C2)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3002_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C2: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),C2)) != 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_3003_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C2: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(R2),C2)) != aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3004_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C2: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C2)) != 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_3005_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C2: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C2)) != 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_3006_frac__eq__eq,axiom,
    ! [Yb: complex,Z: complex,Xb: complex,W2: complex] :
      ( ( Yb != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(Xb),Yb) = aa_complex_complex(divide1717551699836669952omplex(W2),Z) )
        <=> ( aa_complex_complex(times_times_complex(Xb),Z) = aa_complex_complex(times_times_complex(W2),Yb) ) ) ) ) ).

% frac_eq_eq
tff(fact_3007_frac__eq__eq,axiom,
    ! [Yb: real,Z: real,Xb: real,W2: real] :
      ( ( Yb != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Yb) ) ) ) ) ).

% frac_eq_eq
tff(fact_3008_frac__eq__eq,axiom,
    ! [Yb: rat,Z: rat,Xb: rat,W2: rat] :
      ( ( Yb != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(Xb),Yb) = aa_rat_rat(divide_divide_rat(W2),Z) )
        <=> ( aa_rat_rat(times_times_rat(Xb),Z) = aa_rat_rat(times_times_rat(W2),Yb) ) ) ) ) ).

% frac_eq_eq
tff(fact_3009_divide__eq__eq,axiom,
    ! [B: complex,C2: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = A )
    <=> $ite(C2 != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),C2),A = zero_zero_complex) ) ).

% divide_eq_eq
tff(fact_3010_divide__eq__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) = A )
    <=> $ite(C2 != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2),A = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_3011_divide__eq__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),C2) = A )
    <=> $ite(C2 != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),C2),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_3012_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(times_times_complex(A),C2) = B,A = zero_zero_complex) ) ).

% eq_divide_eq
tff(fact_3013_eq__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) )
    <=> $ite(C2 != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = B,A = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_3014_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),C2) )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(times_times_rat(A),C2) = B,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_3015_divide__eq__imp,axiom,
    ! [C2: complex,B: complex,A: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( B = aa_complex_complex(times_times_complex(A),C2) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = A ) ) ) ).

% divide_eq_imp
tff(fact_3016_divide__eq__imp,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( C2 != zero_zero_real )
     => ( ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) = A ) ) ) ).

% divide_eq_imp
tff(fact_3017_divide__eq__imp,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C2) )
       => ( aa_rat_rat(divide_divide_rat(B),C2) = A ) ) ) ).

% divide_eq_imp
tff(fact_3018_eq__divide__imp,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C2) = B )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B),C2) ) ) ) ).

% eq_divide_imp
tff(fact_3019_eq__divide__imp,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = B )
       => ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) ) ) ) ).

% eq_divide_imp
tff(fact_3020_eq__divide__imp,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C2) = B )
       => ( A = aa_rat_rat(divide_divide_rat(B),C2) ) ) ) ).

% eq_divide_imp
tff(fact_3021_nonzero__divide__eq__eq,axiom,
    ! [C2: complex,B: complex,A: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = A )
      <=> ( B = aa_complex_complex(times_times_complex(A),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3022_nonzero__divide__eq__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) = A )
      <=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3023_nonzero__divide__eq__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B),C2) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3024_nonzero__eq__divide__eq,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
      <=> ( aa_complex_complex(times_times_complex(A),C2) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3025_nonzero__eq__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3026_nonzero__eq__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( A = aa_rat_rat(divide_divide_rat(B),C2) )
      <=> ( aa_rat_rat(times_times_rat(A),C2) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3027_list__update__code_I3_J,axiom,
    ! [Xb: int,Xs: list_int,I: nat,Yb: int] : list_update_int(cons_int(Xb,Xs),aa_nat_nat(suc,I),Yb) = cons_int(Xb,list_update_int(Xs,I,Yb)) ).

% list_update_code(3)
tff(fact_3028_list__update__code_I3_J,axiom,
    ! [Xb: nat,Xs: list_nat,I: nat,Yb: nat] : list_update_nat(cons_nat(Xb,Xs),aa_nat_nat(suc,I),Yb) = cons_nat(Xb,list_update_nat(Xs,I,Yb)) ).

% list_update_code(3)
tff(fact_3029_list__update__code_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,I: nat,Yb: vEBT_VEBT] : list_u1324408373059187874T_VEBT(cons_VEBT_VEBT(Xb,Xs),aa_nat_nat(suc,I),Yb) = cons_VEBT_VEBT(Xb,list_u1324408373059187874T_VEBT(Xs,I,Yb)) ).

% list_update_code(3)
tff(fact_3030_list__update__code_I2_J,axiom,
    ! [Xb: int,Xs: list_int,Yb: int] : list_update_int(cons_int(Xb,Xs),zero_zero_nat,Yb) = cons_int(Yb,Xs) ).

% list_update_code(2)
tff(fact_3031_list__update__code_I2_J,axiom,
    ! [Xb: nat,Xs: list_nat,Yb: nat] : list_update_nat(cons_nat(Xb,Xs),zero_zero_nat,Yb) = cons_nat(Yb,Xs) ).

% list_update_code(2)
tff(fact_3032_list__update__code_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,Yb: vEBT_VEBT] : list_u1324408373059187874T_VEBT(cons_VEBT_VEBT(Xb,Xs),zero_zero_nat,Yb) = cons_VEBT_VEBT(Yb,Xs) ).

% list_update_code(2)
tff(fact_3033_power__Suc,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_Suc
tff(fact_3034_power__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_Suc
tff(fact_3035_power__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_Suc
tff(fact_3036_power__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_Suc
tff(fact_3037_power__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb)) ).

% power_Suc
tff(fact_3038_power__Suc2,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),A) ).

% power_Suc2
tff(fact_3039_power__Suc2,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) ).

% power_Suc2
tff(fact_3040_power__Suc2,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),A) ).

% power_Suc2
tff(fact_3041_power__Suc2,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) ).

% power_Suc2
tff(fact_3042_power__Suc2,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),A) ).

% power_Suc2
tff(fact_3043_set__update__subsetI,axiom,
    ! [Xs: list_complex,A4: set_complex,Xb: complex,I: nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),A4)
     => ( aa_set_complex_o(member_complex(Xb),A4)
       => aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(list_update_complex(Xs,I,Xb))),A4) ) ) ).

% set_update_subsetI
tff(fact_3044_set__update__subsetI,axiom,
    ! [Xs: list_real,A4: set_real,Xb: real,I: nat] :
      ( ord_less_eq_set_real(set_real2(Xs),A4)
     => ( member_real(Xb,A4)
       => ord_less_eq_set_real(set_real2(list_update_real(Xs,I,Xb)),A4) ) ) ).

% set_update_subsetI
tff(fact_3045_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A4: set_set_nat,Xb: set_nat,I: nat] :
      ( ord_le6893508408891458716et_nat(set_set_nat2(Xs),A4)
     => ( member_set_nat(Xb,A4)
       => ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I,Xb)),A4) ) ) ).

% set_update_subsetI
tff(fact_3046_set__update__subsetI,axiom,
    ! [Xs: list_nat,A4: set_nat,Xb: nat,I: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Xs)),A4)
     => ( aa_set_nat_o(member_nat(Xb),A4)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I,Xb))),A4) ) ) ).

% set_update_subsetI
tff(fact_3047_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A4: set_VEBT_VEBT,Xb: vEBT_VEBT,I: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A4)
     => ( member_VEBT_VEBT(Xb,A4)
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,Xb)),A4) ) ) ).

% set_update_subsetI
tff(fact_3048_set__update__subsetI,axiom,
    ! [Xs: list_int,A4: set_int,Xb: int,I: nat] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),A4)
     => ( aa_set_int_o(member_int(Xb),A4)
       => aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,Xb))),A4) ) ) ).

% set_update_subsetI
tff(fact_3049_mod__eqE,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2) )
     => ~ ! [D2: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C2),D2)) ) ).

% mod_eqE
tff(fact_3050_mod__eqE,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C2) = modulo364778990260209775nteger(B,C2) )
     => ~ ! [D2: code_integer] : B != plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),D2)) ) ).

% mod_eqE
tff(fact_3051_Suc__mult__less__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_3052_power__add,axiom,
    ! [A: complex,Mb: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_add
tff(fact_3053_power__add,axiom,
    ! [A: real,Mb: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_add
tff(fact_3054_power__add,axiom,
    ! [A: rat,Mb: nat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_add
tff(fact_3055_power__add,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_add
tff(fact_3056_power__add,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ).

% power_add
tff(fact_3057_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K2)) ) ) ).

% mult_less_mono1
tff(fact_3058_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),J)) ) ) ).

% mult_less_mono2
tff(fact_3059_nat__mult__less__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_3060_nat__mult__eq__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_3061_Suc__mult__le__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_3062_mult__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc
tff(fact_3063_mult__eq__self__implies__10,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
     => ( ( Nb = one_one_nat )
        | ( Mb = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_3064_div__times__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb)),Mb) ).

% div_times_less_eq_dividend
tff(fact_3065_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))),Mb) ).

% times_div_less_eq_dividend
tff(fact_3066_mod__eq__0D,axiom,
    ! [Mb: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),D) = zero_zero_nat )
     => ? [Q2: nat] : Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q2) ) ).

% mod_eq_0D
tff(fact_3067_nat__mod__eq__iff,axiom,
    ! [Xb: nat,Nb: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Yb),Nb) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Yb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_3068_power__odd__eq,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3069_power__odd__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3070_power__odd__eq,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3071_power__odd__eq,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3072_power__odd__eq,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3073_mult__le__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3074_mult__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3075_mult__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3076_mult__le__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3077_mult__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3078_mult__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3079_mult__left__less__imp__less,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3080_mult__left__less__imp__less,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3081_mult__left__less__imp__less,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3082_mult__left__less__imp__less,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3083_mult__strict__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3084_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3085_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3086_mult__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3087_mult__less__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3088_mult__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3089_mult__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3090_mult__right__less__imp__less,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3091_mult__right__less__imp__less,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3092_mult__right__less__imp__less,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3093_mult__right__less__imp__less,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3094_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3095_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3096_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3097_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3098_mult__less__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3099_mult__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3100_mult__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3101_mult__le__cancel__left__neg,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3102_mult__le__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3103_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3104_mult__le__cancel__left__pos,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3105_mult__le__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3106_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3107_mult__left__le__imp__le,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3108_mult__left__le__imp__le,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3109_mult__left__le__imp__le,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3110_mult__left__le__imp__le,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3111_mult__right__le__imp__le,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3112_mult__right__le__imp__le,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3113_mult__right__le__imp__le,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3114_mult__right__le__imp__le,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3115_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3116_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3117_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3118_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3119_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3120_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3121_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3122_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3123_mult__left__le__one__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_3124_mult__left__le__one__le,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Yb),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_3125_mult__left__le__one__le,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_3126_mult__right__le__one__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Yb)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_3127_mult__right__le__one__le,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xb),Yb)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_3128_mult__right__le__one__le,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Yb)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_3129_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_3130_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_3131_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_3132_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_3133_mult__left__le,axiom,
    ! [C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),A) ) ) ).

% mult_left_le
tff(fact_3134_mult__left__le,axiom,
    ! [C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),A) ) ) ).

% mult_left_le
tff(fact_3135_mult__left__le,axiom,
    ! [C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),A) ) ) ).

% mult_left_le
tff(fact_3136_mult__left__le,axiom,
    ! [C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),A) ) ) ).

% mult_left_le
tff(fact_3137_sum__squares__le__zero__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Yb))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Yb = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3138_sum__squares__le__zero__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Yb),Yb))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Yb = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3139_sum__squares__le__zero__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Yb))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Yb = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3140_sum__squares__ge__zero,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Yb))) ).

% sum_squares_ge_zero
tff(fact_3141_sum__squares__ge__zero,axiom,
    ! [Xb: rat,Yb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Yb),Yb))) ).

% sum_squares_ge_zero
tff(fact_3142_sum__squares__ge__zero,axiom,
    ! [Xb: int,Yb: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Yb))) ).

% sum_squares_ge_zero
tff(fact_3143_sum__squares__gt__zero__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Yb)))
    <=> ( ( Xb != zero_zero_real )
        | ( Yb != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3144_sum__squares__gt__zero__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Yb),Yb)))
    <=> ( ( Xb != zero_zero_rat )
        | ( Yb != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3145_sum__squares__gt__zero__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Yb)))
    <=> ( ( Xb != zero_zero_int )
        | ( Yb != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3146_not__sum__squares__lt__zero,axiom,
    ! [Xb: real,Yb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Yb))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_3147_not__sum__squares__lt__zero,axiom,
    ! [Xb: rat,Yb: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Yb),Yb))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_3148_not__sum__squares__lt__zero,axiom,
    ! [Xb: int,Yb: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Yb))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_3149_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3150_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3151_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3152_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C2),A)),aa_rat_rat(divide_divide_rat(C2),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3153_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_3154_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C2),A)),aa_rat_rat(divide_divide_rat(C2),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_3155_mult__imp__less__div__pos,axiom,
    ! [Yb: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Yb)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)) ) ) ).

% mult_imp_less_div_pos
tff(fact_3156_mult__imp__less__div__pos,axiom,
    ! [Yb: rat,Z: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Z),Yb)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% mult_imp_less_div_pos
tff(fact_3157_mult__imp__div__pos__less,axiom,
    ! [Yb: real,Xb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Yb))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_3158_mult__imp__div__pos__less,axiom,
    ! [Yb: rat,Xb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(times_times_rat(Z),Yb))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_3159_pos__less__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B) ) ) ).

% pos_less_divide_eq
tff(fact_3160_pos__less__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),B) ) ) ).

% pos_less_divide_eq
tff(fact_3161_pos__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) ) ) ).

% pos_divide_less_eq
tff(fact_3162_pos__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C2)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C2)) ) ) ).

% pos_divide_less_eq
tff(fact_3163_neg__less__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) ) ) ).

% neg_less_divide_eq
tff(fact_3164_neg__less__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C2)) ) ) ).

% neg_less_divide_eq
tff(fact_3165_neg__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B) ) ) ).

% neg_divide_less_eq
tff(fact_3166_neg__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C2)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),B) ) ) ).

% neg_divide_less_eq
tff(fact_3167_less__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_3168_less__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_3169_divide__less__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% divide_less_eq
tff(fact_3170_divide__less__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C2)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_3171_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C2: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = numera6690914467698888265omplex(W2) )
    <=> $ite(C2 != zero_zero_complex,B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C2),numera6690914467698888265omplex(W2) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3172_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) = aa_num_real(numeral_numeral_real,W2) )
    <=> $ite(C2 != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2),aa_num_real(numeral_numeral_real,W2) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3173_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C2) = aa_num_rat(numeral_numeral_rat,W2) )
    <=> $ite(C2 != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2),aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3174_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: complex,C2: complex] :
      ( ( numera6690914467698888265omplex(W2) = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C2) = B,numera6690914467698888265omplex(W2) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3175_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( ( aa_num_real(numeral_numeral_real,W2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) )
    <=> $ite(C2 != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2) = B,aa_num_real(numeral_numeral_real,W2) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3176_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W2) = aa_rat_rat(divide_divide_rat(B),C2) )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2) = B,aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3177_divide__add__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Yb: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Z)),Yb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Yb),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3178_divide__add__eq__iff,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),Yb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3179_divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Z)),Yb) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Yb),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3180_add__divide__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Yb: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(Yb),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xb),Z)),Yb)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3181_add__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Yb),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),Yb)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3182_add__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(divide_divide_rat(Yb),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Z)),Yb)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3183_add__num__frac,axiom,
    ! [Yb: complex,Z: complex,Xb: complex] :
      ( ( Yb != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(Xb),Yb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Z),Yb))),Yb) ) ) ).

% add_num_frac
tff(fact_3184_add__num__frac,axiom,
    ! [Yb: real,Z: real,Xb: real] :
      ( ( Yb != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Yb))),Yb) ) ) ).

% add_num_frac
tff(fact_3185_add__num__frac,axiom,
    ! [Yb: rat,Z: rat,Xb: rat] :
      ( ( Yb != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(Xb),Yb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Z),Yb))),Yb) ) ) ).

% add_num_frac
tff(fact_3186_add__frac__num,axiom,
    ! [Yb: complex,Xb: complex,Z: complex] :
      ( ( Yb != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Yb)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Z),Yb))),Yb) ) ) ).

% add_frac_num
tff(fact_3187_add__frac__num,axiom,
    ! [Yb: real,Xb: real,Z: real] :
      ( ( Yb != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Yb))),Yb) ) ) ).

% add_frac_num
tff(fact_3188_add__frac__num,axiom,
    ! [Yb: rat,Xb: rat,Z: rat] :
      ( ( Yb != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Yb)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Z),Yb))),Yb) ) ) ).

% add_frac_num
tff(fact_3189_add__frac__eq,axiom,
    ! [Yb: complex,Z: complex,Xb: complex,W2: complex] :
      ( ( Yb != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Yb)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xb),Z)),aa_complex_complex(times_times_complex(W2),Yb))),aa_complex_complex(times_times_complex(Yb),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3190_add__frac__eq,axiom,
    ! [Yb: real,Z: real,Xb: real,W2: real] :
      ( ( Yb != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3191_add__frac__eq,axiom,
    ! [Yb: rat,Z: rat,Xb: rat,W2: rat] :
      ( ( Yb != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Yb)),aa_rat_rat(divide_divide_rat(W2),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W2),Yb))),aa_rat_rat(times_times_rat(Yb),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3192_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = $ite(Z = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3193_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3194_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(plus_plus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3195_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = $ite(Z = zero_zero_complex,B,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3196_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,B,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3197_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3198_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2))),C2)) ).

% div_mult1_eq
tff(fact_3199_div__mult1__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2))),C2)) ).

% div_mult1_eq
tff(fact_3200_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C2)),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C2)),C2)) ).

% div_mult1_eq
tff(fact_3201_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ).

% cancel_div_mod_rules(2)
tff(fact_3202_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),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))),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) ).

% cancel_div_mod_rules(2)
tff(fact_3203_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)),C2) = plus_p5714425477246183910nteger(A,C2) ).

% cancel_div_mod_rules(2)
tff(fact_3204_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ).

% cancel_div_mod_rules(1)
tff(fact_3205_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) ).

% cancel_div_mod_rules(1)
tff(fact_3206_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)),C2) = plus_p5714425477246183910nteger(A,C2) ).

% cancel_div_mod_rules(1)
tff(fact_3207_mod__div__decomp,axiom,
    ! [A: nat,B: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ).

% mod_div_decomp
tff(fact_3208_mod__div__decomp,axiom,
    ! [A: int,B: int] : A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_div_decomp
tff(fact_3209_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) ).

% mod_div_decomp
tff(fact_3210_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% div_mult_mod_eq
tff(fact_3211_div__mult__mod__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% div_mult_mod_eq
tff(fact_3212_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) = A ).

% div_mult_mod_eq
tff(fact_3213_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_3214_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_3215_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_3216_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = A ).

% mod_mult_div_eq
tff(fact_3217_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = A ).

% mod_mult_div_eq
tff(fact_3218_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = A ).

% mod_mult_div_eq
tff(fact_3219_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% mult_div_mod_eq
tff(fact_3220_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),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)) = A ).

% mult_div_mod_eq
tff(fact_3221_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)) = A ).

% mult_div_mod_eq
tff(fact_3222_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_complex,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => aa_set_complex_o(member_complex(Xb),set_complex2(list_update_complex(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_3223_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => member_real(Xb,set_real2(list_update_real(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_3224_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_set_nat,Xb: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
     => member_set_nat(Xb,set_set_nat2(list_update_set_nat(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_3225_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => member_VEBT_VEBT(Xb,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_3226_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_list_int_nat(size_size_list_int,Xs))
     => aa_set_int_o(member_int(Xb),set_int2(list_update_int(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_3227_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_list_nat_nat(size_size_list_nat,Xs))
     => aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_3228_n__less__n__mult__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)) ) ) ).

% n_less_n_mult_m
tff(fact_3229_n__less__m__mult__n,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_3230_one__less__mult,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% one_less_mult
tff(fact_3231_nat__mult__le__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_3232_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,Xb) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_3233_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs))
     => ( ( list_update_int(Xs,I,Xb) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_3234_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( ( list_update_nat(Xs,I,Xb) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_3235_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),J) = $ite(I = J,Xb,nth_VEBT_VEBT(Xs,J)) ) ) ).

% nth_list_update
tff(fact_3236_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,Xb: int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),J) = $ite(I = J,Xb,aa_nat_int(nth_int(Xs),J)) ) ) ).

% nth_list_update
tff(fact_3237_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,Xb: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),J) = $ite(I = J,Xb,aa_nat_nat(nth_nat(Xs),J)) ) ) ).

% nth_list_update
tff(fact_3238_div__less__iff__less__mult,axiom,
    ! [Q3: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Q3)),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ) ) ).

% div_less_iff_less_mult
tff(fact_3239_nat__mult__div__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_3240_mod__eq__nat1E,axiom,
    ! [Mb: nat,Q3: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ~ ! [S: nat] : Mb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S)) ) ) ).

% mod_eq_nat1E
tff(fact_3241_mod__eq__nat2E,axiom,
    ! [Mb: nat,Q3: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ~ ! [S: nat] : Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S)) ) ) ).

% mod_eq_nat2E
tff(fact_3242_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Nb: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Yb),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
       => ? [Q2: nat] : Xb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Yb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ) ) ).

% nat_mod_eq_lemma
tff(fact_3243_mod__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Q3))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% mod_mult2_eq
tff(fact_3244_div__mod__decomp,axiom,
    ! [A4: nat,Nb: nat] : A4 = 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,A4),Nb)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),Nb)) ).

% div_mod_decomp
tff(fact_3245_field__le__mult__one__interval,axiom,
    ! [Xb: real,Yb: real] :
      ( ! [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),Xb)),Yb) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ).

% field_le_mult_one_interval
tff(fact_3246_field__le__mult__one__interval,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ! [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z2),Xb)),Yb) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ).

% field_le_mult_one_interval
tff(fact_3247_mult__less__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),C2)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3248_mult__less__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3249_mult__less__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3250_mult__less__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3251_mult__less__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(times_times_rat(B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3252_mult__less__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3253_mult__less__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),C2)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3254_mult__less__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3255_mult__less__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3256_mult__less__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3257_mult__less__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(times_times_rat(C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3258_mult__less__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3259_mult__le__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),C2)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3260_mult__le__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3261_mult__le__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3262_mult__le__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3263_mult__le__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(times_times_rat(B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3264_mult__le__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3265_mult__le__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),C2)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3266_mult__le__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3267_mult__le__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3268_mult__le__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3269_mult__le__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(times_times_rat(C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3270_mult__le__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3271_divide__le__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% divide_le_eq
tff(fact_3272_divide__le__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C2)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_3273_le__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_3274_le__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_3275_divide__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B)) ) ) ) ).

% divide_left_mono
tff(fact_3276_divide__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C2),A)),aa_rat_rat(divide_divide_rat(C2),B)) ) ) ) ).

% divide_left_mono
tff(fact_3277_neg__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3278_neg__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C2)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3279_neg__le__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) ) ) ).

% neg_le_divide_eq
tff(fact_3280_neg__le__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C2)) ) ) ).

% neg_le_divide_eq
tff(fact_3281_pos__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) ) ) ).

% pos_divide_le_eq
tff(fact_3282_pos__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C2)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C2)) ) ) ).

% pos_divide_le_eq
tff(fact_3283_pos__le__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3284_pos__le__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3285_mult__imp__div__pos__le,axiom,
    ! [Yb: real,Xb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Yb))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3286_mult__imp__div__pos__le,axiom,
    ! [Yb: rat,Xb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(times_times_rat(Z),Yb))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3287_mult__imp__le__div__pos,axiom,
    ! [Yb: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Yb)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3288_mult__imp__le__div__pos,axiom,
    ! [Yb: rat,Z: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Yb)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3289_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3290_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C2),A)),aa_rat_rat(divide_divide_rat(C2),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3291_convex__bound__le,axiom,
    ! [Xb: real,A: real,Yb: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Yb))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3292_convex__bound__le,axiom,
    ! [Xb: rat,A: rat,Yb: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),Xb)),aa_rat_rat(times_times_rat(V),Yb))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3293_convex__bound__le,axiom,
    ! [Xb: int,A: int,Yb: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Yb))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3294_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3295_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(divide_divide_rat(B),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3296_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),aa_num_real(numeral_numeral_real,W2))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3297_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C2)),aa_num_rat(numeral_numeral_rat,W2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3298_power__Suc__less,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_3299_power__Suc__less,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_3300_power__Suc__less,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_3301_power__Suc__less,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb))),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_3302_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2
tff(fact_3303_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_3304_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2
tff(fact_3305_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_3306_mult__2,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2
tff(fact_3307_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2_right
tff(fact_3308_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_3309_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2_right
tff(fact_3310_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_3311_mult__2__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2_right
tff(fact_3312_left__add__twice,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3313_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3314_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3315_left__add__twice,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3316_left__add__twice,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,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3317_Suc__double__not__eq__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_double_not_eq_double
tff(fact_3318_double__not__eq__Suc__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_3319_div__nat__eqI,axiom,
    ! [Nb: nat,Q3: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q3)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_3320_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Q3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3321_dividend__less__times__div,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)))) ) ).

% dividend_less_times_div
tff(fact_3322_dividend__less__div__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_3323_split__div,axiom,
    ! [P2: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P2,zero_zero_nat) )
        & ( ( Nb != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I3)),J3) )
               => aa_nat_o(P2,I3) ) ) ) ) ) ).

% split_div
tff(fact_3324_split__mod,axiom,
    ! [P2: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P2,Mb) )
        & ( ( Nb != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I3)),J3) )
               => aa_nat_o(P2,J3) ) ) ) ) ) ).

% split_mod
tff(fact_3325_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_3326_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xc: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,Xaa),Xba),Xc) = Yb )
     => ( Yb = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xba),Xaa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xba),aa_nat_nat(aa_nat_fun_nat_nat(Xb,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_3327_convex__bound__lt,axiom,
    ! [Xb: real,A: real,Yb: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Yb))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3328_convex__bound__lt,axiom,
    ! [Xb: rat,A: rat,Yb: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),Xb)),aa_rat_rat(times_times_rat(V),Yb))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3329_convex__bound__lt,axiom,
    ! [Xb: int,A: int,Yb: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Yb))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3330_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),aa_num_real(numeral_numeral_real,W2))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3331_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C2)),aa_num_rat(numeral_numeral_rat,W2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3332_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3333_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(divide_divide_rat(B),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3334_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
     => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),modulo364778990260209775nteger(divide6298287555418463151nteger(A,B),C2)),modulo364778990260209775nteger(A,B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3335_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3336_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3337_sum__squares__bound,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Yb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_3338_sum__squares__bound,axiom,
    ! [Xb: rat,Yb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Yb)),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_3339_split__div_H,axiom,
    ! [P2: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
          & aa_nat_o(P2,zero_zero_nat) )
        | ? [Q4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q4)),Mb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q4)))
            & aa_nat_o(P2,Q4) ) ) ) ).

% split_div'
tff(fact_3340_Suc__times__mod__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))),Mb) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_3341_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3342_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3343_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B)
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3344_power2__sum,axiom,
    ! [Xb: complex,Yb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)),Yb)) ).

% power2_sum
tff(fact_3345_power2__sum,axiom,
    ! [Xb: real,Yb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Yb)) ).

% power2_sum
tff(fact_3346_power2__sum,axiom,
    ! [Xb: rat,Yb: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Yb)) ).

% power2_sum
tff(fact_3347_power2__sum,axiom,
    ! [Xb: nat,Yb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb)),Yb)) ).

% power2_sum
tff(fact_3348_power2__sum,axiom,
    ! [Xb: int,Yb: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)),Yb)) ).

% power2_sum
tff(fact_3349_zero__le__even__power_H,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_3350_zero__le__even__power_H,axiom,
    ! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_3351_zero__le__even__power_H,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_3352_nat__bit__induct,axiom,
    ! [P2: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P2,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(P2,N2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ) )
       => ( ! [N2: nat] :
              ( aa_nat_o(P2,N2)
             => aa_nat_o(P2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) )
         => aa_nat_o(P2,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_3353_arith__geo__mean,axiom,
    ! [U: real,Xb: real,Yb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Yb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_3354_arith__geo__mean,axiom,
    ! [U: rat,Xb: rat,Yb: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(Xb),Yb) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),Yb)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_3355_triangle__def,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_3356_realpow__pos__nth2,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ? [R3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,Nb)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_3357_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3358_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3359_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3360_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3361_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3362_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3363_odd__power__less__zero,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_3364_odd__power__less__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_3365_odd__power__less__zero,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_3366_mod__double__modulus,axiom,
    ! [Mb: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Mb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
       => ( ( modulo364778990260209775nteger(Xb,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)) = modulo364778990260209775nteger(Xb,Mb) )
          | ( modulo364778990260209775nteger(Xb,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(Xb,Mb),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_3367_mod__double__modulus,axiom,
    ! [Mb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_3368_mod__double__modulus,axiom,
    ! [Mb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Mb) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_3369_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)))
         => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3370_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3371_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3372_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X3),Nb) = A )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),Nb) = A ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_3373_set__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3374_set__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3375_max_Obounded__iff,axiom,
    ! [B: extended_enat,C2: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(ord_ma741700101516333627d_enat(B,C2),A)
    <=> ( ord_le2932123472753598470d_enat(B,A)
        & ord_le2932123472753598470d_enat(C2,A) ) ) ).

% max.bounded_iff
tff(fact_3376_max_Obounded__iff,axiom,
    ! [B: code_integer,C2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C2)),A)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A) ) ) ).

% max.bounded_iff
tff(fact_3377_max_Obounded__iff,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C2)),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% max.bounded_iff
tff(fact_3378_max_Obounded__iff,axiom,
    ! [B: num,C2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B),C2)),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A) ) ) ).

% max.bounded_iff
tff(fact_3379_max_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% max.bounded_iff
tff(fact_3380_max_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% max.bounded_iff
tff(fact_3381_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(A,B)
     => ( ord_ma741700101516333627d_enat(A,B) = B ) ) ).

% max.absorb2
tff(fact_3382_max_Oabsorb2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( ord_max_Code_integer(A,B) = B ) ) ).

% max.absorb2
tff(fact_3383_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb2
tff(fact_3384_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_num(ord_max_num(A),B) = B ) ) ).

% max.absorb2
tff(fact_3385_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_3386_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb2
tff(fact_3387_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
     => ( ord_ma741700101516333627d_enat(A,B) = A ) ) ).

% max.absorb1
tff(fact_3388_max_Oabsorb1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( ord_max_Code_integer(A,B) = A ) ) ).

% max.absorb1
tff(fact_3389_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb1
tff(fact_3390_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_num(ord_max_num(A),B) = A ) ) ).

% max.absorb1
tff(fact_3391_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_3392_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb1
tff(fact_3393_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2)))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys2),Nb) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2)))) ) ) ).

% product_nth
tff(fact_3394_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_list_int_nat(size_size_list_int,Ys2)))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys2),Nb) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_list_int_nat(size_size_list_int,Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_list_int_nat(size_size_list_int,Ys2)))) ) ) ).

% product_nth
tff(fact_3395_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_list_nat_nat(size_size_list_nat,Ys2)))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys2),Nb) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_list_nat_nat(size_size_list_nat,Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_list_nat_nat(size_size_list_nat,Ys2)))) ) ) ).

% product_nth
tff(fact_3396_product__nth,axiom,
    ! [Nb: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_int_nat(size_size_list_int,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2)))
     => ( nth_Pr3474266648193625910T_VEBT(produc662631939642741121T_VEBT(Xs,Ys2),Nb) = produc3329399203697025711T_VEBT(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2)))) ) ) ).

% product_nth
tff(fact_3397_product__nth,axiom,
    ! [Nb: nat,Xs: list_int,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_int_nat(size_size_list_int,Xs)),aa_list_int_nat(size_size_list_int,Ys2)))
     => ( nth_Pr4439495888332055232nt_int(product_int_int(Xs,Ys2),Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_list_int_nat(size_size_list_int,Ys2)))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_list_int_nat(size_size_list_int,Ys2)))) ) ) ).

% product_nth
tff(fact_3398_product__nth,axiom,
    ! [Nb: nat,Xs: list_int,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_int_nat(size_size_list_int,Xs)),aa_list_nat_nat(size_size_list_nat,Ys2)))
     => ( nth_Pr8617346907841251940nt_nat(product_int_nat(Xs,Ys2),Nb) = product_Pair_int_nat(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_list_nat_nat(size_size_list_nat,Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_list_nat_nat(size_size_list_nat,Ys2)))) ) ) ).

% product_nth
tff(fact_3399_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,Ys2: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2)))
     => ( nth_Pr744662078594809490T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys2),Nb) = produc599794634098209291T_VEBT(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2)))) ) ) ).

% product_nth
tff(fact_3400_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_int_nat(size_size_list_int,Ys2)))
     => ( nth_Pr3440142176431000676at_int(product_nat_int(Xs,Ys2),Nb) = product_Pair_nat_int(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_list_int_nat(size_size_list_int,Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_list_int_nat(size_size_list_int,Ys2)))) ) ) ).

% product_nth
tff(fact_3401_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_nat_nat(size_size_list_nat,Ys2)))
     => ( nth_Pr7617993195940197384at_nat(product_nat_nat(Xs,Ys2),Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_list_nat_nat(size_size_list_nat,Ys2)))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_list_nat_nat(size_size_list_nat,Ys2)))) ) ) ).

% product_nth
tff(fact_3402_product__nth,axiom,
    ! [Nb: nat,Xs: list_f8851413139013959296e_term,Ys2: list_P5578671422887162913nteger] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s502031211236231956e_term(Xs)),size_s6527587076508218509nteger(Ys2)))
     => ( nth_Pr2854691652959048409nteger(produc4042828510177926308nteger(Xs,Ys2),Nb) = produc7264323995540650962nteger(nth_fu2298125310669257513e_term(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6527587076508218509nteger(Ys2))),nth_Pr2304437835452373666nteger(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6527587076508218509nteger(Ys2)))) ) ) ).

% product_nth
tff(fact_3403_unset__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_3404_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_3405_set__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_3406_set__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_3407_set__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_3408_flip__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_3409_flip__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_3410_flip__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_3411_unset__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_3412_unset__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_3413_unset__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_3414_i0__less,axiom,
    ! [Nb: extended_enat] :
      ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,Nb)
    <=> ( Nb != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_3415_max__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : ord_ma741700101516333627d_enat(Q3,zero_z5237406670263579293d_enat) = Q3 ).

% max_enat_simps(2)
tff(fact_3416_max__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,Q3) = Q3 ).

% max_enat_simps(3)
tff(fact_3417_not__real__square__gt__zero,axiom,
    ! [Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb))
    <=> ( Xb = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_3418_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_3419_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% set_bit_nonnegative_int_iff
tff(fact_3420_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_3421_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_3422_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_3423_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_3424_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] : size_s7466405169056248089T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2)) ).

% length_product
tff(fact_3425_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_int] : size_s3661962791536183091BT_int(produc7292646706713671643BT_int(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_list_int_nat(size_size_list_int,Ys2)) ).

% length_product
tff(fact_3426_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_nat] : size_s6152045936467909847BT_nat(produc7295137177222721919BT_nat(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_list_nat_nat(size_size_list_nat,Ys2)) ).

% length_product
tff(fact_3427_length__product,axiom,
    ! [Xs: list_int,Ys2: list_VEBT_VEBT] : size_s6639371672096860321T_VEBT(produc662631939642741121T_VEBT(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_int_nat(size_size_list_int,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2)) ).

% length_product
tff(fact_3428_length__product,axiom,
    ! [Xs: list_int,Ys2: list_int] : size_s5157815400016825771nt_int(product_int_int(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_int_nat(size_size_list_int,Xs)),aa_list_int_nat(size_size_list_int,Ys2)) ).

% length_product
tff(fact_3429_length__product,axiom,
    ! [Xs: list_int,Ys2: list_nat] : size_s7647898544948552527nt_nat(product_int_nat(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_int_nat(size_size_list_int,Xs)),aa_list_nat_nat(size_size_list_nat,Ys2)) ).

% length_product
tff(fact_3430_length__product,axiom,
    ! [Xs: list_nat,Ys2: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2)) ).

% length_product
tff(fact_3431_length__product,axiom,
    ! [Xs: list_nat,Ys2: list_int] : size_s2970893825323803983at_int(product_nat_int(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_int_nat(size_size_list_int,Ys2)) ).

% length_product
tff(fact_3432_length__product,axiom,
    ! [Xs: list_nat,Ys2: list_nat] : size_s5460976970255530739at_nat(product_nat_nat(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_nat_nat(size_size_list_nat,Ys2)) ).

% length_product
tff(fact_3433_div__mod__decomp__int,axiom,
    ! [A4: int,Nb: int] : A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),Nb)),Nb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),Nb)) ).

% div_mod_decomp_int
tff(fact_3434_int__gr__induct,axiom,
    ! [K2: int,I: int,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),I)
     => ( aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),I2)
             => ( aa_int_o(P2,I2)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P2,I) ) ) ) ).

% int_gr_induct
tff(fact_3435_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
tff(fact_3436_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_3437_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_3438_not__iless0,axiom,
    ! [Nb: extended_enat] : ~ ord_le72135733267957522d_enat(Nb,zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_3439_enat__0__less__mult__iff,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,times_7803423173614009249d_enat(Mb,Nb))
    <=> ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,Mb)
        & ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_3440_imult__is__0,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( times_7803423173614009249d_enat(Mb,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Mb = zero_z5237406670263579293d_enat )
        | ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_3441_int__distrib_I2_J,axiom,
    ! [W2: int,Z12: int,Z23: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z12),Z23)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z12)),aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z23)) ).

% int_distrib(2)
tff(fact_3442_int__distrib_I1_J,axiom,
    ! [Z12: int,Z23: int,W2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z12),Z23)),W2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z12),W2)),aa_int_int(aa_int_fun_int_int(times_times_int,Z23),W2)) ).

% int_distrib(1)
tff(fact_3443_iadd__is__0,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(Mb,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Mb = zero_z5237406670263579293d_enat )
        & ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_3444_zmod__eq__0__iff,axiom,
    ! [Mb: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),D) = zero_zero_int )
    <=> ? [Q4: int] : Mb = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q4) ) ).

% zmod_eq_0_iff
tff(fact_3445_zmod__eq__0D,axiom,
    ! [Mb: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),D) = zero_zero_int )
     => ? [Q2: int] : Mb = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q2) ) ).

% zmod_eq_0D
tff(fact_3446_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_3447_times__int__code_I1_J,axiom,
    ! [K2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K2),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_3448_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),J)) ) ) ).

% zmult_zless_mono2
tff(fact_3449_pos__zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
      <=> ( ( Mb = one_one_int )
          & ( Nb = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_3450_unset__bit__less__eq,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K2)),K2) ).

% unset_bit_less_eq
tff(fact_3451_set__bit__greater__eq,axiom,
    ! [K2: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K2)) ).

% set_bit_greater_eq
tff(fact_3452_zdiv__zmult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_3453_split__zdiv,axiom,
    ! [P2: fun_int_o,Nb: int,K2: int] :
      ( aa_int_o(P2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K2))
    <=> ( ( ( K2 = zero_zero_int )
         => aa_int_o(P2,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K2)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I3)),J3) ) )
             => aa_int_o(P2,I3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I3)),J3) ) )
             => aa_int_o(P2,I3) ) ) ) ) ).

% split_zdiv
tff(fact_3454_q__pos__lemma,axiom,
    ! [B2: int,Q5: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_3455_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q3: 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),Q3)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3456_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q3: 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),Q3)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3457_zdiv__mono2__lemma,axiom,
    ! [B: int,Q3: int,R2: int,B2: 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),Q3)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3458_incr__mult__lemma,axiom,
    ! [D: int,P2: fun_int_o,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int] :
            ( aa_int_o(P2,X3)
           => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
         => ! [X: int] :
              ( aa_int_o(P2,X)
             => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3459_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q3: int,R2: int,B2: 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),Q3)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R4)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q3) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3460_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,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),Q3)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q3) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3461_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,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),Q3)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3462_split__pos__lemma,axiom,
    ! [K2: int,P2: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(P2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K2))
      <=> ! [I3: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K2)
              & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I3)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P2,I3),J3) ) ) ) ).

% split_pos_lemma
tff(fact_3463_split__neg__lemma,axiom,
    ! [K2: int,P2: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K2))
      <=> ! [I3: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
              & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I3)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P2,I3),J3) ) ) ) ).

% split_neg_lemma
tff(fact_3464_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q3: 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),Q3)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_3465_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q3: 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),Q3)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_3466_split__zmod,axiom,
    ! [P2: fun_int_o,Nb: int,K2: int] :
      ( aa_int_o(P2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K2))
    <=> ( ( ( K2 = zero_zero_int )
         => aa_int_o(P2,Nb) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K2)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I3)),J3) ) )
             => aa_int_o(P2,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I3)),J3) ) )
             => aa_int_o(P2,J3) ) ) ) ) ).

% split_zmod
tff(fact_3467_zmod__zmult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_3468_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_3469_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_3470_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_3471_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C2: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_3472_max_OcoboundedI2,axiom,
    ! [C2: extended_enat,B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(C2,B)
     => ord_le2932123472753598470d_enat(C2,ord_ma741700101516333627d_enat(A,B)) ) ).

% max.coboundedI2
tff(fact_3473_max_OcoboundedI2,axiom,
    ! [C2: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),ord_max_Code_integer(A,B)) ) ).

% max.coboundedI2
tff(fact_3474_max_OcoboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.coboundedI2
tff(fact_3475_max_OcoboundedI2,axiom,
    ! [C2: num,B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),aa_num_num(ord_max_num(A),B)) ) ).

% max.coboundedI2
tff(fact_3476_max_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI2
tff(fact_3477_max_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI2
tff(fact_3478_max_OcoboundedI1,axiom,
    ! [C2: extended_enat,A: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(C2,A)
     => ord_le2932123472753598470d_enat(C2,ord_ma741700101516333627d_enat(A,B)) ) ).

% max.coboundedI1
tff(fact_3479_max_OcoboundedI1,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),ord_max_Code_integer(A,B)) ) ).

% max.coboundedI1
tff(fact_3480_max_OcoboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.coboundedI1
tff(fact_3481_max_OcoboundedI1,axiom,
    ! [C2: num,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),aa_num_num(ord_max_num(A),B)) ) ).

% max.coboundedI1
tff(fact_3482_max_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI1
tff(fact_3483_max_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI1
tff(fact_3484_max_Oabsorb__iff2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(A,B)
    <=> ( ord_ma741700101516333627d_enat(A,B) = B ) ) ).

% max.absorb_iff2
tff(fact_3485_max_Oabsorb__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
    <=> ( ord_max_Code_integer(A,B) = B ) ) ).

% max.absorb_iff2
tff(fact_3486_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3487_max_Oabsorb__iff2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_num(ord_max_num(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3488_max_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3489_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3490_max_Oabsorb__iff1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
    <=> ( ord_ma741700101516333627d_enat(A,B) = A ) ) ).

% max.absorb_iff1
tff(fact_3491_max_Oabsorb__iff1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
    <=> ( ord_max_Code_integer(A,B) = A ) ) ).

% max.absorb_iff1
tff(fact_3492_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3493_max_Oabsorb__iff1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_num(ord_max_num(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3494_max_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3495_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3496_le__max__iff__disj,axiom,
    ! [Z: extended_enat,Xb: extended_enat,Yb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Z,ord_ma741700101516333627d_enat(Xb,Yb))
    <=> ( ord_le2932123472753598470d_enat(Z,Xb)
        | ord_le2932123472753598470d_enat(Z,Yb) ) ) ).

% le_max_iff_disj
tff(fact_3497_le__max__iff__disj,axiom,
    ! [Z: code_integer,Xb: code_integer,Yb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),ord_max_Code_integer(Xb,Yb))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),Xb)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),Yb) ) ) ).

% le_max_iff_disj
tff(fact_3498_le__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(ord_max_rat(Xb),Yb))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Yb) ) ) ).

% le_max_iff_disj
tff(fact_3499_le__max__iff__disj,axiom,
    ! [Z: num,Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),aa_num_num(ord_max_num(Xb),Yb))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Yb) ) ) ).

% le_max_iff_disj
tff(fact_3500_le__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Yb) ) ) ).

% le_max_iff_disj
tff(fact_3501_le__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Yb) ) ) ).

% le_max_iff_disj
tff(fact_3502_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : ord_le2932123472753598470d_enat(B,ord_ma741700101516333627d_enat(A,B)) ).

% max.cobounded2
tff(fact_3503_max_Ocobounded2,axiom,
    ! [B: code_integer,A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),ord_max_Code_integer(A,B)) ).

% max.cobounded2
tff(fact_3504_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(ord_max_rat(A),B)) ).

% max.cobounded2
tff(fact_3505_max_Ocobounded2,axiom,
    ! [B: num,A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),aa_num_num(ord_max_num(A),B)) ).

% max.cobounded2
tff(fact_3506_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded2
tff(fact_3507_max_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded2
tff(fact_3508_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : ord_le2932123472753598470d_enat(A,ord_ma741700101516333627d_enat(A,B)) ).

% max.cobounded1
tff(fact_3509_max_Ocobounded1,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),ord_max_Code_integer(A,B)) ).

% max.cobounded1
tff(fact_3510_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(ord_max_rat(A),B)) ).

% max.cobounded1
tff(fact_3511_max_Ocobounded1,axiom,
    ! [A: num,B: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(ord_max_num(A),B)) ).

% max.cobounded1
tff(fact_3512_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded1
tff(fact_3513_max_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded1
tff(fact_3514_max_Oorder__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
    <=> ( A = ord_ma741700101516333627d_enat(A,B) ) ) ).

% max.order_iff
tff(fact_3515_max_Oorder__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
    <=> ( A = ord_max_Code_integer(A,B) ) ) ).

% max.order_iff
tff(fact_3516_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.order_iff
tff(fact_3517_max_Oorder__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( A = aa_num_num(ord_max_num(A),B) ) ) ).

% max.order_iff
tff(fact_3518_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.order_iff
tff(fact_3519_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.order_iff
tff(fact_3520_max_OboundedI,axiom,
    ! [B: extended_enat,A: extended_enat,C2: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
     => ( ord_le2932123472753598470d_enat(C2,A)
       => ord_le2932123472753598470d_enat(ord_ma741700101516333627d_enat(B,C2),A) ) ) ).

% max.boundedI
tff(fact_3521_max_OboundedI,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C2)),A) ) ) ).

% max.boundedI
tff(fact_3522_max_OboundedI,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C2)),A) ) ) ).

% max.boundedI
tff(fact_3523_max_OboundedI,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B),C2)),A) ) ) ).

% max.boundedI
tff(fact_3524_max_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A) ) ) ).

% max.boundedI
tff(fact_3525_max_OboundedI,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A) ) ) ).

% max.boundedI
tff(fact_3526_max_OboundedE,axiom,
    ! [B: extended_enat,C2: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(ord_ma741700101516333627d_enat(B,C2),A)
     => ~ ( ord_le2932123472753598470d_enat(B,A)
         => ~ ord_le2932123472753598470d_enat(C2,A) ) ) ).

% max.boundedE
tff(fact_3527_max_OboundedE,axiom,
    ! [B: code_integer,C2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C2)),A)
     => ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
         => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A) ) ) ).

% max.boundedE
tff(fact_3528_max_OboundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C2)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% max.boundedE
tff(fact_3529_max_OboundedE,axiom,
    ! [B: num,C2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B),C2)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A) ) ) ).

% max.boundedE
tff(fact_3530_max_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% max.boundedE
tff(fact_3531_max_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% max.boundedE
tff(fact_3532_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A = ord_ma741700101516333627d_enat(A,B) )
     => ord_le2932123472753598470d_enat(B,A) ) ).

% max.orderI
tff(fact_3533_max_OorderI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = ord_max_Code_integer(A,B) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ).

% max.orderI
tff(fact_3534_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(ord_max_rat(A),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% max.orderI
tff(fact_3535_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A = aa_num_num(ord_max_num(A),B) )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% max.orderI
tff(fact_3536_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% max.orderI
tff(fact_3537_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% max.orderI
tff(fact_3538_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
     => ( A = ord_ma741700101516333627d_enat(A,B) ) ) ).

% max.orderE
tff(fact_3539_max_OorderE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( A = ord_max_Code_integer(A,B) ) ) ).

% max.orderE
tff(fact_3540_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.orderE
tff(fact_3541_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( A = aa_num_num(ord_max_num(A),B) ) ) ).

% max.orderE
tff(fact_3542_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_3543_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.orderE
tff(fact_3544_max_Omono,axiom,
    ! [C2: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(C2,A)
     => ( ord_le2932123472753598470d_enat(D,B)
       => ord_le2932123472753598470d_enat(ord_ma741700101516333627d_enat(C2,D),ord_ma741700101516333627d_enat(A,B)) ) ) ).

% max.mono
tff(fact_3545_max_Omono,axiom,
    ! [C2: code_integer,A: code_integer,D: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,D),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(C2,D)),ord_max_Code_integer(A,B)) ) ) ).

% max.mono
tff(fact_3546_max_Omono,axiom,
    ! [C2: rat,A: rat,D: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(C2),D)),aa_rat_rat(ord_max_rat(A),B)) ) ) ).

% max.mono
tff(fact_3547_max_Omono,axiom,
    ! [C2: num,A: num,D: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,D),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(C2),D)),aa_num_num(ord_max_num(A),B)) ) ) ).

% max.mono
tff(fact_3548_max_Omono,axiom,
    ! [C2: nat,A: nat,D: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C2),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ) ).

% max.mono
tff(fact_3549_max_Omono,axiom,
    ! [C2: int,A: int,D: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C2),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ) ).

% max.mono
tff(fact_3550_mult__le__cancel__iff2,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ).

% mult_le_cancel_iff2
tff(fact_3551_mult__le__cancel__iff2,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Xb)),aa_rat_rat(times_times_rat(Z),Yb))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ) ).

% mult_le_cancel_iff2
tff(fact_3552_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Yb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ) ).

% mult_le_cancel_iff2
tff(fact_3553_mult__le__cancel__iff1,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ).

% mult_le_cancel_iff1
tff(fact_3554_mult__le__cancel__iff1,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Yb),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ) ).

% mult_le_cancel_iff1
tff(fact_3555_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ) ).

% mult_le_cancel_iff1
tff(fact_3556_even__succ__mod__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3557_even__succ__mod__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3558_even__succ__mod__exp,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3559_length__Cons,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] : aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,cons_VEBT_VEBT(Xb,Xs)) = aa_nat_nat(suc,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)) ).

% length_Cons
tff(fact_3560_length__Cons,axiom,
    ! [Xb: int,Xs: list_int] : aa_list_int_nat(size_size_list_int,cons_int(Xb,Xs)) = aa_nat_nat(suc,aa_list_int_nat(size_size_list_int,Xs)) ).

% length_Cons
tff(fact_3561_length__Cons,axiom,
    ! [Xb: nat,Xs: list_nat] : aa_list_nat_nat(size_size_list_nat,cons_nat(Xb,Xs)) = aa_nat_nat(suc,aa_list_nat_nat(size_size_list_nat,Xs)) ).

% length_Cons
tff(fact_3562_even__succ__div__exp,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_3563_even__succ__div__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_3564_even__succ__div__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_3565_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_3566_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_3567_nat__dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),one_one_nat)
    <=> ( Mb = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_3568_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
tff(fact_3569_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,zero_zero_complex),A)
    <=> ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
tff(fact_3570_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
    <=> ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_3571_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
    <=> ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_3572_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_3573_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_3574_dvd__0__right,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),zero_z3403309356797280102nteger) ).

% dvd_0_right
tff(fact_3575_dvd__0__right,axiom,
    ! [A: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),zero_zero_complex) ).

% dvd_0_right
tff(fact_3576_dvd__0__right,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),zero_zero_real) ).

% dvd_0_right
tff(fact_3577_dvd__0__right,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),zero_zero_rat) ).

% dvd_0_right
tff(fact_3578_dvd__0__right,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% dvd_0_right
tff(fact_3579_dvd__0__right,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),zero_zero_int) ).

% dvd_0_right
tff(fact_3580_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3581_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3582_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3583_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3584_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3585_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3586_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3587_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3588_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3589_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3590_dvd__1__left,axiom,
    ! [K2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K2) ).

% dvd_1_left
tff(fact_3591_dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Mb = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_3592_nat__mult__dvd__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
    <=> ( ( K2 = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_3593_signed__take__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_3594_dvd__mult__cancel__left,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B))
    <=> ( ( C2 = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3595_dvd__mult__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(C2),B))
    <=> ( ( C2 = zero_zero_complex )
        | aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3596_dvd__mult__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
    <=> ( ( C2 = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3597_dvd__mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))
    <=> ( ( C2 = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3598_dvd__mult__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( C2 = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3599_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2))
    <=> ( ( C2 = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3600_dvd__mult__cancel__right,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(B),C2))
    <=> ( ( C2 = zero_zero_complex )
        | aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3601_dvd__mult__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
    <=> ( ( C2 = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3602_dvd__mult__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2))
    <=> ( ( C2 = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3603_dvd__mult__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( C2 = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3604_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C2) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3605_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C2) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3606_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C2) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3607_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C2) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3608_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C2) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3609_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C2) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3610_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3611_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3612_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(C2),A)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3613_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3614_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3615_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3616_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3617_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(C2),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3618_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3619_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3620_div__add,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C2),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C2),B)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C2) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C2),divide6298287555418463151nteger(B,C2)) ) ) ) ).

% div_add
tff(fact_3621_div__add,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2)) ) ) ) ).

% div_add
tff(fact_3622_div__add,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2)) ) ) ) ).

% div_add
tff(fact_3623_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_3624_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_3625_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_3626_signed__take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_3627_even__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc
tff(fact_3628_even__Suc__Suc__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_3629_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_3630_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_3631_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_3632_even__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_3633_even__add,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_3634_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_3635_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_3636_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_3637_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_3638_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_3639_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_3640_zero__le__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3641_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3642_zero__le__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3643_power__less__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_3644_power__less__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_3645_power__less__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_3646_power__less__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3647_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3648_power__less__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3649_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,one_one_Code_integer))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_3650_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_3651_even__plus__one__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_3652_zero__less__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3653_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3654_zero__less__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3655_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3656_even__succ__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3657_even__succ__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3658_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_3659_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_3660_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_3661_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3662_even__succ__div__2,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3663_even__succ__div__2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3664_even__power,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_3665_even__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_3666_even__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A),Nb))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_3667_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3668_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3669_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3670_power__le__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3671_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3672_power__le__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3673_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
tff(fact_3674_dvd__0__left,axiom,
    ! [A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,zero_zero_complex),A)
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
tff(fact_3675_dvd__0__left,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_3676_dvd__0__left,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_3677_dvd__0__left,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_3678_dvd__0__left,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_3679_dvd__field__iff,axiom,
    ! [A: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B)
    <=> ( ( A = zero_zero_complex )
       => ( B = zero_zero_complex ) ) ) ).

% dvd_field_iff
tff(fact_3680_dvd__field__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
    <=> ( ( A = zero_zero_real )
       => ( B = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_3681_dvd__field__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
    <=> ( ( A = zero_zero_rat )
       => ( B = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_3682_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C2) ) ) ).

% dvd_add_right_iff
tff(fact_3683_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C2) ) ) ).

% dvd_add_right_iff
tff(fact_3684_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C2) ) ) ).

% dvd_add_right_iff
tff(fact_3685_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C2) ) ) ).

% dvd_add_right_iff
tff(fact_3686_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C2) ) ) ).

% dvd_add_right_iff
tff(fact_3687_dvd__add__left__iff,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3688_dvd__add__left__iff,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3689_dvd__add__left__iff,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3690_dvd__add__left__iff,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3691_dvd__add__left__iff,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3692_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C2)) ) ) ).

% dvd_add
tff(fact_3693_dvd__add,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C2)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ) ) ).

% dvd_add
tff(fact_3694_dvd__add,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C2)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C2)) ) ) ).

% dvd_add
tff(fact_3695_dvd__add,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ) ) ).

% dvd_add
tff(fact_3696_dvd__add,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ) ) ).

% dvd_add
tff(fact_3697_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_3698_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat)
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_3699_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_3700_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_3701_gcd__nat_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% gcd_nat.extremum
tff(fact_3702_zdvd__mult__cancel,axiom,
    ! [K2: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),Nb))
     => ( ( K2 != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_3703_zdvd__mono,axiom,
    ! [K2: int,Mb: int,Ta: int] :
      ( ( K2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Ta)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),Ta)) ) ) ).

% zdvd_mono
tff(fact_3704_signed__take__bit__add,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K2)),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),L))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ).

% signed_take_bit_add
tff(fact_3705_zdvd__reduce,axiom,
    ! [K2: int,Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K2),Mb)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),Nb) ) ).

% zdvd_reduce
tff(fact_3706_zdvd__period,axiom,
    ! [A: int,D: int,Xb: int,Ta: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),D)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Ta))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,C2),D))),Ta)) ) ) ).

% zdvd_period
tff(fact_3707_not__is__unit__0,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% not_is_unit_0
tff(fact_3708_not__is__unit__0,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),one_one_nat) ).

% not_is_unit_0
tff(fact_3709_not__is__unit__0,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),one_one_int) ).

% not_is_unit_0
tff(fact_3710_pinf_I9_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z2: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S3))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S3)) ) ) ).

% pinf(9)
tff(fact_3711_pinf_I9_J,axiom,
    ! [D: real,S3: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S3))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S3)) ) ) ).

% pinf(9)
tff(fact_3712_pinf_I9_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S3))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S3)) ) ) ).

% pinf(9)
tff(fact_3713_pinf_I9_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S3)) ) ) ).

% pinf(9)
tff(fact_3714_pinf_I9_J,axiom,
    ! [D: int,S3: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S3))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S3)) ) ) ).

% pinf(9)
tff(fact_3715_pinf_I10_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z2: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S3))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S3)) ) ) ).

% pinf(10)
tff(fact_3716_pinf_I10_J,axiom,
    ! [D: real,S3: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S3))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S3)) ) ) ).

% pinf(10)
tff(fact_3717_pinf_I10_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S3))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S3)) ) ) ).

% pinf(10)
tff(fact_3718_pinf_I10_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S3))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S3)) ) ) ).

% pinf(10)
tff(fact_3719_pinf_I10_J,axiom,
    ! [D: int,S3: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S3))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S3)) ) ) ).

% pinf(10)
tff(fact_3720_minf_I9_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z2: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S3))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S3)) ) ) ).

% minf(9)
tff(fact_3721_minf_I9_J,axiom,
    ! [D: real,S3: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S3))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S3)) ) ) ).

% minf(9)
tff(fact_3722_minf_I9_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S3))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S3)) ) ) ).

% minf(9)
tff(fact_3723_minf_I9_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S3)) ) ) ).

% minf(9)
tff(fact_3724_minf_I9_J,axiom,
    ! [D: int,S3: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S3))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S3)) ) ) ).

% minf(9)
tff(fact_3725_minf_I10_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z2: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z2)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S3))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S3)) ) ) ).

% minf(10)
tff(fact_3726_minf_I10_J,axiom,
    ! [D: real,S3: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S3))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S3)) ) ) ).

% minf(10)
tff(fact_3727_minf_I10_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S3))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S3)) ) ) ).

% minf(10)
tff(fact_3728_minf_I10_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S3))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S3)) ) ) ).

% minf(10)
tff(fact_3729_minf_I10_J,axiom,
    ! [D: int,S3: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S3))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S3)) ) ) ).

% minf(10)
tff(fact_3730_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3731_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A)
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3732_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3733_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3734_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3735_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3736_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C2),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C2) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C2),divide6298287555418463151nteger(B,C2)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3737_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3738_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3739_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C2),B)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C2) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C2),divide6298287555418463151nteger(B,C2)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3740_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3741_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3742_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3743_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3744_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3745_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3746_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3747_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
    <=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3748_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3749_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3750_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3751_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3752_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3753_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3754_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3755_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3756_power__le__dvd,axiom,
    ! [A: code_integer,Nb: nat,B: code_integer,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3757_power__le__dvd,axiom,
    ! [A: nat,Nb: nat,B: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3758_power__le__dvd,axiom,
    ! [A: real,Nb: nat,B: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3759_power__le__dvd,axiom,
    ! [A: int,Nb: nat,B: int,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3760_power__le__dvd,axiom,
    ! [A: complex,Nb: nat,B: complex,Mb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3761_dvd__power__le,axiom,
    ! [Xb: code_integer,Yb: code_integer,Nb: nat,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Yb),Mb)) ) ) ).

% dvd_power_le
tff(fact_3762_dvd__power__le,axiom,
    ! [Xb: nat,Yb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Yb),Mb)) ) ) ).

% dvd_power_le
tff(fact_3763_dvd__power__le,axiom,
    ! [Xb: real,Yb: real,Nb: nat,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),Mb)) ) ) ).

% dvd_power_le
tff(fact_3764_dvd__power__le,axiom,
    ! [Xb: int,Yb: int,Nb: nat,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Yb),Mb)) ) ) ).

% dvd_power_le
tff(fact_3765_dvd__power__le,axiom,
    ! [Xb: complex,Yb: complex,Nb: nat,Mb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Yb),Mb)) ) ) ).

% dvd_power_le
tff(fact_3766_dvd__pos__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb) ) ) ).

% dvd_pos_nat
tff(fact_3767_nat__dvd__not__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ) ).

% nat_dvd_not_less
tff(fact_3768_zdvd__antisym__nonneg,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Nb)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),Mb)
           => ( Mb = Nb ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_3769_zdvd__not__zless,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Mb),Nb)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),Mb) ) ) ).

% zdvd_not_zless
tff(fact_3770_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),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)),D2) )
        | ( 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)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_3771_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Yb)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Yb)),D) ) )
         => ? [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),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),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Y)),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)),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) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_3772_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) ) ) ).

% unit_dvdE
tff(fact_3773_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) ) ) ).

% unit_dvdE
tff(fact_3774_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C) ) ) ).

% unit_dvdE
tff(fact_3775_unity__coeff__ex,axiom,
    ! [P2: fun_Code_integer_o,L: code_integer] :
      ( ? [X4: code_integer] : aa_Code_integer_o(P2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X4))
    <=> ? [X4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),plus_p5714425477246183910nteger(X4,zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P2,X4) ) ) ).

% unity_coeff_ex
tff(fact_3776_unity__coeff__ex,axiom,
    ! [P2: fun_complex_o,L: complex] :
      ( ? [X4: complex] : aa_complex_o(P2,aa_complex_complex(times_times_complex(L),X4))
    <=> ? [X4: complex] :
          ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,L),aa_complex_complex(plus_plus_complex(X4),zero_zero_complex))
          & aa_complex_o(P2,X4) ) ) ).

% unity_coeff_ex
tff(fact_3777_unity__coeff__ex,axiom,
    ! [P2: fun_real_o,L: real] :
      ( ? [X4: real] : aa_real_o(P2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X4))
    <=> ? [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),zero_zero_real))
          & aa_real_o(P2,X4) ) ) ).

% unity_coeff_ex
tff(fact_3778_unity__coeff__ex,axiom,
    ! [P2: fun_rat_o,L: rat] :
      ( ? [X4: rat] : aa_rat_o(P2,aa_rat_rat(times_times_rat(L),X4))
    <=> ? [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(plus_plus_rat(X4),zero_zero_rat))
          & aa_rat_o(P2,X4) ) ) ).

% unity_coeff_ex
tff(fact_3779_unity__coeff__ex,axiom,
    ! [P2: fun_nat_o,L: nat] :
      ( ? [X4: nat] : aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X4))
    <=> ? [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),zero_zero_nat))
          & aa_nat_o(P2,X4) ) ) ).

% unity_coeff_ex
tff(fact_3780_unity__coeff__ex,axiom,
    ! [P2: fun_int_o,L: int] :
      ( ? [X4: int] : aa_int_o(P2,aa_int_int(aa_int_fun_int_int(times_times_int,L),X4))
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),zero_zero_int))
          & aa_int_o(P2,X4) ) ) ).

% unity_coeff_ex
tff(fact_3781_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
       => ( ( divide6298287555418463151nteger(B,A) = C2 )
        <=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3782_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C2 )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3783_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = C2 )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3784_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C2)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3785_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3786_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3787_dvd__div__iff__mult,axiom,
    ! [C2: code_integer,B: code_integer,A: code_integer] :
      ( ( C2 != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C2),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C2))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3788_dvd__div__iff__mult,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( C2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3789_dvd__div__iff__mult,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3790_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C2 != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C2),D)
           => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(D,C2) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3791_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C2: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C2 != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C2) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3792_dvd__div__div__eq__mult,axiom,
    ! [A: int,C2: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C2 != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),D)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D),C2) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3793_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3794_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3795_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3796_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_3797_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_3798_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_3799_is__unit__power__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3800_is__unit__power__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3801_is__unit__power__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3802_dvd__imp__le,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ) ).

% dvd_imp_le
tff(fact_3803_dvd__mult__cancel,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_3804_nat__mult__dvd__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_3805_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D2: nat,X3: nat,Y: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),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)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_3806_zdvd__imp__le,axiom,
    ! [Z: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Nb) ) ) ).

% zdvd_imp_le
tff(fact_3807_mod__greater__zero__iff__not__dvd,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_3808_even__zero,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_z3403309356797280102nteger) ).

% even_zero
tff(fact_3809_even__zero,axiom,
    aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat) ).

% even_zero
tff(fact_3810_even__zero,axiom,
    aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int) ).

% even_zero
tff(fact_3811_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3812_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3813_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3814_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3815_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3816_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3817_is__unitE,axiom,
    ! [A: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B3: code_integer] :
              ( ( B3 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B3),one_one_Code_integer)
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B3 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B3) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B3) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C2,A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3818_is__unitE,axiom,
    ! [A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B3),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B3 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B3) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B3) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3819_is__unitE,axiom,
    ! [A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B3),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B3 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B3) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B3) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C2),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3820_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ) ).

% odd_even_add
tff(fact_3821_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% odd_even_add
tff(fact_3822_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% odd_even_add
tff(fact_3823_dvd__power__iff,axiom,
    ! [Xb: code_integer,Mb: nat,Nb: nat] :
      ( ( Xb != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_3824_dvd__power__iff,axiom,
    ! [Xb: nat,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_3825_dvd__power__iff,axiom,
    ! [Xb: int,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Mb)),aa_nat_int(power_power_int(Xb),Nb))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_3826_dvd__power,axiom,
    ! [Nb: nat,Xb: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)) ) ).

% dvd_power
tff(fact_3827_dvd__power,axiom,
    ! [Nb: nat,Xb: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_nat_rat(power_power_rat(Xb),Nb)) ) ).

% dvd_power
tff(fact_3828_dvd__power,axiom,
    ! [Nb: nat,Xb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ).

% dvd_power
tff(fact_3829_dvd__power,axiom,
    ! [Nb: nat,Xb: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) ) ).

% dvd_power
tff(fact_3830_dvd__power,axiom,
    ! [Nb: nat,Xb: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_nat_int(power_power_int(Xb),Nb)) ) ).

% dvd_power
tff(fact_3831_dvd__power,axiom,
    ! [Nb: nat,Xb: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_complex ) )
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),aa_nat_complex(power_power_complex(Xb),Nb)) ) ).

% dvd_power
tff(fact_3832_div2__even__ext__nat,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Yb) )
       => ( Xb = Yb ) ) ) ).

% div2_even_ext_nat
tff(fact_3833_dvd__mult__cancel1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_3834_dvd__mult__cancel2,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_3835_power__dvd__imp__le,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_3836_mod__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_3837_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3838_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3839_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3840_power__mono__odd,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3841_power__mono__odd,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3842_power__mono__odd,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3843_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% odd_pos
tff(fact_3844_dvd__power__iff__le,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_3845_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Mb,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3846_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3847_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Mb),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3848_even__set__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Mb,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3849_even__set__bit__iff,axiom,
    ! [Mb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3850_even__set__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Mb),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3851_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Mb,A))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3852_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),A))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3853_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Mb),A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3854_oddE,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ~ ! [B3: code_integer] : A != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B3),one_one_Code_integer) ) ).

% oddE
tff(fact_3855_oddE,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3)),one_one_nat) ) ).

% oddE
tff(fact_3856_oddE,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)),one_one_int) ) ).

% oddE
tff(fact_3857_parity__cases,axiom,
    ! [A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_3858_parity__cases,axiom,
    ! [A: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_3859_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
         => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_3860_mod2__eq__if,axiom,
    ! [A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_3861_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_3862_mod2__eq__if,axiom,
    ! [A: code_integer] :
      modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_3863_zero__le__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq
tff(fact_3864_zero__le__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_3865_zero__le__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_3866_zero__le__odd__power,axiom,
    ! [Nb: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ).

% zero_le_odd_power
tff(fact_3867_zero__le__odd__power,axiom,
    ! [Nb: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_3868_zero__le__odd__power,axiom,
    ! [Nb: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_3869_zero__le__even__power,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_le_even_power
tff(fact_3870_zero__le__even__power,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_le_even_power
tff(fact_3871_zero__le__even__power,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% zero_le_even_power
tff(fact_3872_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_3873_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_3874_even__set__encode__iff,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A4))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A4) ) ) ).

% even_set_encode_iff
tff(fact_3875_zero__less__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq
tff(fact_3876_zero__less__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_3877_zero__less__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_3878_power__le__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3879_power__le__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3880_power__le__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3881_mult__less__iff1,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ).

% mult_less_iff1
tff(fact_3882_mult__less__iff1,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Yb),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb) ) ) ).

% mult_less_iff1
tff(fact_3883_mult__less__iff1,axiom,
    ! [Z: int,Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb) ) ) ).

% mult_less_iff1
tff(fact_3884_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_3885_concat__bit__Suc,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(suc,Nb)),K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),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(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_3886_flip__bit__0,axiom,
    ! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_3887_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_3888_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_3889_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Yb )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ( Yb != $ite(
                  Xaa = zero_zero_nat,
                  vEBT_Leaf($true,(B3)),
                  $ite(Xaa = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B3))) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts2,S) )
             => ( Yb != vEBT_Node(Info2,zero_zero_nat,Ts2,S) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
               => ( Yb != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) ) )
           => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                 => ( Yb != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                   => ( Yb != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma ) ) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_3890_set__decode__Suc,axiom,
    ! [Nb: nat,Xb: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Nb)),nat_set_decode(Xb))
    <=> aa_set_nat_o(member_nat(Nb),nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_3891_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3892_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3893_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3894_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_3895_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_3896_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect_nat(vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_3897_finite__Collect__conjI,axiom,
    ! [P2: fun_real_o,Q: fun_real_o] :
      ( ( finite_finite_real(collect_real(P2))
        | finite_finite_real(collect_real(Q)) )
     => finite_finite_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_af(P2),Q))) ) ).

% finite_Collect_conjI
tff(fact_3898_finite__Collect__conjI,axiom,
    ! [P2: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ( finite8100373058378681591st_nat(collect_list_nat(P2))
        | finite8100373058378681591st_nat(collect_list_nat(Q)) )
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ag(P2),Q))) ) ).

% finite_Collect_conjI
tff(fact_3899_finite__Collect__conjI,axiom,
    ! [P2: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ( finite1152437895449049373et_nat(collect_set_nat(P2))
        | finite1152437895449049373et_nat(collect_set_nat(Q)) )
     => finite1152437895449049373et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ah(P2),Q))) ) ).

% finite_Collect_conjI
tff(fact_3900_finite__Collect__conjI,axiom,
    ! [P2: fun_nat_o,Q: fun_nat_o] :
      ( ( aa_set_nat_o(finite_finite_nat,collect_nat(P2))
        | aa_set_nat_o(finite_finite_nat,collect_nat(Q)) )
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ai(P2),Q))) ) ).

% finite_Collect_conjI
tff(fact_3901_finite__Collect__conjI,axiom,
    ! [P2: fun_int_o,Q: fun_int_o] :
      ( ( aa_set_int_o(finite_finite_int,collect_int(P2))
        | aa_set_int_o(finite_finite_int,collect_int(Q)) )
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_aj(P2),Q))) ) ).

% finite_Collect_conjI
tff(fact_3902_finite__Collect__conjI,axiom,
    ! [P2: fun_complex_o,Q: fun_complex_o] :
      ( ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P2))
        | aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q)) )
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ak(P2),Q))) ) ).

% finite_Collect_conjI
tff(fact_3903_finite__Collect__conjI,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P2))
        | aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Q)) )
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_al(P2),Q))) ) ).

% finite_Collect_conjI
tff(fact_3904_finite__Collect__disjI,axiom,
    ! [P2: fun_real_o,Q: fun_real_o] :
      ( finite_finite_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_am(P2),Q)))
    <=> ( finite_finite_real(collect_real(P2))
        & finite_finite_real(collect_real(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_3905_finite__Collect__disjI,axiom,
    ! [P2: fun_list_nat_o,Q: fun_list_nat_o] :
      ( finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_an(P2),Q)))
    <=> ( finite8100373058378681591st_nat(collect_list_nat(P2))
        & finite8100373058378681591st_nat(collect_list_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_3906_finite__Collect__disjI,axiom,
    ! [P2: fun_set_nat_o,Q: fun_set_nat_o] :
      ( finite1152437895449049373et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ao(P2),Q)))
    <=> ( finite1152437895449049373et_nat(collect_set_nat(P2))
        & finite1152437895449049373et_nat(collect_set_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_3907_finite__Collect__disjI,axiom,
    ! [P2: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ap(P2),Q)))
    <=> ( aa_set_nat_o(finite_finite_nat,collect_nat(P2))
        & aa_set_nat_o(finite_finite_nat,collect_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_3908_finite__Collect__disjI,axiom,
    ! [P2: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_aq(P2),Q)))
    <=> ( aa_set_int_o(finite_finite_int,collect_int(P2))
        & aa_set_int_o(finite_finite_int,collect_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_3909_finite__Collect__disjI,axiom,
    ! [P2: fun_complex_o,Q: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ar(P2),Q)))
    <=> ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P2))
        & aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_3910_finite__Collect__disjI,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_as(P2),Q)))
    <=> ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P2))
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_3911_finite__interval__int1,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_at(A),B))) ).

% finite_interval_int1
tff(fact_3912_finite__interval__int4,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_au(A),B))) ).

% finite_interval_int4
tff(fact_3913_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_3914_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_3915_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% diff_self
tff(fact_3916_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_3917_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_3918_diff__0__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_0_right
tff(fact_3919_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_3920_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_3921_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_3922_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_3923_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_3924_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_zero
tff(fact_3925_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_3926_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_3927_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3928_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3929_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3930_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_3931_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_3932_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel
tff(fact_3933_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel
tff(fact_3934_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_3935_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ).

% diff_add_cancel
tff(fact_3936_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ).

% diff_add_cancel
tff(fact_3937_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_3938_add__diff__cancel__left,axiom,
    ! [C2: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_left
tff(fact_3939_add__diff__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_left
tff(fact_3940_add__diff__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_3941_add__diff__cancel__left,axiom,
    ! [C2: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_3942_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3943_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3944_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_3945_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_3946_add__diff__cancel__right,axiom,
    ! [A: real,C2: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_right
tff(fact_3947_add__diff__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_right
tff(fact_3948_add__diff__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_3949_add__diff__cancel__right,axiom,
    ! [A: int,C2: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_3950_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3951_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3952_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_3953_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_3954_diff__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% diff_Suc_Suc
tff(fact_3955_Suc__diff__diff,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),K2) ).

% Suc_diff_diff
tff(fact_3956_diff__self__eq__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Mb) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_3957_diff__0__eq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_3958_diff__diff__cancel,axiom,
    ! [I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_3959_diff__diff__left,axiom,
    ! [I: nat,J: nat,K2: 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)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) ).

% diff_diff_left
tff(fact_3960_of__bool__less__eq__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P2))),zero_n2052037380579107095ol_rat((Q)))
    <=> ( (P2)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_3961_of__bool__less__eq__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_n2687167440665602831ol_nat((P2))),zero_n2687167440665602831ol_nat((Q)))
    <=> ( (P2)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_3962_of__bool__less__eq__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P2))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
    <=> ( (P2)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_3963_of__bool__eq_I1_J,axiom,
    zero_n1201886186963655149omplex($false) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_3964_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real($false) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_3965_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat($false) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_3966_of__bool__eq_I1_J,axiom,
    zero_n2687167440665602831ol_nat($false) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_3967_of__bool__eq_I1_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$false) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_3968_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n1201886186963655149omplex((P2)) = zero_zero_complex )
    <=> ~ (P2) ) ).

% of_bool_eq_0_iff
tff(fact_3969_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n3304061248610475627l_real((P2)) = zero_zero_real )
    <=> ~ (P2) ) ).

% of_bool_eq_0_iff
tff(fact_3970_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n2052037380579107095ol_rat((P2)) = zero_zero_rat )
    <=> ~ (P2) ) ).

% of_bool_eq_0_iff
tff(fact_3971_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n2687167440665602831ol_nat((P2)) = zero_zero_nat )
    <=> ~ (P2) ) ).

% of_bool_eq_0_iff
tff(fact_3972_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P2)) = zero_zero_int )
    <=> ~ (P2) ) ).

% of_bool_eq_0_iff
tff(fact_3973_concat__bit__0,axiom,
    ! [K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(zero_zero_nat),K2),L) = L ).

% concat_bit_0
tff(fact_3974_finite__Collect__subsets,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => finite1152437895449049373et_nat(collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_av,A4))) ) ).

% finite_Collect_subsets
tff(fact_3975_finite__Collect__subsets,axiom,
    ! [A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_aw(A4))) ) ).

% finite_Collect_subsets
tff(fact_3976_finite__Collect__subsets,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => finite9047747110432174090at_nat(collec5514110066124741708at_nat(aTP_Lamm_ax(A4))) ) ).

% finite_Collect_subsets
tff(fact_3977_finite__Collect__subsets,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_ay(A4))) ) ).

% finite_Collect_subsets
tff(fact_3978_finite__interval__int3,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_az(A),B))) ).

% finite_interval_int3
tff(fact_3979_finite__interval__int2,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ba(A),B))) ).

% finite_interval_int2
tff(fact_3980_finite__Collect__less__nat,axiom,
    ! [K2: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bb,K2))) ).

% finite_Collect_less_nat
tff(fact_3981_finite__Collect__le__nat,axiom,
    ! [K2: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bc,K2))) ).

% finite_Collect_le_nat
tff(fact_3982_set__decode__inverse,axiom,
    ! [Nb: nat] : aa_set_nat_nat(nat_set_encode,nat_set_decode(Nb)) = Nb ).

% set_decode_inverse
tff(fact_3983_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3984_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3985_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3986_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3987_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3988_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3989_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3990_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3991_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3992_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3993_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3994_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3995_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3996_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3997_diff__numeral__special_I9_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),one_one_complex) = zero_zero_complex ).

% diff_numeral_special(9)
tff(fact_3998_diff__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_3999_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_4000_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_4001_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_4002_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real((P2)))
    <=> (P2) ) ).

% zero_less_of_bool_iff
tff(fact_4003_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P2)))
    <=> (P2) ) ).

% zero_less_of_bool_iff
tff(fact_4004_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_n2687167440665602831ol_nat((P2)))
    <=> (P2) ) ).

% zero_less_of_bool_iff
tff(fact_4005_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> (P2) ) ).

% zero_less_of_bool_iff
tff(fact_4006_zero__less__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% zero_less_diff
tff(fact_4007_diff__is__0__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% diff_is_0_eq
tff(fact_4008_diff__is__0__eq_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_4009_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K2) ) ) ).

% Nat.add_diff_assoc
tff(fact_4010_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),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)),K2) ) ) ).

% Nat.add_diff_assoc2
tff(fact_4011_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_4012_diff__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),one_one_nat) = Nb ).

% diff_Suc_1
tff(fact_4013_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = zero_n2687167440665602831ol_nat(Nb != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_4014_concat__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_4015_concat__bit__negative__iff,axiom,
    ! [Nb: nat,K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_4016_set__decode__zero,axiom,
    nat_set_decode(zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_4017_set__encode__inverse,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( nat_set_decode(aa_set_nat_nat(nat_set_encode,A4)) = A4 ) ) ).

% set_encode_inverse
tff(fact_4018_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% Suc_pred
tff(fact_4019_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2))),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,K2),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_4020_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_4021_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% zle_diff1_eq
tff(fact_4022_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) = Nb ) ) ).

% Suc_diff_1
tff(fact_4023_nth__Cons__numeral,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,V: num] : nth_VEBT_VEBT(cons_VEBT_VEBT(Xb,Xs),aa_num_nat(numeral_numeral_nat,V)) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat)) ).

% nth_Cons_numeral
tff(fact_4024_nth__Cons__numeral,axiom,
    ! [Xb: int,Xs: list_int,V: num] : aa_nat_int(nth_int(cons_int(Xb,Xs)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat)) ).

% nth_Cons_numeral
tff(fact_4025_nth__Cons__numeral,axiom,
    ! [Xb: nat,Xs: list_nat,V: num] : aa_nat_nat(nth_nat(cons_nat(Xb,Xs)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat)) ).

% nth_Cons_numeral
tff(fact_4026_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ).

% even_diff
tff(fact_4027_even__diff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ).

% even_diff
tff(fact_4028_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_n2687167440665602831ol_nat((B))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_4029_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_o_int(zero_n2684676970156552555ol_int,(B))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_4030_nth__Cons__pos,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( nth_VEBT_VEBT(cons_VEBT_VEBT(Xb,Xs),Nb) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_4031_nth__Cons__pos,axiom,
    ! [Nb: nat,Xb: int,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(nth_int(cons_int(Xb,Xs)),Nb) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_4032_nth__Cons__pos,axiom,
    ! [Nb: nat,Xb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(nth_nat(cons_nat(Xb,Xs)),Nb) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_4033_set__decode__0,axiom,
    ! [Xb: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),nat_set_decode(Xb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb) ) ).

% set_decode_0
tff(fact_4034_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% odd_Suc_minus_one
tff(fact_4035_even__diff__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ) ) ).

% even_diff_nat
tff(fact_4036_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb),one_one_Code_integer))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4037_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4038_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4039_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4040_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4041_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4042_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4043_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_4044_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_4045_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_4046_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_bd,I))) ) ).

% finite_divisors_int
tff(fact_4047_small__lazy_H_Ocases,axiom,
    ! [Xb: product_prod_int_int] :
      ~ ! [D2: int,I2: int] : Xb != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D2),I2) ).

% small_lazy'.cases
tff(fact_4048_exhaustive__int_H_Ocases,axiom,
    ! [Xb: produc4788131091073775919nt_int] :
      ~ ! [F2: fun_in5730379987311621695e_term,D2: int,I2: int] : Xb != produc4729913460834754785nt_int(F2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D2),I2)) ).

% exhaustive_int'.cases
tff(fact_4049_full__exhaustive__int_H_Ocases,axiom,
    ! [Xb: produc3043004577601930755nt_int] :
      ~ ! [F2: fun_Pr7763680713840462443e_term,D2: int,I2: int] : Xb != produc7320969046608631221nt_int(F2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D2),I2)) ).

% full_exhaustive_int'.cases
tff(fact_4050_zdvd__zdiffD,axiom,
    ! [K2: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Mb),Nb))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),Mb) ) ) ).

% zdvd_zdiffD
tff(fact_4051_dvd__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% dvd_antisym
tff(fact_4052_dvd__diff__nat,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_4053_subset__CollectI,axiom,
    ! [B4: set_complex,A4: set_complex,Q: fun_complex_o,P2: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),B4)
           => ( aa_complex_o(Q,X3)
             => aa_complex_o(P2,X3) ) )
       => aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(B4),Q))),collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_4054_subset__CollectI,axiom,
    ! [B4: set_real,A4: set_real,Q: fun_real_o,P2: fun_real_o] :
      ( ord_less_eq_set_real(B4,A4)
     => ( ! [X3: real] :
            ( member_real(X3,B4)
           => ( aa_real_o(Q,X3)
             => aa_real_o(P2,X3) ) )
       => ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bf(B4),Q)),collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bf(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_4055_subset__CollectI,axiom,
    ! [B4: set_list_nat,A4: set_list_nat,Q: fun_list_nat_o,P2: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(B4,A4)
     => ( ! [X3: list_nat] :
            ( member_list_nat(X3,B4)
           => ( aa_list_nat_o(Q,X3)
             => aa_list_nat_o(P2,X3) ) )
       => ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bg(B4),Q)),collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bg(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_4056_subset__CollectI,axiom,
    ! [B4: set_set_nat,A4: set_set_nat,Q: fun_set_nat_o,P2: fun_set_nat_o] :
      ( ord_le6893508408891458716et_nat(B4,A4)
     => ( ! [X3: set_nat] :
            ( member_set_nat(X3,B4)
           => ( aa_set_nat_o(Q,X3)
             => aa_set_nat_o(P2,X3) ) )
       => ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bh(B4),Q)),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bh(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_4057_subset__CollectI,axiom,
    ! [B4: set_nat,A4: set_nat,Q: fun_nat_o,P2: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),B4)
           => ( aa_nat_o(Q,X3)
             => aa_nat_o(P2,X3) ) )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bi(B4),Q))),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bi(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_4058_subset__CollectI,axiom,
    ! [B4: set_int,A4: set_int,Q: fun_int_o,P2: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),B4)
           => ( aa_int_o(Q,X3)
             => aa_int_o(P2,X3) ) )
       => aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bj(B4),Q))),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bj(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_4059_subset__Collect__iff,axiom,
    ! [B4: set_complex,A4: set_complex,P2: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(A4),P2)))
      <=> ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),B4)
           => aa_complex_o(P2,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_4060_subset__Collect__iff,axiom,
    ! [B4: set_real,A4: set_real,P2: fun_real_o] :
      ( ord_less_eq_set_real(B4,A4)
     => ( ord_less_eq_set_real(B4,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bf(A4),P2)))
      <=> ! [X4: real] :
            ( member_real(X4,B4)
           => aa_real_o(P2,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_4061_subset__Collect__iff,axiom,
    ! [B4: set_list_nat,A4: set_list_nat,P2: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(B4,A4)
     => ( ord_le6045566169113846134st_nat(B4,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bg(A4),P2)))
      <=> ! [X4: list_nat] :
            ( member_list_nat(X4,B4)
           => aa_list_nat_o(P2,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_4062_subset__Collect__iff,axiom,
    ! [B4: set_set_nat,A4: set_set_nat,P2: fun_set_nat_o] :
      ( ord_le6893508408891458716et_nat(B4,A4)
     => ( ord_le6893508408891458716et_nat(B4,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bh(A4),P2)))
      <=> ! [X4: set_nat] :
            ( member_set_nat(X4,B4)
           => aa_set_nat_o(P2,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_4063_subset__Collect__iff,axiom,
    ! [B4: set_nat,A4: set_nat,P2: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bi(A4),P2)))
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B4)
           => aa_nat_o(P2,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_4064_subset__Collect__iff,axiom,
    ! [B4: set_int,A4: set_int,P2: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bj(A4),P2)))
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),B4)
           => aa_int_o(P2,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_4065_Collect__subset,axiom,
    ! [A4: set_complex,P2: fun_complex_o] : aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(A4),P2))),A4) ).

% Collect_subset
tff(fact_4066_Collect__subset,axiom,
    ! [A4: set_real,P2: fun_real_o] : ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bf(A4),P2)),A4) ).

% Collect_subset
tff(fact_4067_Collect__subset,axiom,
    ! [A4: set_list_nat,P2: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bg(A4),P2)),A4) ).

% Collect_subset
tff(fact_4068_Collect__subset,axiom,
    ! [A4: set_set_nat,P2: fun_set_nat_o] : ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bh(A4),P2)),A4) ).

% Collect_subset
tff(fact_4069_Collect__subset,axiom,
    ! [A4: set_nat,P2: fun_nat_o] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bi(A4),P2))),A4) ).

% Collect_subset
tff(fact_4070_Collect__subset,axiom,
    ! [A4: set_int,P2: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bj(A4),P2))),A4) ).

% Collect_subset
tff(fact_4071_less__eq__set__def,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
    <=> ord_le7644322562001619171plex_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4)) ) ).

% less_eq_set_def
tff(fact_4072_less__eq__set__def,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ord_less_eq_set_real(A4,B4)
    <=> ord_le3605727316730466277real_o(aTP_Lamm_aa(A4),aTP_Lamm_aa(B4)) ) ).

% less_eq_set_def
tff(fact_4073_less__eq__set__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A4,B4)
    <=> ord_le6468474429079055883_nat_o(aTP_Lamm_ac(A4),aTP_Lamm_ac(B4)) ) ).

% less_eq_set_def
tff(fact_4074_less__eq__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
    <=> ord_le7240017405068453697_nat_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4)) ) ).

% less_eq_set_def
tff(fact_4075_less__eq__set__def,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_ae(A4),aTP_Lamm_ae(B4)) ) ).

% less_eq_set_def
tff(fact_4076_prop__restrict,axiom,
    ! [Xb: complex,Z4: set_complex,X5: set_complex,P2: fun_complex_o] :
      ( aa_set_complex_o(member_complex(Xb),Z4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(Z4),collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(X5),P2)))
       => aa_complex_o(P2,Xb) ) ) ).

% prop_restrict
tff(fact_4077_prop__restrict,axiom,
    ! [Xb: real,Z4: set_real,X5: set_real,P2: fun_real_o] :
      ( member_real(Xb,Z4)
     => ( ord_less_eq_set_real(Z4,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bf(X5),P2)))
       => aa_real_o(P2,Xb) ) ) ).

% prop_restrict
tff(fact_4078_prop__restrict,axiom,
    ! [Xb: list_nat,Z4: set_list_nat,X5: set_list_nat,P2: fun_list_nat_o] :
      ( member_list_nat(Xb,Z4)
     => ( ord_le6045566169113846134st_nat(Z4,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bg(X5),P2)))
       => aa_list_nat_o(P2,Xb) ) ) ).

% prop_restrict
tff(fact_4079_prop__restrict,axiom,
    ! [Xb: set_nat,Z4: set_set_nat,X5: set_set_nat,P2: fun_set_nat_o] :
      ( member_set_nat(Xb,Z4)
     => ( ord_le6893508408891458716et_nat(Z4,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bh(X5),P2)))
       => aa_set_nat_o(P2,Xb) ) ) ).

% prop_restrict
tff(fact_4080_prop__restrict,axiom,
    ! [Xb: nat,Z4: set_nat,X5: set_nat,P2: fun_nat_o] :
      ( aa_set_nat_o(member_nat(Xb),Z4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z4),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bi(X5),P2)))
       => aa_nat_o(P2,Xb) ) ) ).

% prop_restrict
tff(fact_4081_prop__restrict,axiom,
    ! [Xb: int,Z4: set_int,X5: set_int,P2: fun_int_o] :
      ( aa_set_int_o(member_int(Xb),Z4)
     => ( aa_set_int_o(ord_less_eq_set_int(Z4),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bj(X5),P2)))
       => aa_int_o(P2,Xb) ) ) ).

% prop_restrict
tff(fact_4082_Collect__restrict,axiom,
    ! [X5: set_complex,P2: fun_complex_o] : aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(X5),P2))),X5) ).

% Collect_restrict
tff(fact_4083_Collect__restrict,axiom,
    ! [X5: set_real,P2: fun_real_o] : ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bf(X5),P2)),X5) ).

% Collect_restrict
tff(fact_4084_Collect__restrict,axiom,
    ! [X5: set_list_nat,P2: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bg(X5),P2)),X5) ).

% Collect_restrict
tff(fact_4085_Collect__restrict,axiom,
    ! [X5: set_set_nat,P2: fun_set_nat_o] : ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bh(X5),P2)),X5) ).

% Collect_restrict
tff(fact_4086_Collect__restrict,axiom,
    ! [X5: set_nat,P2: fun_nat_o] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bi(X5),P2))),X5) ).

% Collect_restrict
tff(fact_4087_Collect__restrict,axiom,
    ! [X5: set_int,P2: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bj(X5),P2))),X5) ).

% Collect_restrict
tff(fact_4088_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B4: set_nat,R: fun_real_fun_nat_o] :
      ( ~ finite_finite_real(A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: real] :
              ( member_real(X3,A4)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B4)
                  & aa_nat_o(aa_real_fun_nat_o(R,X3),Xa) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B4)
              & ~ finite_finite_real(collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_bk(A4),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4089_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B4: set_int,R: fun_real_fun_int_o] :
      ( ~ finite_finite_real(A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X3: real] :
              ( member_real(X3,A4)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B4)
                  & aa_int_o(aa_real_fun_int_o(R,X3),Xa) ) )
         => ? [X3: int] :
              ( aa_set_int_o(member_int(X3),B4)
              & ~ finite_finite_real(collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_bl(A4),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4090_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B4: set_complex,R: fun_re7319081385389557872plex_o] :
      ( ~ finite_finite_real(A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X3: real] :
              ( member_real(X3,A4)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B4)
                  & aa_complex_o(aa_rea2382570592068293209plex_o(R,X3),Xa) ) )
         => ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),B4)
              & ~ finite_finite_real(collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_bm(A4),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4091_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_nat,R: fun_nat_fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A4)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B4)
                  & aa_nat_o(aa_nat_fun_nat_o(R,X3),Xa) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B4)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_bn(A4),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4092_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_int,R: fun_nat_fun_int_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A4)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B4)
                  & aa_int_o(aa_nat_fun_int_o(R,X3),Xa) ) )
         => ? [X3: int] :
              ( aa_set_int_o(member_int(X3),B4)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_bo(A4),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4093_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_complex,R: fun_na1554462459066888140plex_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A4)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B4)
                  & aa_complex_o(aa_nat_fun_complex_o(R,X3),Xa) ) )
         => ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),B4)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_bp(A4),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4094_pigeonhole__infinite__rel,axiom,
    ! [A4: set_int,B4: set_nat,R: fun_int_fun_nat_o] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A4)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B4)
                  & aa_nat_o(aa_int_fun_nat_o(R,X3),Xa) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B4)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_bq(A4),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4095_pigeonhole__infinite__rel,axiom,
    ! [A4: set_int,B4: set_int,R: fun_int_fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A4)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B4)
                  & aa_int_o(aa_int_fun_int_o(R,X3),Xa) ) )
         => ? [X3: int] :
              ( aa_set_int_o(member_int(X3),B4)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_br(A4),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4096_pigeonhole__infinite__rel,axiom,
    ! [A4: set_int,B4: set_complex,R: fun_in5065666251602127472plex_o] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A4)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B4)
                  & aa_complex_o(aa_int_fun_complex_o(R,X3),Xa) ) )
         => ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),B4)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_bs(A4),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4097_pigeonhole__infinite__rel,axiom,
    ! [A4: set_complex,B4: set_nat,R: fun_co5676208753217392844_nat_o] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A4)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B4)
                  & aa_nat_o(aa_complex_fun_nat_o(R,X3),Xa) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B4)
              & ~ aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_bt(A4),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4098_not__finite__existsD,axiom,
    ! [P2: fun_real_o] :
      ( ~ finite_finite_real(collect_real(P2))
     => ? [X_12: real] : aa_real_o(P2,X_12) ) ).

% not_finite_existsD
tff(fact_4099_not__finite__existsD,axiom,
    ! [P2: fun_list_nat_o] :
      ( ~ finite8100373058378681591st_nat(collect_list_nat(P2))
     => ? [X_12: list_nat] : aa_list_nat_o(P2,X_12) ) ).

% not_finite_existsD
tff(fact_4100_not__finite__existsD,axiom,
    ! [P2: fun_set_nat_o] :
      ( ~ finite1152437895449049373et_nat(collect_set_nat(P2))
     => ? [X_12: set_nat] : aa_set_nat_o(P2,X_12) ) ).

% not_finite_existsD
tff(fact_4101_not__finite__existsD,axiom,
    ! [P2: fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,collect_nat(P2))
     => ? [X_12: nat] : aa_nat_o(P2,X_12) ) ).

% not_finite_existsD
tff(fact_4102_not__finite__existsD,axiom,
    ! [P2: fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,collect_int(P2))
     => ? [X_12: int] : aa_int_o(P2,X_12) ) ).

% not_finite_existsD
tff(fact_4103_not__finite__existsD,axiom,
    ! [P2: fun_complex_o] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P2))
     => ? [X_12: complex] : aa_complex_o(P2,X_12) ) ).

% not_finite_existsD
tff(fact_4104_not__finite__existsD,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P2))
     => ? [X_12: product_prod_nat_nat] : aa_Pro6150637604445964947_nat_o(P2,X_12) ) ).

% not_finite_existsD
tff(fact_4105_diff__commute,axiom,
    ! [I: nat,J: nat,K2: 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)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K2)),J) ).

% diff_commute
tff(fact_4106_diff__right__commute,axiom,
    ! [A: real,C2: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C2) ).

% diff_right_commute
tff(fact_4107_diff__right__commute,axiom,
    ! [A: rat,C2: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C2)),B) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C2) ).

% diff_right_commute
tff(fact_4108_diff__right__commute,axiom,
    ! [A: nat,C2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C2)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C2) ).

% diff_right_commute
tff(fact_4109_diff__right__commute,axiom,
    ! [A: int,C2: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) ).

% diff_right_commute
tff(fact_4110_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D) )
     => ( ( A = B )
      <=> ( C2 = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4111_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C2),D) )
     => ( ( A = B )
      <=> ( C2 = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4112_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D) )
     => ( ( A = B )
      <=> ( C2 = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4113_unique__quotient,axiom,
    ! [A: int,B: int,Q3: int,R2: int,Q5: int,R4: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R4))
       => ( Q3 = Q5 ) ) ) ).

% unique_quotient
tff(fact_4114_unique__remainder,axiom,
    ! [A: int,B: int,Q3: int,R2: int,Q5: int,R4: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R4))
       => ( R2 = R4 ) ) ) ).

% unique_remainder
tff(fact_4115_lambda__zero,axiom,
    aTP_Lamm_bu = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_4116_lambda__zero,axiom,
    aTP_Lamm_bv = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_4117_lambda__zero,axiom,
    aTP_Lamm_bw = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_4118_lambda__zero,axiom,
    aTP_Lamm_bx = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_4119_lambda__zero,axiom,
    aTP_Lamm_by = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_4120_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ord_less_eq_set_real(collect_real(aa_real_fun_real_o(aTP_Lamm_bz,A)),collect_real(aa_real_fun_real_o(aTP_Lamm_bz,B)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% subset_divisors_dvd
tff(fact_4121_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ca,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ca,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_4122_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_cb,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_cb,B)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% subset_divisors_dvd
tff(fact_4123_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_bd,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_bd,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_4124_max__def__raw,axiom,
    ! [X: extended_enat,Xa: extended_enat] :
      ord_ma741700101516333627d_enat(X,Xa) = $ite(ord_le2932123472753598470d_enat(X,Xa),Xa,X) ).

% max_def_raw
tff(fact_4125_max__def__raw,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ord_max_Code_integer(X,Xa) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4126_max__def__raw,axiom,
    ! [X: set_int,Xa: set_int] :
      ord_max_set_int(X,Xa) = $ite(aa_set_int_o(ord_less_eq_set_int(X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4127_max__def__raw,axiom,
    ! [X: rat,Xa: rat] :
      aa_rat_rat(ord_max_rat(X),Xa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4128_max__def__raw,axiom,
    ! [X: num,Xa: num] :
      aa_num_num(ord_max_num(X),Xa) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4129_max__def__raw,axiom,
    ! [X: nat,Xa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4130_max__def__raw,axiom,
    ! [X: int,Xa: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,X),Xa) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4131_finite__M__bounded__by__nat,axiom,
    ! [P2: fun_nat_o,I: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cc(P2),I))) ).

% finite_M_bounded_by_nat
tff(fact_4132_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(F,N2))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cd(F),U))) ) ).

% finite_less_ub
tff(fact_4133_diff__mono,axiom,
    ! [A: real,B: real,D: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_mono
tff(fact_4134_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C2)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_mono
tff(fact_4135_diff__mono,axiom,
    ! [A: int,B: int,D: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_mono
tff(fact_4136_diff__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B)) ) ).

% diff_left_mono
tff(fact_4137_diff__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C2),A)),aa_rat_rat(minus_minus_rat(C2),B)) ) ).

% diff_left_mono
tff(fact_4138_diff__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B)) ) ).

% diff_left_mono
tff(fact_4139_diff__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C2)) ) ).

% diff_right_mono
tff(fact_4140_diff__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C2)),aa_rat_rat(minus_minus_rat(B),C2)) ) ).

% diff_right_mono
tff(fact_4141_diff__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) ) ).

% diff_right_mono
tff(fact_4142_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4143_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C2),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4144_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4145_eq__iff__diff__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = B )
    <=> ( aa_complex_complex(minus_minus_complex(A),B) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_4146_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_4147_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(minus_minus_rat(A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_4148_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_4149_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C2)) ) ).

% diff_strict_right_mono
tff(fact_4150_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C2)),aa_rat_rat(minus_minus_rat(B),C2)) ) ).

% diff_strict_right_mono
tff(fact_4151_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) ) ).

% diff_strict_right_mono
tff(fact_4152_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B)) ) ).

% diff_strict_left_mono
tff(fact_4153_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(C2),A)),aa_rat_rat(minus_minus_rat(C2),B)) ) ).

% diff_strict_left_mono
tff(fact_4154_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B)) ) ).

% diff_strict_left_mono
tff(fact_4155_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D) ) ) ).

% diff_eq_diff_less
tff(fact_4156_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C2),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D) ) ) ).

% diff_eq_diff_less
tff(fact_4157_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D) ) ) ).

% diff_eq_diff_less
tff(fact_4158_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_strict_mono
tff(fact_4159_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C2)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_strict_mono
tff(fact_4160_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_strict_mono
tff(fact_4161_add__diff__add,axiom,
    ! [A: real,C2: real,B: real,D: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D)) ).

% add_diff_add
tff(fact_4162_add__diff__add,axiom,
    ! [A: rat,C2: rat,B: rat,D: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),D)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_rat_rat(minus_minus_rat(C2),D)) ).

% add_diff_add
tff(fact_4163_add__diff__add,axiom,
    ! [A: int,C2: int,B: int,D: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D)) ).

% add_diff_add
tff(fact_4164_group__cancel_Osub1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A4),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4165_group__cancel_Osub1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4 = aa_rat_rat(plus_plus_rat(K2),A) )
     => ( aa_rat_rat(minus_minus_rat(A4),B) = aa_rat_rat(plus_plus_rat(K2),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4166_group__cancel_Osub1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A4),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4167_diff__eq__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = C2 )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B) ) ) ).

% diff_eq_eq
tff(fact_4168_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = C2 )
    <=> ( A = aa_rat_rat(plus_plus_rat(C2),B) ) ) ).

% diff_eq_eq
tff(fact_4169_diff__eq__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C2 )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B) ) ) ).

% diff_eq_eq
tff(fact_4170_eq__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_4171_eq__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( A = aa_rat_rat(minus_minus_rat(C2),B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_4172_eq__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_4173_add__diff__eq,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) ).

% add_diff_eq
tff(fact_4174_add__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(minus_minus_rat(B),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) ).

% add_diff_eq
tff(fact_4175_add__diff__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ).

% add_diff_eq
tff(fact_4176_diff__diff__eq2,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B) ).

% diff_diff_eq2
tff(fact_4177_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(minus_minus_rat(B),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C2)),B) ).

% diff_diff_eq2
tff(fact_4178_diff__diff__eq2,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ).

% diff_diff_eq2
tff(fact_4179_diff__add__eq,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B) ).

% diff_add_eq
tff(fact_4180_diff__add__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),C2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C2)),B) ).

% diff_add_eq
tff(fact_4181_diff__add__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ).

% diff_add_eq
tff(fact_4182_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4183_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C2)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4184_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4185_add__implies__diff,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B) = A )
     => ( C2 = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ) ) ).

% add_implies_diff
tff(fact_4186_add__implies__diff,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C2),B) = A )
     => ( C2 = aa_rat_rat(minus_minus_rat(A),B) ) ) ).

% add_implies_diff
tff(fact_4187_add__implies__diff,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B) = A )
     => ( C2 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_4188_add__implies__diff,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B) = A )
     => ( C2 = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_4189_diff__diff__eq,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ).

% diff_diff_eq
tff(fact_4190_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C2) = aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) ).

% diff_diff_eq
tff(fact_4191_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% diff_diff_eq
tff(fact_4192_diff__diff__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% diff_diff_eq
tff(fact_4193_zero__induct__lemma,axiom,
    ! [P2: fun_nat_o,K2: nat,I: nat] :
      ( aa_nat_o(P2,K2)
     => ( ! [N2: nat] :
            ( aa_nat_o(P2,aa_nat_nat(suc,N2))
           => aa_nat_o(P2,N2) )
       => aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),I)) ) ) ).

% zero_induct_lemma
tff(fact_4194_diffs0__imp__equal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb) = zero_zero_nat )
       => ( Mb = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_4195_minus__nat_Odiff__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),zero_zero_nat) = Mb ).

% minus_nat.diff_0
tff(fact_4196_less__imp__diff__less,axiom,
    ! [J: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Nb)),K2) ) ).

% less_imp_diff_less
tff(fact_4197_diff__less__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ) ).

% diff_less_mono2
tff(fact_4198_dvd__minus__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% dvd_minus_self
tff(fact_4199_minus__int__code_I1_J,axiom,
    ! [K2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),zero_zero_int) = K2 ).

% minus_int_code(1)
tff(fact_4200_eq__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2) )
        <=> ( Mb = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_4201_le__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% le_diff_iff
tff(fact_4202_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_4203_diff__le__mono,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),L)) ) ).

% diff_le_mono
tff(fact_4204_diff__le__self,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ).

% diff_le_self
tff(fact_4205_le__diff__iff_H,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),B))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).

% le_diff_iff'
tff(fact_4206_diff__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ).

% diff_le_mono2
tff(fact_4207_dvd__diffD,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Mb) ) ) ) ).

% dvd_diffD
tff(fact_4208_dvd__diffD1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_4209_less__eq__dvd__minus,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% less_eq_dvd_minus
tff(fact_4210_diff__add__inverse2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),Nb) = Mb ).

% diff_add_inverse2
tff(fact_4211_diff__add__inverse,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),Nb) = Mb ).

% diff_add_inverse
tff(fact_4212_diff__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% diff_cancel2
tff(fact_4213_Nat_Odiff__cancel,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% Nat.diff_cancel
tff(fact_4214_diff__mult__distrib2,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) ).

% diff_mult_distrib2
tff(fact_4215_diff__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)) ).

% diff_mult_distrib
tff(fact_4216_int__distrib_I3_J,axiom,
    ! [Z12: int,Z23: int,W2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z12),Z23)),W2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z12),W2)),aa_int_int(aa_int_fun_int_int(times_times_int,Z23),W2)) ).

% int_distrib(3)
tff(fact_4217_int__distrib_I4_J,axiom,
    ! [W2: int,Z12: int,Z23: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z12),Z23)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z12)),aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z23)) ).

% int_distrib(4)
tff(fact_4218_max__diff__distrib__left,axiom,
    ! [Xb: code_integer,Yb: code_integer,Z: code_integer] : minus_8373710615458151222nteger(ord_max_Code_integer(Xb,Yb),Z) = ord_max_Code_integer(minus_8373710615458151222nteger(Xb,Z),minus_8373710615458151222nteger(Yb,Z)) ).

% max_diff_distrib_left
tff(fact_4219_max__diff__distrib__left,axiom,
    ! [Xb: real,Yb: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Yb)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Yb),Z)) ).

% max_diff_distrib_left
tff(fact_4220_max__diff__distrib__left,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(Xb),Yb)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(Xb),Z)),aa_rat_rat(minus_minus_rat(Yb),Z)) ).

% max_diff_distrib_left
tff(fact_4221_max__diff__distrib__left,axiom,
    ! [Xb: int,Yb: 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,Xb),Yb)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Yb),Z)) ).

% max_diff_distrib_left
tff(fact_4222_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(m2),m2) ) ).

% numeral_code(2)
tff(fact_4223_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),m2) ) ).

% numeral_code(2)
tff(fact_4224_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(plus_plus_rat(m2),m2) ) ).

% numeral_code(2)
tff(fact_4225_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2) ) ).

% numeral_code(2)
tff(fact_4226_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2) ) ).

% numeral_code(2)
tff(fact_4227_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect_nat(vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_4228_finite__divisors__nat,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ca,Mb))) ) ).

% finite_divisors_nat
tff(fact_4229_concat__bit__assoc,axiom,
    ! [Nb: nat,K2: int,Mb: nat,L: int,R2: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Mb),L),R2)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),L)),R2) ).

% concat_bit_assoc
tff(fact_4230_set__decode__def,axiom,
    ! [Xb: nat] : nat_set_decode(Xb) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ce,Xb)) ).

% set_decode_def
tff(fact_4231_eucl__rel__int__by0,axiom,
    ! [K2: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K2)) ).

% eucl_rel_int_by0
tff(fact_4232_div__int__unique,axiom,
    ! [K2: int,L: int,Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = Q3 ) ) ).

% div_int_unique
tff(fact_4233_mod__int__unique,axiom,
    ! [K2: int,L: int,Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = R2 ) ) ).

% mod_int_unique
tff(fact_4234_subset__decode__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,nat_set_decode(Mb)),nat_set_decode(Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% subset_decode_imp_le
tff(fact_4235_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real((P2))) ).

% zero_less_eq_of_bool
tff(fact_4236_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P2))) ).

% zero_less_eq_of_bool
tff(fact_4237_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_n2687167440665602831ol_nat((P2))) ).

% zero_less_eq_of_bool
tff(fact_4238_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P2))) ).

% zero_less_eq_of_bool
tff(fact_4239_of__bool__less__eq__one,axiom,
    ! [P2: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_n3304061248610475627l_real((P2))),one_one_real) ).

% of_bool_less_eq_one
tff(fact_4240_of__bool__less__eq__one,axiom,
    ! [P2: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P2))),one_one_rat) ).

% of_bool_less_eq_one
tff(fact_4241_of__bool__less__eq__one,axiom,
    ! [P2: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_n2687167440665602831ol_nat((P2))),one_one_nat) ).

% of_bool_less_eq_one
tff(fact_4242_of__bool__less__eq__one,axiom,
    ! [P2: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P2))),one_one_int) ).

% of_bool_less_eq_one
tff(fact_4243_of__bool__def,axiom,
    ! [P: $o] :
      zero_n1201886186963655149omplex((P)) = $ite((P),one_one_complex,zero_zero_complex) ).

% of_bool_def
tff(fact_4244_of__bool__def,axiom,
    ! [P: $o] :
      zero_n3304061248610475627l_real((P)) = $ite((P),one_one_real,zero_zero_real) ).

% of_bool_def
tff(fact_4245_of__bool__def,axiom,
    ! [P: $o] :
      zero_n2052037380579107095ol_rat((P)) = $ite((P),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_4246_of__bool__def,axiom,
    ! [P: $o] :
      zero_n2687167440665602831ol_nat((P)) = $ite((P),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_4247_of__bool__def,axiom,
    ! [P: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,(P)) = $ite((P),one_one_int,zero_zero_int) ).

% of_bool_def
tff(fact_4248_split__of__bool,axiom,
    ! [P2: fun_complex_o,P: $o] :
      ( aa_complex_o(P2,zero_n1201886186963655149omplex((P)))
    <=> ( ( (P)
         => aa_complex_o(P2,one_one_complex) )
        & ( ~ (P)
         => aa_complex_o(P2,zero_zero_complex) ) ) ) ).

% split_of_bool
tff(fact_4249_split__of__bool,axiom,
    ! [P2: fun_real_o,P: $o] :
      ( aa_real_o(P2,zero_n3304061248610475627l_real((P)))
    <=> ( ( (P)
         => aa_real_o(P2,one_one_real) )
        & ( ~ (P)
         => aa_real_o(P2,zero_zero_real) ) ) ) ).

% split_of_bool
tff(fact_4250_split__of__bool,axiom,
    ! [P2: fun_rat_o,P: $o] :
      ( aa_rat_o(P2,zero_n2052037380579107095ol_rat((P)))
    <=> ( ( (P)
         => aa_rat_o(P2,one_one_rat) )
        & ( ~ (P)
         => aa_rat_o(P2,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_4251_split__of__bool,axiom,
    ! [P2: fun_nat_o,P: $o] :
      ( aa_nat_o(P2,zero_n2687167440665602831ol_nat((P)))
    <=> ( ( (P)
         => aa_nat_o(P2,one_one_nat) )
        & ( ~ (P)
         => aa_nat_o(P2,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_4252_split__of__bool,axiom,
    ! [P2: fun_int_o,P: $o] :
      ( aa_int_o(P2,aa_o_int(zero_n2684676970156552555ol_int,(P)))
    <=> ( ( (P)
         => aa_int_o(P2,one_one_int) )
        & ( ~ (P)
         => aa_int_o(P2,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_4253_split__of__bool__asm,axiom,
    ! [P2: fun_complex_o,P: $o] :
      ( aa_complex_o(P2,zero_n1201886186963655149omplex((P)))
    <=> ~ ( ( (P)
            & ~ aa_complex_o(P2,one_one_complex) )
          | ( ~ (P)
            & ~ aa_complex_o(P2,zero_zero_complex) ) ) ) ).

% split_of_bool_asm
tff(fact_4254_split__of__bool__asm,axiom,
    ! [P2: fun_real_o,P: $o] :
      ( aa_real_o(P2,zero_n3304061248610475627l_real((P)))
    <=> ~ ( ( (P)
            & ~ aa_real_o(P2,one_one_real) )
          | ( ~ (P)
            & ~ aa_real_o(P2,zero_zero_real) ) ) ) ).

% split_of_bool_asm
tff(fact_4255_split__of__bool__asm,axiom,
    ! [P2: fun_rat_o,P: $o] :
      ( aa_rat_o(P2,zero_n2052037380579107095ol_rat((P)))
    <=> ~ ( ( (P)
            & ~ aa_rat_o(P2,one_one_rat) )
          | ( ~ (P)
            & ~ aa_rat_o(P2,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_4256_split__of__bool__asm,axiom,
    ! [P2: fun_nat_o,P: $o] :
      ( aa_nat_o(P2,zero_n2687167440665602831ol_nat((P)))
    <=> ~ ( ( (P)
            & ~ aa_nat_o(P2,one_one_nat) )
          | ( ~ (P)
            & ~ aa_nat_o(P2,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_4257_split__of__bool__asm,axiom,
    ! [P2: fun_int_o,P: $o] :
      ( aa_int_o(P2,aa_o_int(zero_n2684676970156552555ol_int,(P)))
    <=> ~ ( ( (P)
            & ~ aa_int_o(P2,one_one_int) )
          | ( ~ (P)
            & ~ aa_int_o(P2,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_4258_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_4259_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_4260_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_4261_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_4262_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_4263_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_4264_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C2 )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_4265_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_4266_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_4267_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_4268_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_4269_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_4270_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_4271_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_4272_le__add__diff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A)) ) ).

% le_add_diff
tff(fact_4273_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% diff_add
tff(fact_4274_le__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) ) ).

% le_diff_eq
tff(fact_4275_le__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),C2) ) ).

% le_diff_eq
tff(fact_4276_le__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ) ).

% le_diff_eq
tff(fact_4277_diff__le__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)) ) ).

% diff_le_eq
tff(fact_4278_diff__le__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),C2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(C2),B)) ) ).

% diff_le_eq
tff(fact_4279_diff__le__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% diff_le_eq
tff(fact_4280_add__le__add__imp__diff__le,axiom,
    ! [I: real,K2: real,Nb: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K2))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4281_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K2: rat,Nb: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K2)),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J),K2))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K2)),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J),K2))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4282_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K2: nat,Nb: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4283_add__le__add__imp__diff__le,axiom,
    ! [I: int,K2: int,Nb: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K2))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K2))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4284_add__le__imp__le__diff,axiom,
    ! [I: real,K2: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_4285_add__le__imp__le__diff,axiom,
    ! [I: rat,K2: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K2)),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),aa_rat_rat(minus_minus_rat(Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_4286_add__le__imp__le__diff,axiom,
    ! [I: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_4287_add__le__imp__le__diff,axiom,
    ! [I: int,K2: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_4288_finite__set__decode,axiom,
    ! [Nb: nat] : aa_set_nat_o(finite_finite_nat,nat_set_decode(Nb)) ).

% finite_set_decode
tff(fact_4289_less__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) ) ).

% less_diff_eq
tff(fact_4290_less__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),C2) ) ).

% less_diff_eq
tff(fact_4291_less__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ) ).

% less_diff_eq
tff(fact_4292_diff__less__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)) ) ).

% diff_less_eq
tff(fact_4293_diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(C2),B)) ) ).

% diff_less_eq
tff(fact_4294_diff__less__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% diff_less_eq
tff(fact_4295_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4296_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4297_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4298_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4299_mult__diff__mult,axiom,
    ! [Xb: real,Yb: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Yb)),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,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Yb),B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A)),B)) ).

% mult_diff_mult
tff(fact_4300_mult__diff__mult,axiom,
    ! [Xb: rat,Yb: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Yb)),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),aa_rat_rat(minus_minus_rat(Yb),B))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),A)),B)) ).

% mult_diff_mult
tff(fact_4301_mult__diff__mult,axiom,
    ! [Xb: int,Yb: 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,Xb),Yb)),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,Xb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Yb),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A)),B)) ).

% mult_diff_mult
tff(fact_4302_eq__add__iff1,axiom,
    ! [A: real,E2: real,C2: 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)),C2) = 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(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C2) = D ) ) ).

% eq_add_iff1
tff(fact_4303_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C2) = D ) ) ).

% eq_add_iff1
tff(fact_4304_eq__add__iff1,axiom,
    ! [A: int,E2: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C2) = D ) ) ).

% eq_add_iff1
tff(fact_4305_eq__add__iff2,axiom,
    ! [A: real,E2: real,C2: 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)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( C2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4306_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( C2 = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4307_eq__add__iff2,axiom,
    ! [A: int,E2: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D) )
    <=> ( C2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4308_square__diff__square__factored,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Yb)) ).

% square_diff_square_factored
tff(fact_4309_square__diff__square__factored,axiom,
    ! [Xb: rat,Yb: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Yb),Yb)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xb),Yb)),aa_rat_rat(minus_minus_rat(Xb),Yb)) ).

% square_diff_square_factored
tff(fact_4310_square__diff__square__factored,axiom,
    ! [Xb: int,Yb: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Yb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Yb)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Yb)) ).

% square_diff_square_factored
tff(fact_4311_diff__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(suc,Mb)) ).

% diff_less_Suc
tff(fact_4312_Suc__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_4313_diff__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ) ) ).

% diff_less
tff(fact_4314_Suc__diff__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% Suc_diff_le
tff(fact_4315_less__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ) ).

% less_diff_iff
tff(fact_4316_diff__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C2)) ) ) ).

% diff_less_mono
tff(fact_4317_diff__add__0,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = zero_zero_nat ).

% diff_add_0
tff(fact_4318_add__diff__inverse__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = Mb ) ) ).

% add_diff_inverse_nat
tff(fact_4319_less__diff__conv,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),J) ) ).

% less_diff_conv
tff(fact_4320_le__diff__conv,axiom,
    ! [J: nat,K2: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),I)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)) ) ).

% le_diff_conv
tff(fact_4321_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_4322_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)) ) ) ).

% Nat.diff_add_assoc
tff(fact_4323_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_4324_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K2 )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_4325_diff__Suc__eq__diff__pred,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_4326_int__le__induct,axiom,
    ! [I: int,K2: int,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K2)
     => ( aa_int_o(P2,K2)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K2)
             => ( aa_int_o(P2,I2)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P2,I) ) ) ) ).

% int_le_induct
tff(fact_4327_int__less__induct,axiom,
    ! [I: int,K2: int,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K2)
     => ( aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K2)
             => ( aa_int_o(P2,I2)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P2,I) ) ) ) ).

% int_less_induct
tff(fact_4328_le__mod__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).

% le_mod_geq
tff(fact_4329_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Mb: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_4330_nat__minus__add__max,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Mb) ).

% nat_minus_add_max
tff(fact_4331_finite__lists__length__eq,axiom,
    ! [A4: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_cf(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4332_finite__lists__length__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Nb: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1970911183850433186_nat_o(aTP_Lamm_cg(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4333_finite__lists__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,Nb: nat] :
      ( finite5795047828879050333T_VEBT(A4)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_ch(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4334_finite__lists__length__eq,axiom,
    ! [A4: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_ci(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4335_finite__lists__length__eq,axiom,
    ! [A4: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_cj(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4336_finite__lists__length__le,axiom,
    ! [A4: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_ck(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_4337_finite__lists__length__le,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Nb: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1970911183850433186_nat_o(aTP_Lamm_cl(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_4338_finite__lists__length__le,axiom,
    ! [A4: set_VEBT_VEBT,Nb: nat] :
      ( finite5795047828879050333T_VEBT(A4)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_cm(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_4339_finite__lists__length__le,axiom,
    ! [A4: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_cn(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_4340_finite__lists__length__le,axiom,
    ! [A4: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_co(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_4341_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4342_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4343_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4344_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C2)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4345_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C2)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4346_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C2)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4347_less__add__iff1,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C2)),D) ) ).

% less_add_iff1
tff(fact_4348_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C2)),D) ) ).

% less_add_iff1
tff(fact_4349_less__add__iff1,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C2)),D) ) ).

% less_add_iff1
tff(fact_4350_less__add__iff2,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_4351_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_4352_less__add__iff2,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_4353_divide__diff__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Yb: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Z)),Yb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Xb),aa_complex_complex(times_times_complex(Yb),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4354_divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),Yb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4355_divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xb),Z)),Yb) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(times_times_rat(Yb),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4356_diff__divide__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Yb: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(Yb),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Z)),Yb)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4357_diff__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Yb),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),Yb)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4358_diff__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(Yb),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),Yb)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4359_diff__frac__eq,axiom,
    ! [Yb: complex,Z: complex,Xb: complex,W2: complex] :
      ( ( Yb != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Yb)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Z)),aa_complex_complex(times_times_complex(W2),Yb))),aa_complex_complex(times_times_complex(Yb),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4360_diff__frac__eq,axiom,
    ! [Yb: real,Z: real,Xb: real,W2: real] :
      ( ( Yb != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4361_diff__frac__eq,axiom,
    ! [Yb: rat,Z: rat,Xb: rat,W2: rat] :
      ( ( Yb != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xb),Yb)),aa_rat_rat(divide_divide_rat(W2),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W2),Yb))),aa_rat_rat(times_times_rat(Yb),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4362_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = $ite(Z = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_4363_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_4364_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_4365_square__diff__one__factored,axiom,
    ! [Xb: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Xb)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Xb),one_one_complex)),aa_complex_complex(minus_minus_complex(Xb),one_one_complex)) ).

% square_diff_one_factored
tff(fact_4366_square__diff__one__factored,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ).

% square_diff_one_factored
tff(fact_4367_square__diff__one__factored,axiom,
    ! [Xb: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xb),one_one_rat)),aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) ).

% square_diff_one_factored
tff(fact_4368_square__diff__one__factored,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),one_one_int)) ).

% square_diff_one_factored
tff(fact_4369_inf__period_I4_J,axiom,
    ! [D: code_integer,D4: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D4)
     => ! [X: code_integer,K4: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Ta))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D4)),Ta)) ) ) ).

% inf_period(4)
tff(fact_4370_inf__period_I4_J,axiom,
    ! [D: real,D4: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D4)
     => ! [X: real,K4: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Ta))
        <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))),Ta)) ) ) ).

% inf_period(4)
tff(fact_4371_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D4)
     => ! [X: rat,K4: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Ta))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D4))),Ta)) ) ) ).

% inf_period(4)
tff(fact_4372_inf__period_I4_J,axiom,
    ! [D: int,D4: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: int,K4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))),Ta)) ) ) ).

% inf_period(4)
tff(fact_4373_inf__period_I3_J,axiom,
    ! [D: code_integer,D4: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D4)
     => ! [X: code_integer,K4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Ta))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D4)),Ta)) ) ) ).

% inf_period(3)
tff(fact_4374_inf__period_I3_J,axiom,
    ! [D: real,D4: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D4)
     => ! [X: real,K4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Ta))
        <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))),Ta)) ) ) ).

% inf_period(3)
tff(fact_4375_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D4)
     => ! [X: rat,K4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Ta))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D4))),Ta)) ) ) ).

% inf_period(3)
tff(fact_4376_inf__period_I3_J,axiom,
    ! [D: int,D4: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: int,K4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))),Ta)) ) ) ).

% inf_period(3)
tff(fact_4377_eucl__rel__int__dividesI,axiom,
    ! [L: int,K2: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K2 = aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L) )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_4378_diff__Suc__less,axiom,
    ! [Nb: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,I))),Nb) ) ).

% diff_Suc_less
tff(fact_4379_nat__diff__split__asm,axiom,
    ! [P2: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
            & ~ aa_nat_o(P2,zero_zero_nat) )
          | ? [D5: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
              & ~ aa_nat_o(P2,D5) ) ) ) ).

% nat_diff_split_asm
tff(fact_4380_nat__diff__split,axiom,
    ! [P2: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => aa_nat_o(P2,zero_zero_nat) )
        & ! [D5: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
           => aa_nat_o(P2,D5) ) ) ) ).

% nat_diff_split
tff(fact_4381_less__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),I)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)) ) ) ).

% less_diff_conv2
tff(fact_4382_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_4383_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_4384_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_4385_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_4386_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_4387_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_4388_dvd__minus__add,axiom,
    ! [Q3: nat,Nb: nat,R2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Q3))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb)),Q3))) ) ) ) ).

% dvd_minus_add
tff(fact_4389_plusinfinity,axiom,
    ! [D: int,P3: fun_int_o,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int,K: int] :
            ( aa_int_o(P3,X3)
          <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
             => ( aa_int_o(P2,X3)
              <=> aa_int_o(P3,X3) ) )
         => ( ? [X_1: int] : aa_int_o(P3,X_1)
           => ? [X_12: int] : aa_int_o(P2,X_12) ) ) ) ) ).

% plusinfinity
tff(fact_4390_minusinfinity,axiom,
    ! [D: int,P1: fun_int_o,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int,K: int] :
            ( aa_int_o(P1,X3)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
             => ( aa_int_o(P2,X3)
              <=> aa_int_o(P1,X3) ) )
         => ( ? [X_1: int] : aa_int_o(P1,X_1)
           => ? [X_12: int] : aa_int_o(P2,X_12) ) ) ) ) ).

% minusinfinity
tff(fact_4391_int__induct,axiom,
    ! [P2: fun_int_o,K2: int,I: int] :
      ( aa_int_o(P2,K2)
     => ( ! [I2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),I2)
           => ( aa_int_o(P2,I2)
             => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K2)
             => ( aa_int_o(P2,I2)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P2,I) ) ) ) ).

% int_induct
tff(fact_4392_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),R2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_4393_eucl__rel__int,axiom,
    ! [K2: int,L: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L))) ).

% eucl_rel_int
tff(fact_4394_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          zero_n2687167440665602831ol_nat(( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_4395_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_4396_frac__le__eq,axiom,
    ! [Yb: real,Z: real,Xb: real,W2: real] :
      ( ( Yb != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_4397_frac__le__eq,axiom,
    ! [Yb: rat,Z: rat,Xb: rat,W2: rat] :
      ( ( Yb != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),aa_rat_rat(divide_divide_rat(W2),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W2),Yb))),aa_rat_rat(times_times_rat(Yb),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_4398_frac__less__eq,axiom,
    ! [Yb: real,Z: real,Xb: real,W2: real] :
      ( ( Yb != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_4399_frac__less__eq,axiom,
    ! [Yb: rat,Z: rat,Xb: rat,W2: rat] :
      ( ( Yb != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),aa_rat_rat(divide_divide_rat(W2),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W2),Yb))),aa_rat_rat(times_times_rat(Yb),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_4400_power__diff,axiom,
    ! [A: complex,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ) ) ) ).

% power_diff
tff(fact_4401_power__diff,axiom,
    ! [A: real,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_diff
tff(fact_4402_power__diff,axiom,
    ! [A: rat,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_diff
tff(fact_4403_power__diff,axiom,
    ! [A: nat,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_diff
tff(fact_4404_power__diff,axiom,
    ! [A: int,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_diff
tff(fact_4405_even__diff__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_diff_iff
tff(fact_4406_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_4407_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_4408_div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_4409_div__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ).

% div_if
tff(fact_4410_add__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,Nb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% add_eq_if
tff(fact_4411_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_4412_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_4413_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_4414_mult__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% mult_eq_if
tff(fact_4415_nth__Cons_H,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,Nb: nat] :
      nth_VEBT_VEBT(cons_VEBT_VEBT(Xb,Xs),Nb) = $ite(Nb = zero_zero_nat,Xb,nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ).

% nth_Cons'
tff(fact_4416_nth__Cons_H,axiom,
    ! [Xb: int,Xs: list_int,Nb: nat] :
      aa_nat_int(nth_int(cons_int(Xb,Xs)),Nb) = $ite(Nb = zero_zero_nat,Xb,aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ).

% nth_Cons'
tff(fact_4417_nth__Cons_H,axiom,
    ! [Xb: nat,Xs: list_nat,Nb: nat] :
      aa_nat_nat(nth_nat(cons_nat(Xb,Xs)),Nb) = $ite(Nb = zero_zero_nat,Xb,aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ).

% nth_Cons'
tff(fact_4418_decr__mult__lemma,axiom,
    ! [D: int,P2: fun_int_o,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int] :
            ( aa_int_o(P2,X3)
           => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
         => ! [X: int] :
              ( aa_int_o(P2,X)
             => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_4419_mod__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K2)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_4420_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_4421_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),V)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S3))),V) ) ) ) ).

% scaling_mono
tff(fact_4422_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R2),S3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(U),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(R2),aa_rat_rat(minus_minus_rat(V),U))),S3))),V) ) ) ) ).

% scaling_mono
tff(fact_4423_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4424_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4425_power__diff__power__eq,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_4426_power__diff__power__eq,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_4427_power__eq__if,axiom,
    ! [P: complex,Mb: nat] :
      aa_nat_complex(power_power_complex(P),Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(P),aa_nat_complex(power_power_complex(P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_4428_power__eq__if,axiom,
    ! [P: real,Mb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P),Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P),aa_nat_real(aa_real_fun_nat_real(power_power_real,P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_4429_power__eq__if,axiom,
    ! [P: rat,Mb: nat] :
      aa_nat_rat(power_power_rat(P),Mb) = $ite(Mb = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(P),aa_nat_rat(power_power_rat(P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_4430_power__eq__if,axiom,
    ! [P: nat,Mb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P),Mb) = $ite(Mb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_4431_power__eq__if,axiom,
    ! [P: int,Mb: nat] :
      aa_nat_int(power_power_int(P),Mb) = $ite(Mb = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P),aa_nat_int(power_power_int(P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_4432_vebt__buildup_Oelims,axiom,
    ! [Xb: nat,Yb: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Yb )
     => ( ( ( Xb = zero_zero_nat )
         => ( Yb != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Yb != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Yb != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_4433_power__minus__mult,axiom,
    ! [Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),Nb) ) ) ).

% power_minus_mult
tff(fact_4434_power__minus__mult,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% power_minus_mult
tff(fact_4435_power__minus__mult,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% power_minus_mult
tff(fact_4436_power__minus__mult,axiom,
    ! [Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) ) ) ).

% power_minus_mult
tff(fact_4437_power__minus__mult,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% power_minus_mult
tff(fact_4438_diff__le__diff__pow,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Nb))) ) ).

% diff_le_diff_pow
tff(fact_4439_le__div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_4440_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_4441_nth__non__equal__first__eq,axiom,
    ! [Xb: vEBT_VEBT,Yb: vEBT_VEBT,Xs: list_VEBT_VEBT,Nb: nat] :
      ( ( Xb != Yb )
     => ( ( nth_VEBT_VEBT(cons_VEBT_VEBT(Xb,Xs),Nb) = Yb )
      <=> ( ( nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) = Yb )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_4442_nth__non__equal__first__eq,axiom,
    ! [Xb: int,Yb: int,Xs: list_int,Nb: nat] :
      ( ( Xb != Yb )
     => ( ( aa_nat_int(nth_int(cons_int(Xb,Xs)),Nb) = Yb )
      <=> ( ( aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) = Yb )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_4443_nth__non__equal__first__eq,axiom,
    ! [Xb: nat,Yb: nat,Xs: list_nat,Nb: nat] :
      ( ( Xb != Yb )
     => ( ( aa_nat_nat(nth_nat(cons_nat(Xb,Xs)),Nb) = Yb )
      <=> ( ( aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) = Yb )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_4444_Cons__replicate__eq,axiom,
    ! [Xb: int,Xs: list_int,Nb: nat,Yb: int] :
      ( ( cons_int(Xb,Xs) = replicate_int(Nb,Yb) )
    <=> ( ( Xb = Yb )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( Xs = replicate_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),Xb) ) ) ) ).

% Cons_replicate_eq
tff(fact_4445_Cons__replicate__eq,axiom,
    ! [Xb: nat,Xs: list_nat,Nb: nat,Yb: nat] :
      ( ( cons_nat(Xb,Xs) = replicate_nat(Nb,Yb) )
    <=> ( ( Xb = Yb )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( Xs = replicate_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),Xb) ) ) ) ).

% Cons_replicate_eq
tff(fact_4446_Cons__replicate__eq,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,Nb: nat,Yb: vEBT_VEBT] :
      ( ( cons_VEBT_VEBT(Xb,Xs) = replicate_VEBT_VEBT(Nb,Yb) )
    <=> ( ( Xb = Yb )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( Xs = replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),Xb) ) ) ) ).

% Cons_replicate_eq
tff(fact_4447_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeLista: list_VEBT_VEBT,S3: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeLista,S3)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_4448_bits__induct,axiom,
    ! [P2: fun_nat_o,A: nat] :
      ( ! [A3: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
         => aa_nat_o(P2,A3) )
     => ( ! [A3: nat,B3: $o] :
            ( aa_nat_o(P2,A3)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat((B3))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
             => aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat((B3))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))) ) )
       => aa_nat_o(P2,A) ) ) ).

% bits_induct
tff(fact_4449_bits__induct,axiom,
    ! [P2: fun_int_o,A: int] :
      ( ! [A3: int] :
          ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
         => aa_int_o(P2,A3) )
     => ( ! [A3: int,B3: $o] :
            ( aa_int_o(P2,A3)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
             => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))) ) )
       => aa_int_o(P2,A) ) ) ).

% bits_induct
tff(fact_4450_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Vd2: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeLista,Vd2)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_4451_signed__take__bit__int__less__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_4452_div__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),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,K2),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_4453_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V: nat,TreeLista: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,V),TreeLista,Vc)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_4454_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),Xb)
    <=> $ite(
          Xb = Mia,
          $true,
          $ite(
            Xb = Maa,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_4455_power2__diff,axiom,
    ! [Xb: complex,Yb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)),Yb)) ).

% power2_diff
tff(fact_4456_power2__diff,axiom,
    ! [Xb: real,Yb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Yb)) ).

% power2_diff
tff(fact_4457_power2__diff,axiom,
    ! [Xb: rat,Yb: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Yb)) ).

% power2_diff
tff(fact_4458_power2__diff,axiom,
    ! [Xb: int,Yb: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)),Yb)) ).

% power2_diff
tff(fact_4459_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4460_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4461_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4462_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
      <=> (Yb) )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ( (Yb)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => (Yb) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S)
               => ( (Yb)
                <=> ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_4463_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S)
             => ~ $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_4464_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S)
               => $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_4465_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% even_mod_4_div_2
tff(fact_4466_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
    <=> ( ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q3)),R2) )
        & $ite(
            aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R2),L) ),
            $ite(
              aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int) ),
              Q3 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_4467_int__power__div__base,axiom,
    ! [Mb: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(K2),Mb)),K2) = aa_nat_int(power_power_int(K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_4468_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( ! [Mi: nat,Ma: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma ) ) )
       => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd)
               => ~ $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_4469_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4470_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4471_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4472_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_4473_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_4474_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_4475_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_4476_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
      <=> (Yb) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => (Yb) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => (Yb) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
               => ( (Yb)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( (Yb)
                  <=> ~ ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd)
                   => ( (Yb)
                    <=> ~ $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_4477_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xb != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd)
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_4478_vebt__insert_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Xb,Mia)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Maa))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeLista,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),vEBT_vebt_insert(Summarya,h),Summarya)),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_4479_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
                   => $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_4480_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
      <=> (Yb) )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ( (Yb)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => (Yb) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => (Yb) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => (Yb) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
                   => ( (Yb)
                    <=> ~ $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_4481_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_4482_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_4483_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_4484_divmod__step__eq,axiom,
    ! [L: num,Q3: nat,R2: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q3)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),aa_num_nat(numeral_numeral_nat,L))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q3)),R2)) ).

% divmod_step_eq
tff(fact_4485_divmod__step__eq,axiom,
    ! [L: num,Q3: int,R2: int] :
      aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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))),Q3)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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))),Q3)),R2)) ).

% divmod_step_eq
tff(fact_4486_divmod__step__eq,axiom,
    ! [L: num,Q3: code_integer,R2: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q3,R2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q3),one_one_Code_integer),minus_8373710615458151222nteger(R2,numera6620942414471956472nteger(L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q3),R2)) ).

% divmod_step_eq
tff(fact_4487_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,vEBT_VEBT_set_vebt(Ta)),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ) ).

% inrange
tff(fact_4488_finite__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_cp(Nb),C2))) ) ).

% finite_nth_roots
tff(fact_4489_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => finite_finite_real(collect_real(aa_nat_fun_real_o(aTP_Lamm_cq,Nb))) ) ).

% finite_roots_unity
tff(fact_4490_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cr,Nb))) ) ).

% finite_roots_unity
tff(fact_4491_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Yb )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( ( Yb = $ite(
                      Xaa = zero_zero_nat,
                      vEBT_Leaf($true,(B3)),
                      $ite(Xaa = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B3))) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts2,S) )
               => ( ( Yb = vEBT_Node(Info2,zero_zero_nat,Ts2,S) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
                 => ( ( Yb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S),Xaa)) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                   => ( ( Yb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( ( Yb = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma ) ) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_4492_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_complex,Yb: fun_real_complex] :
      ( finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_cs(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_cs(I6),Yb)))
       => finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_ct(I6),Xb),Yb))) ) ) ).

% prod.finite_Collect_op
tff(fact_4493_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_complex,Yb: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_cu(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_cu(I6),Yb)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_cv(I6),Xb),Yb))) ) ) ).

% prod.finite_Collect_op
tff(fact_4494_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_complex,Yb: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_cw(I6),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_cw(I6),Yb)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_cx(I6),Xb),Yb))) ) ) ).

% prod.finite_Collect_op
tff(fact_4495_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_complex,Yb: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_cy(I6),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_cy(I6),Yb)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_cz(I6),Xb),Yb))) ) ) ).

% prod.finite_Collect_op
tff(fact_4496_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_real,Yb: fun_real_real] :
      ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_da(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_da(I6),Yb)))
       => finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_db(I6),Xb),Yb))) ) ) ).

% prod.finite_Collect_op
tff(fact_4497_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_real,Yb: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_dc(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_dc(I6),Yb)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_dd(I6),Xb),Yb))) ) ) ).

% prod.finite_Collect_op
tff(fact_4498_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_real,Yb: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_de(I6),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_de(I6),Yb)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_df(I6),Xb),Yb))) ) ) ).

% prod.finite_Collect_op
tff(fact_4499_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_real,Yb: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_dg(I6),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_dg(I6),Yb)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_dh(I6),Xb),Yb))) ) ) ).

% prod.finite_Collect_op
tff(fact_4500_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_rat,Yb: fun_real_rat] :
      ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_di(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_di(I6),Yb)))
       => finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_dj(I6),Xb),Yb))) ) ) ).

% prod.finite_Collect_op
tff(fact_4501_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_rat,Yb: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_dk(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_dk(I6),Yb)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_dl(I6),Xb),Yb))) ) ) ).

% prod.finite_Collect_op
tff(fact_4502_finite__Diff2,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(finite_finite_int,minus_minus_set_int(A4,B4))
      <=> aa_set_int_o(finite_finite_int,A4) ) ) ).

% finite_Diff2
tff(fact_4503_finite__Diff2,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,B4))
      <=> aa_set_complex_o(finite3207457112153483333omplex,A4) ) ) ).

% finite_Diff2
tff(fact_4504_finite__Diff2,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A4,B4))
      <=> aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ) ).

% finite_Diff2
tff(fact_4505_finite__Diff2,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A4,B4))
      <=> aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_Diff2
tff(fact_4506_finite__Diff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => aa_set_int_o(finite_finite_int,minus_minus_set_int(A4,B4)) ) ).

% finite_Diff
tff(fact_4507_finite__Diff,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,B4)) ) ).

% finite_Diff
tff(fact_4508_finite__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A4,B4)) ) ).

% finite_Diff
tff(fact_4509_finite__Diff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A4,B4)) ) ).

% finite_Diff
tff(fact_4510_idiff__0,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,Nb) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_4511_idiff__0__right,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(Nb,zero_z5237406670263579293d_enat) = Nb ).

% idiff_0_right
tff(fact_4512_Diff__eq__empty__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat(A4,B4) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_4513_Diff__eq__empty__iff,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ( minus_minus_set_real(A4,B4) = bot_bot_set_real )
    <=> ord_less_eq_set_real(A4,B4) ) ).

% Diff_eq_empty_iff
tff(fact_4514_Diff__eq__empty__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( minus_minus_set_nat(A4,B4) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_4515_Diff__eq__empty__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( minus_minus_set_int(A4,B4) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_4516_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),I)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4517_atLeastAtMost__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( member_set_int(I,set_or370866239135849197et_int(L,U))
    <=> ( aa_set_int_o(ord_less_eq_set_int(L),I)
        & aa_set_int_o(ord_less_eq_set_int(I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4518_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( member_rat(I,set_or633870826150836451st_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4519_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( member_num(I,set_or7049704709247886629st_num(L,U))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4520_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I),set_or1269000886237332187st_nat(L,U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4521_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( aa_set_int_o(member_int(I),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4522_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( member_real(I,set_or1222579329274155063t_real(L,U))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),I)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4523_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L2: set_int,H2: set_int] :
      ( ( set_or370866239135849197et_int(L,H) = set_or370866239135849197et_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_set_int_o(ord_less_eq_set_int(L),H)
          & ~ aa_set_int_o(ord_less_eq_set_int(L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4524_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L2: rat,H2: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),H)
          & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4525_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L2: num,H2: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),H)
          & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4526_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( set_or1269000886237332187st_nat(L,H) = set_or1269000886237332187st_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4527_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),H) = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L2),H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
          & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4528_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L2: real,H2: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),H)
          & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4529_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or1269000886237332187st_nat(L,U)) ).

% finite_atLeastAtMost
tff(fact_4530_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( set_or370866239135849197et_int(A,B) = bot_bot_set_set_int )
    <=> ~ aa_set_int_o(ord_less_eq_set_int(A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4531_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4532_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4533_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( set_or1269000886237332187st_nat(A,B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4534_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4535_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4536_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_set_int = set_or370866239135849197et_int(A,B) )
    <=> ~ aa_set_int_o(ord_less_eq_set_int(A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4537_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4538_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4539_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = set_or1269000886237332187st_nat(A,B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4540_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4541_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4542_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,C2: set_int,D: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(C2,D))
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(A),B)
        | ( aa_set_int_o(ord_less_eq_set_int(C2),A)
          & aa_set_int_o(ord_less_eq_set_int(B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4543_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B),set_or633870826150836451st_rat(C2,D))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4544_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(C2,D))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4545_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_or1269000886237332187st_nat(A,B)),set_or1269000886237332187st_nat(C2,D))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4546_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,C2),D))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4547_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ord_less_eq_set_real(set_or1222579329274155063t_real(A,B),set_or1222579329274155063t_real(C2,D))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4548_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_4549_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_4550_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( set_or1269000886237332187st_nat(A,B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_4551_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_4552_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_4553_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ finite_finite_rat(set_or633870826150836451st_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% infinite_Icc_iff
tff(fact_4554_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ~ finite_finite_real(set_or1222579329274155063t_real(A,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% infinite_Icc_iff
tff(fact_4555_Diff__infinite__finite,axiom,
    ! [T: set_int,S2: set_int] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( ~ aa_set_int_o(finite_finite_int,S2)
       => ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S2,T)) ) ) ).

% Diff_infinite_finite
tff(fact_4556_Diff__infinite__finite,axiom,
    ! [T: set_complex,S2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,T)) ) ) ).

% Diff_infinite_finite
tff(fact_4557_Diff__infinite__finite,axiom,
    ! [T: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,T)
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S2)
       => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(S2,T)) ) ) ).

% Diff_infinite_finite
tff(fact_4558_Diff__infinite__finite,axiom,
    ! [T: set_nat,S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( ~ aa_set_nat_o(finite_finite_nat,S2)
       => ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(S2,T)) ) ) ).

% Diff_infinite_finite
tff(fact_4559_Diff__mono,axiom,
    ! [A4: set_nat,C4: set_nat,D4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,D4),B4)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(A4,B4)),minus_minus_set_nat(C4,D4)) ) ) ).

% Diff_mono
tff(fact_4560_Diff__mono,axiom,
    ! [A4: set_int,C4: set_int,D4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),C4)
     => ( aa_set_int_o(ord_less_eq_set_int(D4),B4)
       => aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A4,B4)),minus_minus_set_int(C4,D4)) ) ) ).

% Diff_mono
tff(fact_4561_Diff__subset,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(A4,B4)),A4) ).

% Diff_subset
tff(fact_4562_Diff__subset,axiom,
    ! [A4: set_int,B4: set_int] : aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A4,B4)),A4) ).

% Diff_subset
tff(fact_4563_double__diff,axiom,
    ! [A4: set_nat,B4: set_nat,C4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C4)
       => ( minus_minus_set_nat(B4,minus_minus_set_nat(C4,A4)) = A4 ) ) ) ).

% double_diff
tff(fact_4564_double__diff,axiom,
    ! [A4: set_int,B4: set_int,C4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),C4)
       => ( minus_minus_set_int(B4,minus_minus_set_int(C4,A4)) = A4 ) ) ) ).

% double_diff
tff(fact_4565_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ finite_finite_rat(set_or633870826150836451st_rat(A,B)) ) ).

% infinite_Icc
tff(fact_4566_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ finite_finite_real(set_or1222579329274155063t_real(A,B)) ) ).

% infinite_Icc
tff(fact_4567_ex__nat__less,axiom,
    ! [Nb: nat,P2: fun_nat_o] :
      ( ? [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),Nb)
          & aa_nat_o(P2,M4) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or1269000886237332187st_nat(zero_zero_nat,Nb))
          & aa_nat_o(P2,X4) ) ) ).

% ex_nat_less
tff(fact_4568_all__nat__less,axiom,
    ! [Nb: nat,P2: fun_nat_o] :
      ( ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),Nb)
         => aa_nat_o(P2,M4) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or1269000886237332187st_nat(zero_zero_nat,Nb))
         => aa_nat_o(P2,X4) ) ) ).

% all_nat_less
tff(fact_4569_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set_nat,Nb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N4),set_or1269000886237332187st_nat(zero_zero_nat,Nb))
     => aa_set_nat_o(finite_finite_nat,N4) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_4570_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Yb: extended_enat,Xb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Z,Yb)
     => ( plus_p3455044024723400733d_enat(Xb,minus_3235023915231533773d_enat(Yb,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(Xb,Yb),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_4571_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,C2: set_int,D: set_int] :
      ( ord_less_set_set_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(C2,D))
    <=> ( ( ~ aa_set_int_o(ord_less_eq_set_int(A),B)
          | ( aa_set_int_o(ord_less_eq_set_int(C2),A)
            & aa_set_int_o(ord_less_eq_set_int(B),D)
            & ( aa_set_int_o(ord_less_set_int(C2),A)
              | aa_set_int_o(ord_less_set_int(B),D) ) ) )
        & aa_set_int_o(ord_less_eq_set_int(C2),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4572_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ord_less_set_rat(set_or633870826150836451st_rat(A,B),set_or633870826150836451st_rat(C2,D))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
          | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),D) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4573_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( ord_less_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(C2,D))
    <=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
          | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D)
            & ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A)
              | aa_num_o(aa_num_fun_num_o(ord_less_num,B),D) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4574_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,set_or1269000886237332187st_nat(A,B)),set_or1269000886237332187st_nat(C2,D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4575_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_set_int_o(ord_less_set_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,C2),D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B),D) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4576_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ord_less_set_real(set_or1222579329274155063t_real(A,B),set_or1222579329274155063t_real(C2,D))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
          | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),A)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D)
            & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A)
              | aa_real_o(aa_real_fun_real_o(ord_less_real,B),D) ) ) )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4577_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_complex,Yb: fun_real_complex] :
      ( finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_dm(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_dm(I6),Yb)))
       => finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_dn(I6),Xb),Yb))) ) ) ).

% sum.finite_Collect_op
tff(fact_4578_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_complex,Yb: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_do(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_do(I6),Yb)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_dp(I6),Xb),Yb))) ) ) ).

% sum.finite_Collect_op
tff(fact_4579_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_complex,Yb: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_dq(I6),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_dq(I6),Yb)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_dr(I6),Xb),Yb))) ) ) ).

% sum.finite_Collect_op
tff(fact_4580_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_complex,Yb: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_ds(I6),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_ds(I6),Yb)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_dt(I6),Xb),Yb))) ) ) ).

% sum.finite_Collect_op
tff(fact_4581_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_real,Yb: fun_real_real] :
      ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_du(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_du(I6),Yb)))
       => finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_dv(I6),Xb),Yb))) ) ) ).

% sum.finite_Collect_op
tff(fact_4582_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_real,Yb: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_dw(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_dw(I6),Yb)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_dx(I6),Xb),Yb))) ) ) ).

% sum.finite_Collect_op
tff(fact_4583_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_real,Yb: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_dy(I6),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_dy(I6),Yb)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_dz(I6),Xb),Yb))) ) ) ).

% sum.finite_Collect_op
tff(fact_4584_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_real,Yb: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ea(I6),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ea(I6),Yb)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_eb(I6),Xb),Yb))) ) ) ).

% sum.finite_Collect_op
tff(fact_4585_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_rat,Yb: fun_real_rat] :
      ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_ec(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_ec(I6),Yb)))
       => finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_ed(I6),Xb),Yb))) ) ) ).

% sum.finite_Collect_op
tff(fact_4586_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_rat,Yb: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_ee(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_ee(I6),Yb)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_ef(I6),Xb),Yb))) ) ) ).

% sum.finite_Collect_op
tff(fact_4587_real__average__minus__first,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_4588_real__average__minus__second,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_4589_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xaa)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa))
                       => $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_4590_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
      <=> (Yb) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( ( (Yb)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ (Yb)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ (Yb)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ (Yb)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( ( (Yb)
                        <=> $ite(
                              Xaa = Mi,
                              $true,
                              $ite(
                                Xaa = Ma,
                                $true,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                  $false,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_4591_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa))
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_4592_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_4593_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
      <=> (Yb) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( ( (Yb)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ (Yb)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S) )
                 => ( ( (Yb)
                    <=> $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4594_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U)) ).

% finite_atLeastAtMost_int
tff(fact_4595_aset_I2_J,axiom,
    ! [D4: int,A4: set_int,P2: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P2,X3)
           => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A4)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P2,X)
                | aa_int_o(Q,X) )
             => ( aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)) ) ) ) ) ) ).

% aset(2)
tff(fact_4596_aset_I1_J,axiom,
    ! [D4: int,A4: set_int,P2: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P2,X3)
           => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A4)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P2,X)
                & aa_int_o(Q,X) )
             => ( aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)) ) ) ) ) ) ).

% aset(1)
tff(fact_4597_bset_I2_J,axiom,
    ! [D4: int,B4: set_int,P2: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P2,X3)
           => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P2,X)
                | aa_int_o(Q,X) )
             => ( aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)) ) ) ) ) ) ).

% bset(2)
tff(fact_4598_bset_I1_J,axiom,
    ! [D4: int,B4: set_int,P2: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P2,X3)
           => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P2,X)
                & aa_int_o(Q,X) )
             => ( aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)) ) ) ) ) ) ).

% bset(1)
tff(fact_4599_bset_I9_J,axiom,
    ! [D: int,D4: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B4)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)),Ta)) ) ) ) ).

% bset(9)
tff(fact_4600_bset_I10_J,axiom,
    ! [D: int,D4: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B4)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)),Ta)) ) ) ) ).

% bset(10)
tff(fact_4601_aset_I9_J,axiom,
    ! [D: int,D4: int,A4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A4)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)),Ta)) ) ) ) ).

% aset(9)
tff(fact_4602_aset_I10_J,axiom,
    ! [D: int,D4: int,A4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A4)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)),Ta)) ) ) ) ).

% aset(10)
tff(fact_4603_periodic__finite__ex,axiom,
    ! [D: int,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int,K: int] :
            ( aa_int_o(P2,X3)
          <=> aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) )
       => ( ? [X_13: int] : aa_int_o(P2,X_13)
        <=> ? [X4: int] :
              ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D))
              & aa_int_o(P2,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_4604_aset_I7_J,axiom,
    ! [D4: int,A4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A4)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)) ) ) ) ).

% aset(7)
tff(fact_4605_aset_I5_J,axiom,
    ! [D4: int,Ta: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(Ta),A4)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A4)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_4606_aset_I4_J,axiom,
    ! [D4: int,Ta: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(Ta),A4)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A4)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( X != Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_4607_aset_I3_J,axiom,
    ! [D4: int,Ta: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A4)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A4)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( X = Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_4608_bset_I7_J,axiom,
    ! [D4: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(Ta),B4)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)) ) ) ) ) ).

% bset(7)
tff(fact_4609_bset_I5_J,axiom,
    ! [D4: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B4)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)),Ta) ) ) ) ).

% bset(5)
tff(fact_4610_bset_I4_J,axiom,
    ! [D4: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(Ta),B4)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( X != Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_4611_bset_I3_J,axiom,
    ! [D4: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B4)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( X = Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_4612_aset_I8_J,axiom,
    ! [D4: int,A4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A4)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)) ) ) ) ).

% aset(8)
tff(fact_4613_aset_I6_J,axiom,
    ! [D4: int,Ta: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A4)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A4)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_4614_bset_I8_J,axiom,
    ! [D4: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B4)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)) ) ) ) ) ).

% bset(8)
tff(fact_4615_bset_I6_J,axiom,
    ! [D4: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B4)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)),Ta) ) ) ) ).

% bset(6)
tff(fact_4616_cpmi,axiom,
    ! [D4: int,P2: fun_int_o,P3: fun_int_o,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
           => ( aa_int_o(P2,X3)
            <=> aa_int_o(P3,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
                 => ! [Xb2: int] :
                      ( aa_set_int_o(member_int(Xb2),B4)
                     => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
             => ( aa_int_o(P2,X3)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
         => ( ! [X3: int,K: int] :
                ( aa_int_o(P3,X3)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D4))) )
           => ( ? [X_13: int] : aa_int_o(P2,X_13)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
                    & aa_int_o(P3,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),B4)
                        & aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X4)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_4617_cppi,axiom,
    ! [D4: int,P2: fun_int_o,P3: fun_int_o,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
           => ( aa_int_o(P2,X3)
            <=> aa_int_o(P3,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
                 => ! [Xb2: int] :
                      ( aa_set_int_o(member_int(Xb2),A4)
                     => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
             => ( aa_int_o(P2,X3)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
         => ( ! [X3: int,K: int] :
                ( aa_int_o(P3,X3)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D4))) )
           => ( ? [X_13: int] : aa_int_o(P2,X_13)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
                    & aa_int_o(P3,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),A4)
                        & aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X4)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_4618_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_4619_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
      <=> (Yb) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Yb)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ (Yb)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xaa)) ) )
           => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
                 => ( ( (Yb)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa)) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( ( (Yb)
                      <=> ( ( Xaa = Mi )
                          | ( Xaa = Ma )
                          | $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa)) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd) )
                     => ( ( (Yb)
                        <=> $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4620_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xaa)) )
           => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd),Xaa))
                       => $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_4621_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma ) ) ) )
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_4622_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_4623_Bolzano,axiom,
    ! [A: real,B: real,P2: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [A3: real,B3: real,C: real] :
            ( aa_real_o(aa_real_fun_real_o(P2,A3),B3)
           => ( aa_real_o(aa_real_fun_real_o(P2,B3),C)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),B3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),C)
                 => aa_real_o(aa_real_fun_real_o(P2,A3),C) ) ) ) )
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
               => ? [D3: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
                    & ! [A3: real,B3: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),X3)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B3)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A3)),D3) )
                       => aa_real_o(aa_real_fun_real_o(P2,A3),B3) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P2,A),B) ) ) ) ).

% Bolzano
tff(fact_4624_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_4625_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_4626_artanh__def,axiom,
    ! [Xb: real] : aa_real_real(artanh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_4627_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R2: int] : aa_Pro6799258434465214121nt_int(adjust_div,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q3),aa_o_int(zero_n2684676970156552555ol_int,R2 != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_4628_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(Nb,A) = $ite(Nb = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_4629_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),A) = $ite(Nb = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_4630_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Yb: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Yb )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Yb = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Yb = vEBT_Leaf($false,$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Yb = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_4631_diff__shunt__var,axiom,
    ! [Xb: set_Pr1261947904930325089at_nat,Yb: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat(Xb,Yb) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Xb),Yb) ) ).

% diff_shunt_var
tff(fact_4632_diff__shunt__var,axiom,
    ! [Xb: set_real,Yb: set_real] :
      ( ( minus_minus_set_real(Xb,Yb) = bot_bot_set_real )
    <=> ord_less_eq_set_real(Xb,Yb) ) ).

% diff_shunt_var
tff(fact_4633_diff__shunt__var,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( ( minus_minus_set_nat(Xb,Yb) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Yb) ) ).

% diff_shunt_var
tff(fact_4634_diff__shunt__var,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( ( minus_minus_set_int(Xb,Yb) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(Xb),Yb) ) ).

% diff_shunt_var
tff(fact_4635_Sum__Icc__int,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),Nb)
     => ( groups4538972089207619220nt_int(aTP_Lamm_eg,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,Mb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Mb),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_4636_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_eh(L)),Qr) ).

% divmod_step_def
tff(fact_4637_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_ei(L)),Qr) ).

% divmod_step_def
tff(fact_4638_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_ej(L),Qr) ).

% divmod_step_def
tff(fact_4639_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_4640_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_4641_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_4642_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_4643_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4644_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_4645_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_4646_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_4647_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_4648_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_4649_Compl__subset__Compl__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(A4)),uminus1532241313380277803et_int(B4))
    <=> aa_set_int_o(ord_less_eq_set_int(B4),A4) ) ).

% Compl_subset_Compl_iff
tff(fact_4650_Compl__anti__mono,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(B4)),uminus1532241313380277803et_int(A4)) ) ).

% Compl_anti_mono
tff(fact_4651_compl__le__compl__iff,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Xb)),uminus1532241313380277803et_int(Yb))
    <=> aa_set_int_o(ord_less_eq_set_int(Yb),Xb) ) ).

% compl_le_compl_iff
tff(fact_4652_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% neg_le_iff_le
tff(fact_4653_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% neg_le_iff_le
tff(fact_4654_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% neg_le_iff_le
tff(fact_4655_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% neg_le_iff_le
tff(fact_4656_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_4657_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_4658_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_4659_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_4660_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_4661_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_4662_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_4663_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_4664_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_4665_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4666_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_4667_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_4668_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_4669_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_4670_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_4671_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_4672_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_4673_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_4674_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_4675_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_4676_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_4677_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_4678_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_4679_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% neg_less_iff_less
tff(fact_4680_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% neg_less_iff_less
tff(fact_4681_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% neg_less_iff_less
tff(fact_4682_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% neg_less_iff_less
tff(fact_4683_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_4684_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_4685_add__minus__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B)) = B ).

% add_minus_cancel
tff(fact_4686_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).

% add_minus_cancel
tff(fact_4687_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) = B ).

% add_minus_cancel
tff(fact_4688_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_4689_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_4690_minus__add__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(plus_plus_complex(A),B)) = B ).

% minus_add_cancel
tff(fact_4691_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(plus_plus_rat(A),B)) = B ).

% minus_add_cancel
tff(fact_4692_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),plus_p5714425477246183910nteger(A,B)) = B ).

% minus_add_cancel
tff(fact_4693_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_4694_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_4695_minus__add__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_add_distrib
tff(fact_4696_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_4697_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_add_distrib
tff(fact_4698_minus__diff__eq,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% minus_diff_eq
tff(fact_4699_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_4700_minus__diff__eq,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) = aa_complex_complex(minus_minus_complex(B),A) ).

% minus_diff_eq
tff(fact_4701_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(minus_minus_rat(B),A) ).

% minus_diff_eq
tff(fact_4702_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) = minus_8373710615458151222nteger(B,A) ).

% minus_diff_eq
tff(fact_4703_ln__less__cancel__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ) ).

% ln_less_cancel_iff
tff(fact_4704_ln__inj__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => ( ( aa_real_real(ln_ln_real,Xb) = aa_real_real(ln_ln_real,Yb) )
        <=> ( Xb = Yb ) ) ) ) ).

% ln_inj_iff
tff(fact_4705_real__add__minus__iff,axiom,
    ! [Xb: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( Xb = A ) ) ).

% real_add_minus_iff
tff(fact_4706_sum_Oneutral__const,axiom,
    ! [A4: set_int] : groups4538972089207619220nt_int(aTP_Lamm_by,A4) = zero_zero_int ).

% sum.neutral_const
tff(fact_4707_sum_Oneutral__const,axiom,
    ! [A4: set_complex] : groups7754918857620584856omplex(aTP_Lamm_bu,A4) = zero_zero_complex ).

% sum.neutral_const
tff(fact_4708_sum_Oneutral__const,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bx),A4) = zero_zero_nat ).

% sum.neutral_const
tff(fact_4709_sum_Oneutral__const,axiom,
    ! [A4: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ek),A4) = zero_zero_real ).

% sum.neutral_const
tff(fact_4710_case__prod__conv,axiom,
    ! [F: fun_na1668100129766719005at_nat,A: nat,B: nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,A),B) ).

% case_prod_conv
tff(fact_4711_case__prod__conv,axiom,
    ! [F: fun_na5914473681949881652_nat_o,A: nat,B: nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,A),B) ).

% case_prod_conv
tff(fact_4712_case__prod__conv,axiom,
    ! [F: fun_in1696471567231710611nt_int,A: int,B: int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,A),B) ).

% case_prod_conv
tff(fact_4713_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prod_conv
tff(fact_4714_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_int,A: int,B: int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int_int(aa_int_fun_int_int(F,A),B) ).

% case_prod_conv
tff(fact_4715_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_less_eq_nonneg
tff(fact_4716_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_eq_nonneg
tff(fact_4717_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_less_eq_nonneg
tff(fact_4718_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_less_eq_nonneg
tff(fact_4719_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% less_eq_neg_nonpos
tff(fact_4720_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_4721_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_4722_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_4723_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_le_0_iff_le
tff(fact_4724_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_le_0_iff_le
tff(fact_4725_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_le_0_iff_le
tff(fact_4726_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_le_0_iff_le
tff(fact_4727_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% neg_0_le_iff_le
tff(fact_4728_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_4729_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_4730_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_4731_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_0_iff_less
tff(fact_4732_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_0_iff_less
tff(fact_4733_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_0_iff_less
tff(fact_4734_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_0_iff_less
tff(fact_4735_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% neg_0_less_iff_less
tff(fact_4736_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_4737_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_4738_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_4739_neg__less__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_pos
tff(fact_4740_neg__less__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_pos
tff(fact_4741_neg__less__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_pos
tff(fact_4742_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_pos
tff(fact_4743_less__neg__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% less_neg_neg
tff(fact_4744_less__neg__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% less_neg_neg
tff(fact_4745_less__neg__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_4746_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_4747_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_4748_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_4749_add_Oright__inverse,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,A)) = zero_zero_complex ).

% add.right_inverse
tff(fact_4750_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_4751_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_4752_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_4753_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_4754_ab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_left_minus
tff(fact_4755_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_4756_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_4757_diff__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_4758_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_4759_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_4760_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_4761_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_4762_verit__minus__simplify_I3_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ).

% verit_minus_simplify(3)
tff(fact_4763_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_4764_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% verit_minus_simplify(3)
tff(fact_4765_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_4766_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_4767_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4768_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4769_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4770_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4771_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4772_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) ).

% diff_minus_eq_add
tff(fact_4773_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_4774_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(plus_plus_complex(A),B) ).

% diff_minus_eq_add
tff(fact_4775_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(plus_plus_rat(A),B) ).

% diff_minus_eq_add
tff(fact_4776_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = plus_p5714425477246183910nteger(A,B) ).

% diff_minus_eq_add
tff(fact_4777_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% uminus_add_conv_diff
tff(fact_4778_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_4779_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(minus_minus_complex(B),A) ).

% uminus_add_conv_diff
tff(fact_4780_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(minus_minus_rat(B),A) ).

% uminus_add_conv_diff
tff(fact_4781_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = minus_8373710615458151222nteger(B,A) ).

% uminus_add_conv_diff
tff(fact_4782_sum_Oempty,axiom,
    ! [G: fun_real_complex] : groups5754745047067104278omplex(G,bot_bot_set_real) = zero_zero_complex ).

% sum.empty
tff(fact_4783_sum_Oempty,axiom,
    ! [G: fun_real_real] : groups8097168146408367636l_real(G,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_4784_sum_Oempty,axiom,
    ! [G: fun_real_rat] : groups1300246762558778688al_rat(G,bot_bot_set_real) = zero_zero_rat ).

% sum.empty
tff(fact_4785_sum_Oempty,axiom,
    ! [G: fun_real_nat] : groups1935376822645274424al_nat(G,bot_bot_set_real) = zero_zero_nat ).

% sum.empty
tff(fact_4786_sum_Oempty,axiom,
    ! [G: fun_real_int] : groups1932886352136224148al_int(G,bot_bot_set_real) = zero_zero_int ).

% sum.empty
tff(fact_4787_sum_Oempty,axiom,
    ! [G: fun_nat_complex] : aa_set_nat_complex(groups2073611262835488442omplex(G),bot_bot_set_nat) = zero_zero_complex ).

% sum.empty
tff(fact_4788_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : groups2906978787729119204at_rat(G,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_4789_sum_Oempty,axiom,
    ! [G: fun_nat_int] : groups3539618377306564664at_int(G,bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_4790_sum_Oempty,axiom,
    ! [G: fun_int_complex] : groups3049146728041665814omplex(G,bot_bot_set_int) = zero_zero_complex ).

% sum.empty
tff(fact_4791_sum_Oempty,axiom,
    ! [G: fun_int_real] : groups8778361861064173332t_real(G,bot_bot_set_int) = zero_zero_real ).

% sum.empty
tff(fact_4792_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( ( groups4541462559716669496nt_nat(F,F3) = zero_zero_nat )
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),F3)
           => ( aa_int_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4793_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( ( groups5693394587270226106ex_nat(F,F3) = zero_zero_nat )
      <=> ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),F3)
           => ( aa_complex_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4794_sum__eq__0__iff,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => ( ( groups977919841031483927at_nat(F,F3) = zero_zero_nat )
      <=> ! [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),F3)
           => ( aa_Pro4243173770044397973at_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4795_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),F3) = zero_zero_nat )
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),F3)
           => ( aa_nat_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4796_sum_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_complex] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(G),A4) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_4797_sum_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_complex] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( groups3049146728041665814omplex(G,A4) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_4798_sum_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_real] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( groups8778361861064173332t_real(G,A4) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_4799_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_real] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5808333547571424918x_real(G,A4) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_4800_sum_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( groups2906978787729119204at_rat(G,A4) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_4801_sum_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_rat] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(G,A4) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_4802_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_rat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5058264527183730370ex_rat(G,A4) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_4803_sum_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_nat] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( groups4541462559716669496nt_nat(G,A4) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_4804_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_nat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5693394587270226106ex_nat(G,A4) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_4805_sum_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_int] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( groups3539618377306564664at_int(G,A4) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_4806_ln__le__cancel__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ) ).

% ln_le_cancel_iff
tff(fact_4807_ln__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% ln_less_zero_iff
tff(fact_4808_ln__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% ln_gt_zero_iff
tff(fact_4809_ln__eq__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( ( aa_real_real(ln_ln_real,Xb) = zero_zero_real )
      <=> ( Xb = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_4810_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_4811_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( finite_finite_real(S2)
     => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_el(A),B),S2) = $ite(member_real(A,S2),aa_real_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_4812_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_em(A),B)),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_4813_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_en(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_4814_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_eo(A),B),S2) = $ite(member_real(A,S2),aa_real_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_4815_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ep(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_4816_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_eq(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_4817_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( finite_finite_real(S2)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_er(A),B),S2) = $ite(member_real(A,S2),aa_real_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_4818_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_es(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_4819_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_et(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_4820_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_eu(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_4821_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( finite_finite_real(S2)
     => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_ev(A),B),S2) = $ite(member_real(A,S2),aa_real_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_4822_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ew(A),B)),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_4823_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_ex(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_4824_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ey(A),B),S2) = $ite(member_real(A,S2),aa_real_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_4825_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ez(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_4826_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_fa(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_4827_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( finite_finite_real(S2)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fb(A),B),S2) = $ite(member_real(A,S2),aa_real_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_4828_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fc(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_4829_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fd(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_4830_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fe(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_4831_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_4832_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_4833_add__neg__numeral__special_I8_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = zero_zero_complex ).

% add_neg_numeral_special(8)
tff(fact_4834_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_4835_add__neg__numeral__special_I8_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_4836_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_4837_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_4838_add__neg__numeral__special_I7_J,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% add_neg_numeral_special(7)
tff(fact_4839_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_4840_add__neg__numeral__special_I7_J,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_4841_diff__numeral__special_I12_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% diff_numeral_special(12)
tff(fact_4842_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_4843_diff__numeral__special_I12_J,axiom,
    aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% diff_numeral_special(12)
tff(fact_4844_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_4845_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_4846_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_4847_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_4848_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,U)) ).

% max_number_of(2)
tff(fact_4849_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),numera6620942414471956472nteger(U)) ).

% max_number_of(2)
tff(fact_4850_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(2)
tff(fact_4851_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(2)
tff(fact_4852_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,V),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))) ).

% max_number_of(3)
tff(fact_4853_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)),numera6620942414471956472nteger(V),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))) ).

% max_number_of(3)
tff(fact_4854_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).

% max_number_of(3)
tff(fact_4855_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(3)
tff(fact_4856_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))) ).

% max_number_of(4)
tff(fact_4857_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))) ).

% max_number_of(4)
tff(fact_4858_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).

% max_number_of(4)
tff(fact_4859_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(4)
tff(fact_4860_ln__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_4861_ln__ge__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% ln_ge_zero_iff
tff(fact_4862_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Yb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W2)))),Yb) ).

% semiring_norm(168)
tff(fact_4863_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Yb: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W2))),Yb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W2)))),Yb) ).

% semiring_norm(168)
tff(fact_4864_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Yb: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),Yb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W2)))),Yb) ).

% semiring_norm(168)
tff(fact_4865_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Yb: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),Yb)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W2)))),Yb) ).

% semiring_norm(168)
tff(fact_4866_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Yb: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W2)),Yb)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(V),W2))),Yb) ).

% semiring_norm(168)
tff(fact_4867_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4868_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4869_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4870_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4871_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(Mb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4872_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4873_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4874_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4875_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4876_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4877_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_4878_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_4879_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_4880_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_4881_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4882_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4883_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4884_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4885_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4886_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4887_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4888_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4889_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4890_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex,aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4891_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4892_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = A )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4893_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = A )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4894_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = A )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4895_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4896_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4897_add__neg__numeral__special_I9_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4898_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4899_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4900_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_4901_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_4902_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_4903_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_4904_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_4905_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_4906_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_4907_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_4908_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_4909_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_4910_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_4911_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_4912_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_4913_compl__mono,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Yb)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Yb)),uminus1532241313380277803et_int(Xb)) ) ).

% compl_mono
tff(fact_4914_compl__le__swap1,axiom,
    ! [Yb: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Yb),uminus1532241313380277803et_int(Xb))
     => aa_set_int_o(ord_less_eq_set_int(Xb),uminus1532241313380277803et_int(Yb)) ) ).

% compl_le_swap1
tff(fact_4915_compl__le__swap2,axiom,
    ! [Yb: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Yb)),Xb)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Xb)),Yb) ) ).

% compl_le_swap2
tff(fact_4916_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_4917_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_4918_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_4919_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_4920_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) = A ) ) ).

% minus_equation_iff
tff(fact_4921_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_4922_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_4923_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_4924_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_4925_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% equation_minus_iff
tff(fact_4926_prod_Ocase__distrib,axiom,
    ! [H: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(H,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun6133342665515662012_int_o(aTP_Lamm_ff(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_4927_prod_Ocase__distrib,axiom,
    ! [H: fun_o_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_int(H,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun7323606162024495668nt_int(aTP_Lamm_fg(H),F)),Prod) ).

% prod.case_distrib
tff(fact_4928_prod_Ocase__distrib,axiom,
    ! [H: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(H,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun82482788810740534_int_o(aTP_Lamm_fh(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_4929_prod_Ocase__distrib,axiom,
    ! [H: fun_int_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int_int(H,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun6119123525526402106nt_int(aTP_Lamm_fi(H),F)),Prod) ).

% prod.case_distrib
tff(fact_4930_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(H,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun4545072666309863515_int_o(aTP_Lamm_fj(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_4931_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr982841511892225912nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(H,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun7461304428735181397nt_int(aTP_Lamm_fk(H),F)),Prod) ).

% prod.case_distrib
tff(fact_4932_prod_Ocase__distrib,axiom,
    ! [H: fun_o_8988054265482332260nt_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_P3134438092330336605nt_int(H,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun4023131440832735197nt_int(aTP_Lamm_fl(H),F)),Prod) ).

% prod.case_distrib
tff(fact_4933_prod_Ocase__distrib,axiom,
    ! [H: fun_in4211310926729683942nt_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(H,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun1034830265760468567nt_int(aTP_Lamm_fm(H),F)),Prod) ).

% prod.case_distrib
tff(fact_4934_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr2478283820680522243nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(H,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun4152872853588681340nt_int(aTP_Lamm_fn(H),F)),Prod) ).

% prod.case_distrib
tff(fact_4935_prod_Ocase__distrib,axiom,
    ! [H: fun_fu3275390434952296369_nat_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_fun359341089595923442_nat_o(H,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),Prod)) = aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aa_fun824319262034445606_nat_o(aTP_Lamm_fo(H),F)),Prod) ).

% prod.case_distrib
tff(fact_4936_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A4: set_real] :
      ( ( groups5754745047067104278omplex(G,A4) != zero_zero_complex )
     => ~ ! [A3: real] :
            ( member_real(A3,A4)
           => ( aa_real_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4937_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A4: set_nat] :
      ( ( aa_set_nat_complex(groups2073611262835488442omplex(G),A4) != zero_zero_complex )
     => ~ ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A4)
           => ( aa_nat_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4938_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A4: set_int] :
      ( ( groups3049146728041665814omplex(G,A4) != zero_zero_complex )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A4)
           => ( aa_int_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4939_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_real,A4: set_complex] :
      ( ( groups5808333547571424918x_real(G,A4) != zero_zero_real )
     => ~ ! [A3: complex] :
            ( aa_set_complex_o(member_complex(A3),A4)
           => ( aa_complex_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4940_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A4: set_real] :
      ( ( groups8097168146408367636l_real(G,A4) != zero_zero_real )
     => ~ ! [A3: real] :
            ( member_real(A3,A4)
           => ( aa_real_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4941_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A4: set_int] :
      ( ( groups8778361861064173332t_real(G,A4) != zero_zero_real )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A4)
           => ( aa_int_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4942_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_rat,A4: set_complex] :
      ( ( groups5058264527183730370ex_rat(G,A4) != zero_zero_rat )
     => ~ ! [A3: complex] :
            ( aa_set_complex_o(member_complex(A3),A4)
           => ( aa_complex_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4943_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A4: set_real] :
      ( ( groups1300246762558778688al_rat(G,A4) != zero_zero_rat )
     => ~ ! [A3: real] :
            ( member_real(A3,A4)
           => ( aa_real_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4944_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A4: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A4) != zero_zero_rat )
     => ~ ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A4)
           => ( aa_nat_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4945_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A4: set_int] :
      ( ( groups3906332499630173760nt_rat(G,A4) != zero_zero_rat )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A4)
           => ( aa_int_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4946_sum_Oneutral,axiom,
    ! [A4: set_int,G: fun_int_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => ( aa_int_int(G,X3) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A4) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_4947_sum_Oneutral,axiom,
    ! [A4: set_complex,G: fun_complex_complex] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => ( aa_complex_complex(G,X3) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A4) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_4948_sum_Oneutral,axiom,
    ! [A4: set_nat,G: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => ( aa_nat_nat(G,X3) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A4) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_4949_sum_Oneutral,axiom,
    ! [A4: set_nat,G: fun_nat_real] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => ( aa_nat_real(G,X3) = zero_zero_real ) )
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),A4) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_4950_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% le_minus_iff
tff(fact_4951_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_minus_iff
tff(fact_4952_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_minus_iff
tff(fact_4953_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% le_minus_iff
tff(fact_4954_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_le_iff
tff(fact_4955_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_le_iff
tff(fact_4956_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_le_iff
tff(fact_4957_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_le_iff
tff(fact_4958_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% le_imp_neg_le
tff(fact_4959_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_imp_neg_le
tff(fact_4960_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_imp_neg_le
tff(fact_4961_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% le_imp_neg_le
tff(fact_4962_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_less_iff
tff(fact_4963_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_less_iff
tff(fact_4964_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_less_iff
tff(fact_4965_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_less_iff
tff(fact_4966_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% less_minus_iff
tff(fact_4967_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% less_minus_iff
tff(fact_4968_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% less_minus_iff
tff(fact_4969_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% less_minus_iff
tff(fact_4970_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_4971_is__num__normalize_I8_J,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)) ).

% is_num_normalize(8)
tff(fact_4972_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% is_num_normalize(8)
tff(fact_4973_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% is_num_normalize(8)
tff(fact_4974_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% is_num_normalize(8)
tff(fact_4975_group__cancel_Oneg1,axiom,
    ! [A4: real,K2: real,A: real] :
      ( ( A4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),A) )
     => ( aa_real_real(uminus_uminus_real,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K2)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_4976_group__cancel_Oneg1,axiom,
    ! [A4: int,K2: int,A: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(uminus_uminus_int,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K2)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_4977_group__cancel_Oneg1,axiom,
    ! [A4: complex,K2: complex,A: complex] :
      ( ( A4 = aa_complex_complex(plus_plus_complex(K2),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A4) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K2)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_4978_group__cancel_Oneg1,axiom,
    ! [A4: rat,K2: rat,A: rat] :
      ( ( A4 = aa_rat_rat(plus_plus_rat(K2),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A4) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K2)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_4979_group__cancel_Oneg1,axiom,
    ! [A4: code_integer,K2: code_integer,A: code_integer] :
      ( ( A4 = plus_p5714425477246183910nteger(K2,A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A4) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_4980_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_4981_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_4982_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% add.inverse_distrib_swap
tff(fact_4983_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% add.inverse_distrib_swap
tff(fact_4984_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% add.inverse_distrib_swap
tff(fact_4985_minus__diff__commute,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_diff_commute
tff(fact_4986_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_4987_minus__diff__commute,axiom,
    ! [B: complex,A: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_diff_commute
tff(fact_4988_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_diff_commute
tff(fact_4989_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_diff_commute
tff(fact_4990_split__cong,axiom,
    ! [Q3: product_prod_nat_nat,F: fun_na1668100129766719005at_nat,G: fun_na1668100129766719005at_nat,P: product_prod_nat_nat] :
      ( ! [X3: nat,Y: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y) = Q3 )
         => ( aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X3),Y) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(G,X3),Y) ) )
     => ( ( P = Q3 )
       => ( aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),P) = aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,G),Q3) ) ) ) ).

% split_cong
tff(fact_4991_split__cong,axiom,
    ! [Q3: product_prod_nat_nat,F: fun_na5914473681949881652_nat_o,G: fun_na5914473681949881652_nat_o,P: product_prod_nat_nat] :
      ( ! [X3: nat,Y: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y) = Q3 )
         => ( aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X3),Y) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(G,X3),Y) ) )
     => ( ( P = Q3 )
       => ( aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),P) = aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,G),Q3) ) ) ) ).

% split_cong
tff(fact_4992_split__cong,axiom,
    ! [Q3: product_prod_int_int,F: fun_in1696471567231710611nt_int,G: fun_in1696471567231710611nt_int,P: product_prod_int_int] :
      ( ! [X3: int,Y: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) = Q3 )
         => ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X3),Y) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(G,X3),Y) ) )
     => ( ( P = Q3 )
       => ( aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),P) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,G),Q3) ) ) ) ).

% split_cong
tff(fact_4993_split__cong,axiom,
    ! [Q3: product_prod_int_int,F: fun_int_fun_int_o,G: fun_int_fun_int_o,P: product_prod_int_int] :
      ( ! [X3: int,Y: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) = Q3 )
         => ( aa_int_o(aa_int_fun_int_o(F,X3),Y)
          <=> aa_int_o(aa_int_fun_int_o(G,X3),Y) ) )
     => ( ( P = Q3 )
       => ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),P)
        <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,G),Q3) ) ) ) ).

% split_cong
tff(fact_4994_split__cong,axiom,
    ! [Q3: product_prod_int_int,F: fun_int_fun_int_int,G: fun_int_fun_int_int,P: product_prod_int_int] :
      ( ! [X3: int,Y: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) = Q3 )
         => ( aa_int_int(aa_int_fun_int_int(F,X3),Y) = aa_int_int(aa_int_fun_int_int(G,X3),Y) ) )
     => ( ( P = Q3 )
       => ( aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),P) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,G),Q3) ) ) ) ).

% split_cong
tff(fact_4995_old_Oprod_Ocase,axiom,
    ! [F: fun_na1668100129766719005at_nat,X1: nat,X2: nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X2) ).

% old.prod.case
tff(fact_4996_old_Oprod_Ocase,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X1: nat,X2: nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X1),X2) ).

% old.prod.case
tff(fact_4997_old_Oprod_Ocase,axiom,
    ! [F: fun_in1696471567231710611nt_int,X1: int,X2: int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X1),X2) ).

% old.prod.case
tff(fact_4998_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_o,X1: int,X2: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2))
    <=> aa_int_o(aa_int_fun_int_o(F,X1),X2) ) ).

% old.prod.case
tff(fact_4999_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_int,X1: int,X2: int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = aa_int_int(aa_int_fun_int_int(F,X1),X2) ).

% old.prod.case
tff(fact_5000_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_5001_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,K5)),groups5058264527183730370ex_rat(G,K5)) ) ).

% sum_mono
tff(fact_5002_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( member_real(I2,K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,K5)),groups1300246762558778688al_rat(G,K5)) ) ).

% sum_mono
tff(fact_5003_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,K5)),groups2906978787729119204at_rat(G,K5)) ) ).

% sum_mono
tff(fact_5004_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5)) ) ).

% sum_mono
tff(fact_5005_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,K5)),groups5693394587270226106ex_nat(G,K5)) ) ).

% sum_mono
tff(fact_5006_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( member_real(I2,K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5)) ) ).

% sum_mono
tff(fact_5007_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5)) ) ).

% sum_mono
tff(fact_5008_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I2)),aa_complex_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,K5)),groups5690904116761175830ex_int(G,K5)) ) ).

% sum_mono
tff(fact_5009_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( member_real(I2,K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,K5)),groups1932886352136224148al_int(G,K5)) ) ).

% sum_mono
tff(fact_5010_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,K5)),groups3539618377306564664at_int(G,K5)) ) ).

% sum_mono
tff(fact_5011_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A4: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_fp(G),H),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A4)),groups4538972089207619220nt_int(H,A4)) ).

% sum.distrib
tff(fact_5012_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A4: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_fq(G),H),A4) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A4)),groups7754918857620584856omplex(H,A4)) ).

% sum.distrib
tff(fact_5013_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A4: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fr(G),H)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A4)) ).

% sum.distrib
tff(fact_5014_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A4: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_fs(G),H)),A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),A4)),aa_set_nat_real(groups6591440286371151544t_real(H),A4)) ).

% sum.distrib
tff(fact_5015_sum_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_int,G: fun_real_fun_int_int,R: fun_real_fun_int_o] :
      ( finite_finite_real(A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups1932886352136224148al_int(aa_fun7555737461897725707al_int(aa_fun1679075639499870878al_int(aTP_Lamm_fu(B4),G),R),A4) = groups4538972089207619220nt_int(aa_fun3996843141751676939nt_int(aa_fun129049606478405534nt_int(aTP_Lamm_fw(A4),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5016_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_int,G: fun_nat_fun_int_int,R: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups3539618377306564664at_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_fy(B4),G),R),A4) = groups4538972089207619220nt_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_ga(A4),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5017_sum_Oswap__restrict,axiom,
    ! [A4: set_complex,B4: set_int,G: fun_co333780937270390912nt_int,R: fun_co3431063785546946928_int_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups5690904116761175830ex_int(aa_fun5055871773053899019ex_int(aa_fun7361901416012365340ex_int(aTP_Lamm_gc(B4),G),R),A4) = groups4538972089207619220nt_int(aa_fun2158622962795119753nt_int(aa_fun607866698265288346nt_int(aTP_Lamm_gf(A4),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5018_sum_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_complex,G: fun_re7208826350674946562omplex,R: fun_re7319081385389557872plex_o] :
      ( finite_finite_real(A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups5754745047067104278omplex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_gh(B4),G),R),A4) = groups7754918857620584856omplex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_gj(A4),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5019_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_complex,G: fun_na1591161823130732966omplex,R: fun_na1554462459066888140plex_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_gl(B4),G),R)),A4) = groups7754918857620584856omplex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_gn(A4),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5020_sum_Oswap__restrict,axiom,
    ! [A4: set_int,B4: set_complex,G: fun_in1307110748760623106omplex,R: fun_in5065666251602127472plex_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups3049146728041665814omplex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_gp(B4),G),R),A4) = groups7754918857620584856omplex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_gr(A4),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5021_sum_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_nat,G: fun_real_fun_nat_nat,R: fun_real_fun_nat_o] :
      ( finite_finite_real(A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups1935376822645274424al_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_gt(B4),G),R),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_gv(A4),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5022_sum_Oswap__restrict,axiom,
    ! [A4: set_int,B4: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_gx(B4),G),R),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_gz(A4),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5023_sum_Oswap__restrict,axiom,
    ! [A4: set_complex,B4: set_nat,G: fun_co636942507509095880at_nat,R: fun_co5676208753217392844_nat_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups5693394587270226106ex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_hb(B4),G),R),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_hd(A4),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5024_sum_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_nat,G: fun_re4238438383274954274t_real,R: fun_real_fun_nat_o] :
      ( finite_finite_real(A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups8097168146408367636l_real(aa_fun6320755270648597735l_real(aa_fun6240130701872165846l_real(aTP_Lamm_he(B4),G),R),A4) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun6247686994378451851t_real(aa_fun7274880274332377722t_real(aTP_Lamm_hg(A4),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5025_cond__case__prod__eta,axiom,
    ! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
      ( ! [X3: nat,Y: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X3),Y) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y))
     => ( aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5026_cond__case__prod__eta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,G: fun_Pr2028404973800837934_nat_o] :
      ( ! [X3: nat,Y: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X3),Y) = aa_Pro989595756709598047_nat_o(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y))
     => ( aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5027_cond__case__prod__eta,axiom,
    ! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X3: int,Y: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X3),Y) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y))
     => ( aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5028_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
      ( ! [X3: int,Y: int] :
          ( aa_int_o(aa_int_fun_int_o(F,X3),Y)
        <=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y)) )
     => ( aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5029_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_int,G: fun_Pr982841511892225912nt_int] :
      ( ! [X3: int,Y: int] : aa_int_int(aa_int_fun_int_int(F,X3),Y) = aa_Pro6799258434465214121nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y))
     => ( aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5030_case__prod__eta,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_hh(F)) = F ).

% case_prod_eta
tff(fact_5031_case__prod__eta,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_hi(F)) = F ).

% case_prod_eta
tff(fact_5032_case__prod__eta,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_hj(F)) = F ).

% case_prod_eta
tff(fact_5033_case__prod__eta,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_hk(F)) = F ).

% case_prod_eta
tff(fact_5034_case__prod__eta,axiom,
    ! [F: fun_Pr982841511892225912nt_int] : aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aTP_Lamm_hl(F)) = F ).

% case_prod_eta
tff(fact_5035_case__prodE2,axiom,
    ! [Q: fun_fu8391177546919232046_nat_o,P2: fun_na1668100129766719005at_nat,Z: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(Q,aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,P2),Z))
     => ~ ! [X3: nat,Y: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y) )
           => ~ aa_fun3538214411680374423_nat_o(Q,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(P2,X3),Y)) ) ) ).

% case_prodE2
tff(fact_5036_case__prodE2,axiom,
    ! [Q: fun_fu6891447142836604759at_o_o,P2: fun_na5914473681949881652_nat_o,Z: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(Q,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,P2),Z))
     => ~ ! [X3: nat,Y: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y) )
           => ~ aa_fun8408151826566693824at_o_o(Q,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(P2,X3),Y)) ) ) ).

% case_prodE2
tff(fact_5037_case__prodE2,axiom,
    ! [Q: fun_Pr3238374608701176162_int_o,P2: fun_in1696471567231710611nt_int,Z: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Q,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,P2),Z))
     => ~ ! [X3: int,Y: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) )
           => ~ aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P2,X3),Y)) ) ) ).

% case_prodE2
tff(fact_5038_case__prodE2,axiom,
    ! [Q: fun_o_o,P2: fun_int_fun_int_o,Z: product_prod_int_int] :
      ( aa_o_o(Q,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,P2),Z))
     => ~ ! [X3: int,Y: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) )
           => ~ aa_o_o(Q,aa_int_o(aa_int_fun_int_o(P2,X3),Y)) ) ) ).

% case_prodE2
tff(fact_5039_case__prodE2,axiom,
    ! [Q: fun_int_o,P2: fun_int_fun_int_int,Z: product_prod_int_int] :
      ( aa_int_o(Q,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,P2),Z))
     => ~ ! [X3: int,Y: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) )
           => ~ aa_int_o(Q,aa_int_int(aa_int_fun_int_int(P2,X3),Y)) ) ) ).

% case_prodE2
tff(fact_5040_sum__nonpos,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X3)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A4)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_5041_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X3)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A4)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_5042_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A4)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_5043_sum__nonpos,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_5044_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( member_real(X3,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_5045_sum__nonpos,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_5046_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_5047_sum__nonpos,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X3)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A4)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_5048_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X3)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A4)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_5049_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A4)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_5050_sum__nonneg,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups5808333547571424918x_real(F,A4)) ) ).

% sum_nonneg
tff(fact_5051_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A4)) ) ).

% sum_nonneg
tff(fact_5052_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A4)) ) ).

% sum_nonneg
tff(fact_5053_sum__nonneg,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_5054_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( member_real(X3,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_5055_sum__nonneg,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_5056_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_5057_sum__nonneg,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,A4)) ) ).

% sum_nonneg
tff(fact_5058_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A4)) ) ).

% sum_nonneg
tff(fact_5059_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A4)) ) ).

% sum_nonneg
tff(fact_5060_ln__add__one__self__le__self2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self2
tff(fact_5061_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I6: set_real,G: fun_real_rat,I: real] :
      ( ( groups1300246762558778688al_rat(F,I6) = groups1300246762558778688al_rat(G,I6) )
     => ( ! [I2: real] :
            ( member_real(I2,I6)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2)) )
       => ( member_real(I,I6)
         => ( finite_finite_real(I6)
           => ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5062_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I6: set_nat,G: fun_nat_rat,I: nat] :
      ( ( groups2906978787729119204at_rat(F,I6) = groups2906978787729119204at_rat(G,I6) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I6)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) )
       => ( aa_set_nat_o(member_nat(I),I6)
         => ( aa_set_nat_o(finite_finite_nat,I6)
           => ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5063_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I6: set_int,G: fun_int_rat,I: int] :
      ( ( groups3906332499630173760nt_rat(F,I6) = groups3906332499630173760nt_rat(G,I6) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I6)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) )
       => ( aa_set_int_o(member_int(I),I6)
         => ( aa_set_int_o(finite_finite_int,I6)
           => ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5064_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I6: set_complex,G: fun_complex_rat,I: complex] :
      ( ( groups5058264527183730370ex_rat(F,I6) = groups5058264527183730370ex_rat(G,I6) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I6)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2)) )
       => ( aa_set_complex_o(member_complex(I),I6)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I6)
           => ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5065_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I6: set_real,G: fun_real_nat,I: real] :
      ( ( groups1935376822645274424al_nat(F,I6) = groups1935376822645274424al_nat(G,I6) )
     => ( ! [I2: real] :
            ( member_real(I2,I6)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) )
       => ( member_real(I,I6)
         => ( finite_finite_real(I6)
           => ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5066_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I6: set_int,G: fun_int_nat,I: int] :
      ( ( groups4541462559716669496nt_nat(F,I6) = groups4541462559716669496nt_nat(G,I6) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I6)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) )
       => ( aa_set_int_o(member_int(I),I6)
         => ( aa_set_int_o(finite_finite_int,I6)
           => ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5067_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I6: set_complex,G: fun_complex_nat,I: complex] :
      ( ( groups5693394587270226106ex_nat(F,I6) = groups5693394587270226106ex_nat(G,I6) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I6)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) )
       => ( aa_set_complex_o(member_complex(I),I6)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I6)
           => ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5068_sum__mono__inv,axiom,
    ! [F: fun_real_int,I6: set_real,G: fun_real_int,I: real] :
      ( ( groups1932886352136224148al_int(F,I6) = groups1932886352136224148al_int(G,I6) )
     => ( ! [I2: real] :
            ( member_real(I2,I6)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2)) )
       => ( member_real(I,I6)
         => ( finite_finite_real(I6)
           => ( aa_real_int(F,I) = aa_real_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5069_sum__mono__inv,axiom,
    ! [F: fun_nat_int,I6: set_nat,G: fun_nat_int,I: nat] :
      ( ( groups3539618377306564664at_int(F,I6) = groups3539618377306564664at_int(G,I6) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I6)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2)) )
       => ( aa_set_nat_o(member_nat(I),I6)
         => ( aa_set_nat_o(finite_finite_nat,I6)
           => ( aa_nat_int(F,I) = aa_nat_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5070_sum__mono__inv,axiom,
    ! [F: fun_complex_int,I6: set_complex,G: fun_complex_int,I: complex] :
      ( ( groups5690904116761175830ex_int(F,I6) = groups5690904116761175830ex_int(G,I6) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I6)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I2)),aa_complex_int(G,I2)) )
       => ( aa_set_complex_o(member_complex(I),I6)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I6)
           => ( aa_complex_int(F,I) = aa_complex_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5071_ln__less__self,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),Xb) ) ).

% ln_less_self
tff(fact_5072_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5073_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5074_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5075_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5076_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5077_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_le_numeral
tff(fact_5078_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5079_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5080_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5081_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5082_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% zero_neq_neg_numeral
tff(fact_5083_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5084_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% zero_neq_neg_numeral
tff(fact_5085_le__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(4)
tff(fact_5086_le__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(4)
tff(fact_5087_le__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(4)
tff(fact_5088_le__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(4)
tff(fact_5089_le__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% le_minus_one_simps(2)
tff(fact_5090_le__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% le_minus_one_simps(2)
tff(fact_5091_le__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% le_minus_one_simps(2)
tff(fact_5092_le__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% le_minus_one_simps(2)
tff(fact_5093_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_5094_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_5095_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_5096_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_5097_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_5098_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_5099_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_5100_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% add_eq_0_iff
tff(fact_5101_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_5102_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% add_eq_0_iff
tff(fact_5103_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_5104_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_5105_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_5106_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_5107_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_5108_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_5109_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_5110_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = B ) ) ).

% add.inverse_unique
tff(fact_5111_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_5112_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).

% add.inverse_unique
tff(fact_5113_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_5114_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_5115_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5116_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5117_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5118_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_5119_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_5120_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5121_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5122_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5123_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5124_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5125_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5126_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5127_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5128_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5129_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5130_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_5131_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5132_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5133_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5134_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_5135_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_5136_diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% diff_conv_add_uminus
tff(fact_5137_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_5138_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% diff_conv_add_uminus
tff(fact_5139_group__cancel_Osub2,axiom,
    ! [B4: real,K2: real,B: real,A: real] :
      ( ( B4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5140_group__cancel_Osub2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5141_group__cancel_Osub2,axiom,
    ! [B4: complex,K2: complex,B: complex,A: complex] :
      ( ( B4 = aa_complex_complex(plus_plus_complex(K2),B) )
     => ( aa_complex_complex(minus_minus_complex(A),B4) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K2)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5142_group__cancel_Osub2,axiom,
    ! [B4: rat,K2: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(plus_plus_rat(K2),B) )
     => ( aa_rat_rat(minus_minus_rat(A),B4) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K2)),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5143_group__cancel_Osub2,axiom,
    ! [B4: code_integer,K2: code_integer,B: code_integer,A: code_integer] :
      ( ( B4 = plus_p5714425477246183910nteger(K2,B) )
     => ( minus_8373710615458151222nteger(A,B4) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_5144_subset__Compl__self__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A4),uminus6524753893492686040at_nat(A4))
    <=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% subset_Compl_self_eq
tff(fact_5145_subset__Compl__self__eq,axiom,
    ! [A4: set_real] :
      ( ord_less_eq_set_real(A4,uminus612125837232591019t_real(A4))
    <=> ( A4 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_5146_subset__Compl__self__eq,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),uminus5710092332889474511et_nat(A4))
    <=> ( A4 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_5147_subset__Compl__self__eq,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),uminus1532241313380277803et_int(A4))
    <=> ( A4 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_5148_real__minus__mult__self__le,axiom,
    ! [U: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)) ).

% real_minus_mult_self_le
tff(fact_5149_pos__zmult__eq__1__iff__lemma,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
     => ( ( Mb = one_one_int )
        | ( Mb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_5150_zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
    <=> ( ( ( Mb = one_one_int )
          & ( Nb = one_one_int ) )
        | ( ( Mb = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( Nb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_5151_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_5152_sum_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_complex,P2: fun_real_o] :
      ( finite_finite_real(A4)
     => ( groups5754745047067104278omplex(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bf(A4),P2))) = groups5754745047067104278omplex(aa_fun1098856492951973088omplex(aTP_Lamm_hm(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5153_sum_Ointer__filter,axiom,
    ! [A4: set_nat,G: fun_nat_complex,P2: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bi(A4),P2))) = aa_set_nat_complex(groups2073611262835488442omplex(aa_fun1400689748890728928omplex(aTP_Lamm_hn(G),P2)),A4) ) ) ).

% sum.inter_filter
tff(fact_5154_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_complex,P2: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3049146728041665814omplex(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bj(A4),P2))) = groups3049146728041665814omplex(aa_fun6439487526662521568omplex(aTP_Lamm_ho(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5155_sum_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_real,P2: fun_real_o] :
      ( finite_finite_real(A4)
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bf(A4),P2))) = groups8097168146408367636l_real(aa_fun4448911596306054750l_real(aTP_Lamm_hp(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5156_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_real,P2: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bj(A4),P2))) = groups8778361861064173332t_real(aa_fun7506669883404309598t_real(aTP_Lamm_hq(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5157_sum_Ointer__filter,axiom,
    ! [A4: set_complex,G: fun_complex_real,P2: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(A4),P2))) = groups5808333547571424918x_real(aa_fun8261057609414086622x_real(aTP_Lamm_hr(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5158_sum_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_rat,P2: fun_real_o] :
      ( finite_finite_real(A4)
     => ( groups1300246762558778688al_rat(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bf(A4),P2))) = groups1300246762558778688al_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_hs(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5159_sum_Ointer__filter,axiom,
    ! [A4: set_nat,G: fun_nat_rat,P2: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups2906978787729119204at_rat(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bi(A4),P2))) = groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_ht(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5160_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_rat,P2: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bj(A4),P2))) = groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_hu(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5161_sum_Ointer__filter,axiom,
    ! [A4: set_complex,G: fun_complex_rat,P2: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5058264527183730370ex_rat(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(A4),P2))) = groups5058264527183730370ex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_hv(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5162_zmod__zminus2__not__zero,axiom,
    ! [K2: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_5163_zmod__zminus1__not__zero,axiom,
    ! [K2: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K2)),L) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_5164_minus__real__def,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Yb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(uminus_uminus_real,Yb)) ).

% minus_real_def
tff(fact_5165_internal__case__prod__def,axiom,
    produc8104577556401574439at_nat = produc4528391901023682385at_nat ).

% internal_case_prod_def
tff(fact_5166_internal__case__prod__def,axiom,
    produc9189008374811844670_nat_o = produc6395135316500328296_nat_o ).

% internal_case_prod_def
tff(fact_5167_internal__case__prod__def,axiom,
    produc297006045350968285nt_int = produc4245557441103728435nt_int ).

% internal_case_prod_def
tff(fact_5168_internal__case__prod__def,axiom,
    produc8005341501107743676_int_o = produc4947309494688390418_int_o ).

% internal_case_prod_def
tff(fact_5169_internal__case__prod__def,axiom,
    produc7926200574084438792nt_int = produc8211389475949308722nt_int ).

% internal_case_prod_def
tff(fact_5170_ln__one__minus__pos__upper__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))),aa_real_real(uminus_uminus_real,Xb)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_5171_sum__le__included,axiom,
    ! [S3: set_int,Ta: set_int,G: fun_int_real,I: fun_int_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S3)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Ta)
                    & ( aa_int_int(I,Xa) = X3 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),aa_int_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,S3)),groups8778361861064173332t_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5172_sum__le__included,axiom,
    ! [S3: set_int,Ta: set_complex,G: fun_complex_real,I: fun_complex_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S3)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_int(I,Xa) = X3 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,S3)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5173_sum__le__included,axiom,
    ! [S3: set_complex,Ta: set_int,G: fun_int_real,I: fun_int_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X3)) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S3)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Ta)
                    & ( aa_int_complex(I,Xa) = X3 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X3)),aa_int_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,S3)),groups8778361861064173332t_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5174_sum__le__included,axiom,
    ! [S3: set_complex,Ta: set_complex,G: fun_complex_real,I: fun_complex_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X3)) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S3)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_complex(I,Xa) = X3 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X3)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,S3)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5175_sum__le__included,axiom,
    ! [S3: set_nat,Ta: set_nat,G: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S3)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_nat(I,Xa) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S3)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5176_sum__le__included,axiom,
    ! [S3: set_nat,Ta: set_int,G: fun_int_rat,I: fun_int_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S3)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Ta)
                    & ( aa_int_nat(I,Xa) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_int_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S3)),groups3906332499630173760nt_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5177_sum__le__included,axiom,
    ! [S3: set_nat,Ta: set_complex,G: fun_complex_rat,I: fun_complex_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S3)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_nat(I,Xa) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_complex_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S3)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5178_sum__le__included,axiom,
    ! [S3: set_int,Ta: set_nat,G: fun_nat_rat,I: fun_nat_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S3)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_int(I,Xa) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S3)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5179_sum__le__included,axiom,
    ! [S3: set_int,Ta: set_int,G: fun_int_rat,I: fun_int_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S3)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Ta)
                    & ( aa_int_int(I,Xa) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S3)),groups3906332499630173760nt_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5180_sum__le__included,axiom,
    ! [S3: set_int,Ta: set_complex,G: fun_complex_rat,I: fun_complex_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S3)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_int(I,Xa) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_complex_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S3)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5181_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( finite_finite_real(A4)
     => ( ! [X3: real] :
            ( member_real(X3,A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
       => ( ( groups8097168146408367636l_real(F,A4) = zero_zero_real )
        <=> ! [X4: real] :
              ( member_real(X4,A4)
             => ( aa_real_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5182_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3)) )
       => ( ( groups8778361861064173332t_real(F,A4) = zero_zero_real )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A4)
             => ( aa_int_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5183_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3)) )
       => ( ( groups5808333547571424918x_real(F,A4) = zero_zero_real )
        <=> ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A4)
             => ( aa_complex_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5184_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( finite_finite_real(A4)
     => ( ! [X3: real] :
            ( member_real(X3,A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3)) )
       => ( ( groups1300246762558778688al_rat(F,A4) = zero_zero_rat )
        <=> ! [X4: real] :
              ( member_real(X4,A4)
             => ( aa_real_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5185_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3)) )
       => ( ( groups2906978787729119204at_rat(F,A4) = zero_zero_rat )
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A4)
             => ( aa_nat_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5186_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3)) )
       => ( ( groups3906332499630173760nt_rat(F,A4) = zero_zero_rat )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A4)
             => ( aa_int_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5187_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3)) )
       => ( ( groups5058264527183730370ex_rat(F,A4) = zero_zero_rat )
        <=> ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A4)
             => ( aa_complex_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5188_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( finite_finite_real(A4)
     => ( ! [X3: real] :
            ( member_real(X3,A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3)) )
       => ( ( groups1935376822645274424al_nat(F,A4) = zero_zero_nat )
        <=> ! [X4: real] :
              ( member_real(X4,A4)
             => ( aa_real_nat(F,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5189_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3)) )
       => ( ( groups4541462559716669496nt_nat(F,A4) = zero_zero_nat )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A4)
             => ( aa_int_nat(F,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5190_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3)) )
       => ( ( groups5693394587270226106ex_nat(F,A4) = zero_zero_nat )
        <=> ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A4)
             => ( aa_complex_nat(F,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5191_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),aa_int_real(G,X3)) )
       => ( ? [X: int] :
              ( aa_set_int_o(member_int(X),A4)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5192_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X3)),aa_complex_real(G,X3)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A4)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X)),aa_complex_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5193_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5194_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3)) )
       => ( ? [X: int] :
              ( aa_set_int_o(member_int(X),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A4)),groups3906332499630173760nt_rat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5195_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X3)),aa_complex_rat(G,X3)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X)),aa_complex_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5196_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3)) )
       => ( ? [X: int] :
              ( aa_set_int_o(member_int(X),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X)),aa_int_nat(G,X)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5197_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X3)),aa_complex_nat(G,X3)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X)),aa_complex_nat(G,X)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5198_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X3)),aa_nat_int(G,X3)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X)),aa_nat_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,A4)),groups3539618377306564664at_int(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5199_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X3)),aa_complex_int(G,X3)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_complex_int(F,X)),aa_complex_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A4)),groups5690904116761175830ex_int(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5200_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X3)),aa_int_int(G,X3)) )
       => ( ? [X: int] :
              ( aa_set_int_o(member_int(X),A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X)),aa_int_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups4538972089207619220nt_int(F,A4)),groups4538972089207619220nt_int(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5201_sum_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,zero_zero_complex),zero_zero_complex)
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_nat_complex(H,X3)),aa_nat_complex(G,X3)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_set_nat_complex(groups2073611262835488442omplex(H),S2)),aa_set_nat_complex(groups2073611262835488442omplex(G),S2)) ) ) ) ) ).

% sum.related
tff(fact_5202_sum_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S2: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,zero_zero_complex),zero_zero_complex)
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_int_complex(H,X3)),aa_int_complex(G,X3)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,groups3049146728041665814omplex(H,S2)),groups3049146728041665814omplex(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_5203_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_int_real(H,X3)),aa_int_real(G,X3)) )
           => aa_real_o(aa_real_fun_real_o(R,groups8778361861064173332t_real(H,S2)),groups8778361861064173332t_real(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_5204_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_complex_real(H,X3)),aa_complex_real(G,X3)) )
           => aa_real_o(aa_real_fun_real_o(R,groups5808333547571424918x_real(H,S2)),groups5808333547571424918x_real(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_5205_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H,X3)),aa_nat_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups2906978787729119204at_rat(H,S2)),groups2906978787729119204at_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_5206_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_int_rat(H,X3)),aa_int_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups3906332499630173760nt_rat(H,S2)),groups3906332499630173760nt_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_5207_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_complex_rat(H,X3)),aa_complex_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups5058264527183730370ex_rat(H,S2)),groups5058264527183730370ex_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_5208_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,zero_zero_nat),zero_zero_nat)
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_int_nat(H,X3)),aa_int_nat(G,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups4541462559716669496nt_nat(H,S2)),groups4541462559716669496nt_nat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_5209_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,zero_zero_nat),zero_zero_nat)
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_complex_nat(H,X3)),aa_complex_nat(G,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups5693394587270226106ex_nat(H,S2)),groups5693394587270226106ex_nat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_5210_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S2: set_nat,H: fun_nat_int,G: fun_nat_int] :
      ( aa_int_o(aa_int_fun_int_o(R,zero_zero_int),zero_zero_int)
     => ( ! [X15: int,Y15: int,X22: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X15),X22)
              & aa_int_o(aa_int_fun_int_o(R,Y15),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X15),Y15)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => aa_int_o(aa_int_fun_int_o(R,aa_nat_int(H,X3)),aa_nat_int(G,X3)) )
           => aa_int_o(aa_int_fun_int_o(R,groups3539618377306564664at_int(H,S2)),groups3539618377306564664at_int(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_5211_sum__strict__mono,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X3)),aa_complex_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_5212_sum__strict__mono,axiom,
    ! [A4: set_real,F: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( member_real(X3,A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_5213_sum__strict__mono,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_5214_sum__strict__mono,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X3)),aa_complex_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_5215_sum__strict__mono,axiom,
    ! [A4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( member_real(X3,A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_5216_sum__strict__mono,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_5217_sum__strict__mono,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A4)),groups3906332499630173760nt_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_5218_sum__strict__mono,axiom,
    ! [A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X3)),aa_complex_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_5219_sum__strict__mono,axiom,
    ! [A4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( member_real(X3,A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_5220_sum__strict__mono,axiom,
    ! [A4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_5221_ln__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),Xb) ) ).

% ln_bound
tff(fact_5222_ln__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb)) ) ).

% ln_gt_zero
tff(fact_5223_ln__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real) ) ) ).

% ln_less_zero
tff(fact_5224_ln__gt__zero__imp__gt__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_5225_ln__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb)) ) ).

% ln_ge_zero
tff(fact_5226_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(S5)
     => ( finite_finite_real(T3)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S2,S5))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S2,S5))
               => member_real(aa_real_real(J,A3),minus_minus_set_real(T,T3)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T3))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T3))
                   => member_real(aa_real_real(I,B3),minus_minus_set_real(S2,S5)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S5)
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T3)
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S2)
                         => ( aa_real_complex(H,aa_real_real(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S2) = groups5754745047067104278omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5227_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( finite_finite_real(S5)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S2,S5))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S2,S5))
               => aa_set_int_o(member_int(aa_real_int(J,A3)),minus_minus_set_int(T,T3)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(T,T3))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int(B3),minus_minus_set_int(T,T3))
                   => member_real(aa_int_real(I,B3),minus_minus_set_real(S2,S5)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S5)
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int(B3),T3)
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S2)
                         => ( aa_int_complex(H,aa_real_int(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S2) = groups3049146728041665814omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5228_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( aa_set_int_o(finite_finite_int,S5)
     => ( finite_finite_real(T3)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
               => member_real(aa_int_real(J,A3),minus_minus_set_real(T,T3)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T3))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T3))
                   => aa_set_int_o(member_int(aa_real_int(I,B3)),minus_minus_set_int(S2,S5)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S5)
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T3)
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S2)
                         => ( aa_real_complex(H,aa_int_real(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S2) = groups5754745047067104278omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5229_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S5)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
               => aa_set_int_o(member_int(aa_int_int(J,A3)),minus_minus_set_int(T,T3)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(T,T3))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int(B3),minus_minus_set_int(T,T3))
                   => aa_set_int_o(member_int(aa_int_int(I,B3)),minus_minus_set_int(S2,S5)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S5)
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int(B3),T3)
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S2)
                         => ( aa_int_complex(H,aa_int_int(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5230_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(S5)
     => ( finite_finite_real(T3)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S2,S5))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S2,S5))
               => member_real(aa_real_real(J,A3),minus_minus_set_real(T,T3)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T3))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T3))
                   => member_real(aa_real_real(I,B3),minus_minus_set_real(S2,S5)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S5)
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T3)
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S2)
                         => ( aa_real_real(H,aa_real_real(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5231_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T: set_int,G: fun_real_real,H: fun_int_real] :
      ( finite_finite_real(S5)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S2,S5))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S2,S5))
               => aa_set_int_o(member_int(aa_real_int(J,A3)),minus_minus_set_int(T,T3)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(T,T3))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int(B3),minus_minus_set_int(T,T3))
                   => member_real(aa_int_real(I,B3),minus_minus_set_real(S2,S5)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S5)
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int(B3),T3)
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S2)
                         => ( aa_int_real(H,aa_real_int(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups8778361861064173332t_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5232_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_complex,S2: set_real,I: fun_complex_real,J: fun_real_complex,T: set_complex,G: fun_real_real,H: fun_complex_real] :
      ( finite_finite_real(S5)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S2,S5))
             => ( aa_complex_real(I,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S2,S5))
               => aa_set_complex_o(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T,T3)) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T,T3))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T,T3))
                   => member_real(aa_complex_real(I,B3),minus_minus_set_real(S2,S5)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S5)
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: complex] :
                        ( aa_set_complex_o(member_complex(B3),T3)
                       => ( aa_complex_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S2)
                         => ( aa_complex_real(H,aa_real_complex(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups5808333547571424918x_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5233_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T: set_real,G: fun_int_real,H: fun_real_real] :
      ( aa_set_int_o(finite_finite_int,S5)
     => ( finite_finite_real(T3)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
               => member_real(aa_int_real(J,A3),minus_minus_set_real(T,T3)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T3))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T3))
                   => aa_set_int_o(member_int(aa_real_int(I,B3)),minus_minus_set_int(S2,S5)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S5)
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T3)
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S2)
                         => ( aa_real_real(H,aa_int_real(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups8097168146408367636l_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5234_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S5)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
               => aa_set_int_o(member_int(aa_int_int(J,A3)),minus_minus_set_int(T,T3)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(T,T3))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int(B3),minus_minus_set_int(T,T3))
                   => aa_set_int_o(member_int(aa_int_int(I,B3)),minus_minus_set_int(S2,S5)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S5)
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int(B3),T3)
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S2)
                         => ( aa_int_real(H,aa_int_int(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5235_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_complex,S2: set_int,I: fun_complex_int,J: fun_int_complex,T: set_complex,G: fun_int_real,H: fun_complex_real] :
      ( aa_set_int_o(finite_finite_int,S5)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
             => ( aa_complex_int(I,aa_int_complex(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
               => aa_set_complex_o(member_complex(aa_int_complex(J,A3)),minus_811609699411566653omplex(T,T3)) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T,T3))
                 => ( aa_int_complex(J,aa_complex_int(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T,T3))
                   => aa_set_int_o(member_int(aa_complex_int(I,B3)),minus_minus_set_int(S2,S5)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S5)
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: complex] :
                        ( aa_set_complex_o(member_complex(B3),T3)
                       => ( aa_complex_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S2)
                         => ( aa_complex_real(H,aa_int_complex(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups5808333547571424918x_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5236_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5237_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5238_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5239_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5240_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_5241_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_5242_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_5243_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_5244_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_5245_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_5246_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_5247_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_5248_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5249_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5250_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5251_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5252_le__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% le_minus_one_simps(1)
tff(fact_5253_le__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% le_minus_one_simps(1)
tff(fact_5254_le__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% le_minus_one_simps(1)
tff(fact_5255_le__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% le_minus_one_simps(1)
tff(fact_5256_le__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(3)
tff(fact_5257_le__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(3)
tff(fact_5258_le__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(3)
tff(fact_5259_le__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(3)
tff(fact_5260_less__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(3)
tff(fact_5261_less__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(3)
tff(fact_5262_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_5263_less__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(3)
tff(fact_5264_less__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% less_minus_one_simps(1)
tff(fact_5265_less__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% less_minus_one_simps(1)
tff(fact_5266_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_5267_less__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% less_minus_one_simps(1)
tff(fact_5268_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).

% not_one_le_neg_numeral
tff(fact_5269_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).

% not_one_le_neg_numeral
tff(fact_5270_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))) ).

% not_one_le_neg_numeral
tff(fact_5271_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_one_le_neg_numeral
tff(fact_5272_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_5273_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_5274_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_5275_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_5276_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_5277_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_5278_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_5279_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_5280_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Mb)) ).

% neg_one_le_numeral
tff(fact_5281_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Mb)) ).

% neg_one_le_numeral
tff(fact_5282_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Mb)) ).

% neg_one_le_numeral
tff(fact_5283_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Mb)) ).

% neg_one_le_numeral
tff(fact_5284_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) ).

% neg_numeral_le_one
tff(fact_5285_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_5286_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_5287_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) ).

% neg_numeral_le_one
tff(fact_5288_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C2 = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5289_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C2: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C2 = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) )
      <=> ( aa_complex_complex(times_times_complex(C2),B) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5290_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C2 = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) )
      <=> ( aa_rat_rat(times_times_rat(C2),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5291_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = C2 )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5292_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C2: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = C2 )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C2),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5293_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = C2 )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(times_times_rat(C2),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5294_minus__divide__eq__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)) = A )
    <=> $ite(C2 != zero_zero_real,aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2),A = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_5295_minus__divide__eq__eq,axiom,
    ! [B: complex,C2: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C2)) = A )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C2),A = zero_zero_complex) ) ).

% minus_divide_eq_eq
tff(fact_5296_minus__divide__eq__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2)) = A )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(times_times_rat(A),C2),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_5297_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)) )
    <=> $ite(C2 != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = aa_real_real(uminus_uminus_real,B),A = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_5298_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C2)) )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(times_times_complex(A),C2) = aa_complex_complex(uminus1482373934393186551omplex,B),A = zero_zero_complex) ) ).

% eq_minus_divide_eq
tff(fact_5299_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2)) )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(times_times_rat(A),C2) = aa_rat_rat(uminus_uminus_rat,B),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_5300_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B != zero_zero_real )
        & ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5301_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B != zero_zero_complex )
        & ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5302_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5303_sum__nonneg__0,axiom,
    ! [S3: set_real,F: fun_real_real,I: real] :
      ( finite_finite_real(S3)
     => ( ! [I2: real] :
            ( member_real(I2,S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( ( groups8097168146408367636l_real(F,S3) = zero_zero_real )
         => ( member_real(I,S3)
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5304_sum__nonneg__0,axiom,
    ! [S3: set_int,F: fun_int_real,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( ( groups8778361861064173332t_real(F,S3) = zero_zero_real )
         => ( aa_set_int_o(member_int(I),S3)
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5305_sum__nonneg__0,axiom,
    ! [S3: set_complex,F: fun_complex_real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( ( groups5808333547571424918x_real(F,S3) = zero_zero_real )
         => ( aa_set_complex_o(member_complex(I),S3)
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5306_sum__nonneg__0,axiom,
    ! [S3: set_real,F: fun_real_rat,I: real] :
      ( finite_finite_real(S3)
     => ( ! [I2: real] :
            ( member_real(I2,S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
       => ( ( groups1300246762558778688al_rat(F,S3) = zero_zero_rat )
         => ( member_real(I,S3)
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5307_sum__nonneg__0,axiom,
    ! [S3: set_nat,F: fun_nat_rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( groups2906978787729119204at_rat(F,S3) = zero_zero_rat )
         => ( aa_set_nat_o(member_nat(I),S3)
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5308_sum__nonneg__0,axiom,
    ! [S3: set_int,F: fun_int_rat,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,S3) = zero_zero_rat )
         => ( aa_set_int_o(member_int(I),S3)
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5309_sum__nonneg__0,axiom,
    ! [S3: set_complex,F: fun_complex_rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
       => ( ( groups5058264527183730370ex_rat(F,S3) = zero_zero_rat )
         => ( aa_set_complex_o(member_complex(I),S3)
           => ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5310_sum__nonneg__0,axiom,
    ! [S3: set_real,F: fun_real_nat,I: real] :
      ( finite_finite_real(S3)
     => ( ! [I2: real] :
            ( member_real(I2,S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)) )
       => ( ( groups1935376822645274424al_nat(F,S3) = zero_zero_nat )
         => ( member_real(I,S3)
           => ( aa_real_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5311_sum__nonneg__0,axiom,
    ! [S3: set_int,F: fun_int_nat,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
       => ( ( groups4541462559716669496nt_nat(F,S3) = zero_zero_nat )
         => ( aa_set_int_o(member_int(I),S3)
           => ( aa_int_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5312_sum__nonneg__0,axiom,
    ! [S3: set_complex,F: fun_complex_nat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
       => ( ( groups5693394587270226106ex_nat(F,S3) = zero_zero_nat )
         => ( aa_set_complex_o(member_complex(I),S3)
           => ( aa_complex_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5313_sum__nonneg__leq__bound,axiom,
    ! [S3: set_real,F: fun_real_real,B4: real,I: real] :
      ( finite_finite_real(S3)
     => ( ! [I2: real] :
            ( member_real(I2,S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( ( groups8097168146408367636l_real(F,S3) = B4 )
         => ( member_real(I,S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5314_sum__nonneg__leq__bound,axiom,
    ! [S3: set_int,F: fun_int_real,B4: real,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( ( groups8778361861064173332t_real(F,S3) = B4 )
         => ( aa_set_int_o(member_int(I),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5315_sum__nonneg__leq__bound,axiom,
    ! [S3: set_complex,F: fun_complex_real,B4: real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( ( groups5808333547571424918x_real(F,S3) = B4 )
         => ( aa_set_complex_o(member_complex(I),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5316_sum__nonneg__leq__bound,axiom,
    ! [S3: set_real,F: fun_real_rat,B4: rat,I: real] :
      ( finite_finite_real(S3)
     => ( ! [I2: real] :
            ( member_real(I2,S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
       => ( ( groups1300246762558778688al_rat(F,S3) = B4 )
         => ( member_real(I,S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5317_sum__nonneg__leq__bound,axiom,
    ! [S3: set_nat,F: fun_nat_rat,B4: rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( groups2906978787729119204at_rat(F,S3) = B4 )
         => ( aa_set_nat_o(member_nat(I),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5318_sum__nonneg__leq__bound,axiom,
    ! [S3: set_int,F: fun_int_rat,B4: rat,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,S3) = B4 )
         => ( aa_set_int_o(member_int(I),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5319_sum__nonneg__leq__bound,axiom,
    ! [S3: set_complex,F: fun_complex_rat,B4: rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
       => ( ( groups5058264527183730370ex_rat(F,S3) = B4 )
         => ( aa_set_complex_o(member_complex(I),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5320_sum__nonneg__leq__bound,axiom,
    ! [S3: set_real,F: fun_real_nat,B4: nat,I: real] :
      ( finite_finite_real(S3)
     => ( ! [I2: real] :
            ( member_real(I2,S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)) )
       => ( ( groups1935376822645274424al_nat(F,S3) = B4 )
         => ( member_real(I,S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5321_sum__nonneg__leq__bound,axiom,
    ! [S3: set_int,F: fun_int_nat,B4: nat,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
       => ( ( groups4541462559716669496nt_nat(F,S3) = B4 )
         => ( aa_set_int_o(member_int(I),S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5322_sum__nonneg__leq__bound,axiom,
    ! [S3: set_complex,F: fun_complex_nat,B4: nat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
       => ( ( groups5693394587270226106ex_nat(F,S3) = B4 )
         => ( aa_set_complex_o(member_complex(I),S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5323_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_complex] :
      ( finite_finite_real(A4)
     => ( groups5754745047067104278omplex(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_hw(G)))) = groups5754745047067104278omplex(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5324_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3049146728041665814omplex(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_hx(G)))) = groups3049146728041665814omplex(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5325_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_real] :
      ( finite_finite_real(A4)
     => ( groups8097168146408367636l_real(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_hy(G)))) = groups8097168146408367636l_real(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5326_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_hz(G)))) = groups8778361861064173332t_real(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5327_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_ia(G)))) = groups5808333547571424918x_real(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5328_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_rat] :
      ( finite_finite_real(A4)
     => ( groups1300246762558778688al_rat(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_ib(G)))) = groups1300246762558778688al_rat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5329_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_ic(G)))) = groups3906332499630173760nt_rat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5330_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_id(G)))) = groups5058264527183730370ex_rat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5331_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_nat] :
      ( finite_finite_real(A4)
     => ( groups1935376822645274424al_nat(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_ie(G)))) = groups1935376822645274424al_nat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5332_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_if(G)))) = groups4541462559716669496nt_nat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5333_real__0__less__add__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xb)),Yb) ) ).

% real_0_less_add_iff
tff(fact_5334_real__add__less__0__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_less_0_iff
tff(fact_5335_real__add__le__0__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_le_0_iff
tff(fact_5336_real__0__le__add__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xb)),Yb) ) ).

% real_0_le_add_iff
tff(fact_5337_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ).

% zmod_zminus1_eq_if
tff(fact_5338_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B)) ).

% zmod_zminus2_eq_if
tff(fact_5339_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_real] :
      ( finite_finite_real(I6)
     => ( member_real(I,I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_5340_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( aa_set_int_o(member_int(I),I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_5341_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( aa_set_complex_o(member_complex(I),I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_5342_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_rat] :
      ( finite_finite_real(I6)
     => ( member_real(I,I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_5343_sum__pos2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( aa_set_nat_o(member_nat(I),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_5344_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( aa_set_int_o(member_int(I),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_5345_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( aa_set_complex_o(member_complex(I),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_5346_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_nat] :
      ( finite_finite_real(I6)
     => ( member_real(I,I6)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I6)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_5347_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( aa_set_int_o(member_int(I),I6)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I6)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_5348_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( aa_set_complex_o(member_complex(I),I6)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I6)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_5349_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I6)) ) ) ) ).

% sum_pos
tff(fact_5350_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I6)) ) ) ) ).

% sum_pos
tff(fact_5351_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I6)) ) ) ) ).

% sum_pos
tff(fact_5352_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_5353_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_rat] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_5354_sum__pos,axiom,
    ! [I6: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_5355_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_5356_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_5357_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_nat] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,I2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_5358_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I6)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_5359_ln__ge__zero__imp__ge__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_5360_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_complex(G,X3) = zero_zero_complex ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_complex(G,X3) = aa_real_complex(H,X3) ) )
           => ( groups5754745047067104278omplex(G,T) = groups5754745047067104278omplex(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5361_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_real(G,X3) = zero_zero_real ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_real(G,X3) = aa_real_real(H,X3) ) )
           => ( groups8097168146408367636l_real(G,T) = groups8097168146408367636l_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5362_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(G,X3) = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_real(G,X3) = aa_complex_real(H,X3) ) )
           => ( groups5808333547571424918x_real(G,T) = groups5808333547571424918x_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5363_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_rat(G,X3) = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_rat(G,X3) = aa_real_rat(H,X3) ) )
           => ( groups1300246762558778688al_rat(G,T) = groups1300246762558778688al_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5364_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(G,X3) = zero_zero_rat ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_rat(G,X3) = aa_complex_rat(H,X3) ) )
           => ( groups5058264527183730370ex_rat(G,T) = groups5058264527183730370ex_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5365_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_nat(G,X3) = zero_zero_nat ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_nat(G,X3) = aa_real_nat(H,X3) ) )
           => ( groups1935376822645274424al_nat(G,T) = groups1935376822645274424al_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5366_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(G,X3) = zero_zero_nat ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_nat(G,X3) = aa_complex_nat(H,X3) ) )
           => ( groups5693394587270226106ex_nat(G,T) = groups5693394587270226106ex_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5367_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_int(G,X3) = zero_zero_int ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_int(G,X3) = aa_real_int(H,X3) ) )
           => ( groups1932886352136224148al_int(G,T) = groups1932886352136224148al_int(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5368_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(G,X3) = zero_zero_int ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_int(G,X3) = aa_complex_int(H,X3) ) )
           => ( groups5690904116761175830ex_int(G,T) = groups5690904116761175830ex_int(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5369_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T,S2))
             => ( aa_nat_complex(G,X3) = zero_zero_complex ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => ( aa_nat_complex(G,X3) = aa_nat_complex(H,X3) ) )
           => ( aa_set_nat_complex(groups2073611262835488442omplex(G),T) = aa_set_nat_complex(groups2073611262835488442omplex(H),S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5370_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_complex(H,X3) = zero_zero_complex ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_complex(G,X3) = aa_real_complex(H,X3) ) )
           => ( groups5754745047067104278omplex(G,S2) = groups5754745047067104278omplex(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5371_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_real(H,X3) = zero_zero_real ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_real(G,X3) = aa_real_real(H,X3) ) )
           => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5372_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(H,X3) = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_real(G,X3) = aa_complex_real(H,X3) ) )
           => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5373_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_rat(H,X3) = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_rat(G,X3) = aa_real_rat(H,X3) ) )
           => ( groups1300246762558778688al_rat(G,S2) = groups1300246762558778688al_rat(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5374_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(H,X3) = zero_zero_rat ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_rat(G,X3) = aa_complex_rat(H,X3) ) )
           => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5375_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_nat(H,X3) = zero_zero_nat ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_nat(G,X3) = aa_real_nat(H,X3) ) )
           => ( groups1935376822645274424al_nat(G,S2) = groups1935376822645274424al_nat(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5376_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(H,X3) = zero_zero_nat ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_nat(G,X3) = aa_complex_nat(H,X3) ) )
           => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5377_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_int,G: fun_real_int] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_int(H,X3) = zero_zero_int ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_int(G,X3) = aa_real_int(H,X3) ) )
           => ( groups1932886352136224148al_int(G,S2) = groups1932886352136224148al_int(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5378_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(H,X3) = zero_zero_int ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_int(G,X3) = aa_complex_int(H,X3) ) )
           => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5379_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_nat,S2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T,S2))
             => ( aa_nat_complex(H,X3) = zero_zero_complex ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => ( aa_nat_complex(G,X3) = aa_nat_complex(H,X3) ) )
           => ( aa_set_nat_complex(groups2073611262835488442omplex(G),S2) = aa_set_nat_complex(groups2073611262835488442omplex(H),T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5380_sum_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(G,X3) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T) = groups5808333547571424918x_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5381_sum_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(G,X3) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T) = groups5058264527183730370ex_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5382_sum_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(G,X3) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T) = groups5693394587270226106ex_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5383_sum_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(G,X3) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T) = groups5690904116761175830ex_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5384_sum_Omono__neutral__right,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T,S2))
             => ( aa_nat_complex(G,X3) = zero_zero_complex ) )
         => ( aa_set_nat_complex(groups2073611262835488442omplex(G),T) = aa_set_nat_complex(groups2073611262835488442omplex(G),S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5385_sum_Omono__neutral__right,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T,S2))
             => ( aa_nat_rat(G,X3) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,T) = groups2906978787729119204at_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5386_sum_Omono__neutral__right,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T,S2))
             => ( aa_nat_int(G,X3) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,T) = groups3539618377306564664at_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5387_sum_Omono__neutral__right,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T,S2))
             => ( aa_int_complex(G,X3) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,T) = groups3049146728041665814omplex(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5388_sum_Omono__neutral__right,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T,S2))
             => ( aa_int_real(G,X3) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T) = groups8778361861064173332t_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5389_sum_Omono__neutral__right,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T,S2))
             => ( aa_int_rat(G,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T) = groups3906332499630173760nt_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5390_sum_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(G,X3) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5391_sum_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(G,X3) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5392_sum_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(G,X3) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5393_sum_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(G,X3) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5394_sum_Omono__neutral__left,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T,S2))
             => ( aa_nat_complex(G,X3) = zero_zero_complex ) )
         => ( aa_set_nat_complex(groups2073611262835488442omplex(G),S2) = aa_set_nat_complex(groups2073611262835488442omplex(G),T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5395_sum_Omono__neutral__left,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T,S2))
             => ( aa_nat_rat(G,X3) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,S2) = groups2906978787729119204at_rat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5396_sum_Omono__neutral__left,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T,S2))
             => ( aa_nat_int(G,X3) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,S2) = groups3539618377306564664at_int(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5397_sum_Omono__neutral__left,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T,S2))
             => ( aa_int_complex(G,X3) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5398_sum_Omono__neutral__left,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T,S2))
             => ( aa_int_real(G,X3) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5399_sum_Omono__neutral__left,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T,S2))
             => ( aa_int_rat(G,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5400_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) )
               => ( groups5754745047067104278omplex(G,A4) = groups5754745047067104278omplex(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5401_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) )
               => ( groups8097168146408367636l_real(G,A4) = groups8097168146408367636l_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5402_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) )
               => ( groups5808333547571424918x_real(G,A4) = groups5808333547571424918x_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5403_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) )
               => ( groups1300246762558778688al_rat(G,A4) = groups1300246762558778688al_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5404_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) )
               => ( groups5058264527183730370ex_rat(G,A4) = groups5058264527183730370ex_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5405_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) )
               => ( groups1935376822645274424al_nat(G,A4) = groups1935376822645274424al_nat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5406_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,C4) = groups5693394587270226106ex_nat(H,C4) )
               => ( groups5693394587270226106ex_nat(G,A4) = groups5693394587270226106ex_nat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5407_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_int(H,B3) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,C4) = groups1932886352136224148al_int(H,C4) )
               => ( groups1932886352136224148al_int(G,A4) = groups1932886352136224148al_int(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5408_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_int(H,B3) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,C4) = groups5690904116761175830ex_int(H,C4) )
               => ( groups5690904116761175830ex_int(G,A4) = groups5690904116761175830ex_int(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5409_sum_Osame__carrierI,axiom,
    ! [C4: set_nat,A4: set_nat,B4: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,C4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C4)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C4)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),minus_minus_set_nat(C4,A4))
               => ( aa_nat_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(C4,B4))
                 => ( aa_nat_complex(H,B3) = zero_zero_complex ) )
             => ( ( aa_set_nat_complex(groups2073611262835488442omplex(G),C4) = aa_set_nat_complex(groups2073611262835488442omplex(H),C4) )
               => ( aa_set_nat_complex(groups2073611262835488442omplex(G),A4) = aa_set_nat_complex(groups2073611262835488442omplex(H),B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5410_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,A4) = groups5754745047067104278omplex(H,B4) )
              <=> ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5411_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A4) = groups8097168146408367636l_real(H,B4) )
              <=> ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5412_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A4) = groups5808333547571424918x_real(H,B4) )
              <=> ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5413_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A4) = groups1300246762558778688al_rat(H,B4) )
              <=> ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5414_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,A4) = groups5058264527183730370ex_rat(H,B4) )
              <=> ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5415_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,A4) = groups1935376822645274424al_nat(H,B4) )
              <=> ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5416_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,A4) = groups5693394587270226106ex_nat(H,B4) )
              <=> ( groups5693394587270226106ex_nat(G,C4) = groups5693394587270226106ex_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5417_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_int(H,B3) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,A4) = groups1932886352136224148al_int(H,B4) )
              <=> ( groups1932886352136224148al_int(G,C4) = groups1932886352136224148al_int(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5418_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_int(H,B3) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,A4) = groups5690904116761175830ex_int(H,B4) )
              <=> ( groups5690904116761175830ex_int(G,C4) = groups5690904116761175830ex_int(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5419_sum_Osame__carrier,axiom,
    ! [C4: set_nat,A4: set_nat,B4: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,C4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C4)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C4)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),minus_minus_set_nat(C4,A4))
               => ( aa_nat_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(C4,B4))
                 => ( aa_nat_complex(H,B3) = zero_zero_complex ) )
             => ( ( aa_set_nat_complex(groups2073611262835488442omplex(G),A4) = aa_set_nat_complex(groups2073611262835488442omplex(H),B4) )
              <=> ( aa_set_nat_complex(groups2073611262835488442omplex(G),C4) = aa_set_nat_complex(groups2073611262835488442omplex(H),C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5420_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups5808333547571424918x_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,B4))),groups5808333547571424918x_real(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5421_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups5058264527183730370ex_rat(G,A4) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,B4))),groups5058264527183730370ex_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5422_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups5693394587270226106ex_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A4,B4))),groups5693394587270226106ex_nat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5423_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups5690904116761175830ex_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A4,B4))),groups5690904116761175830ex_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5424_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups2906978787729119204at_rat(G,A4) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A4,B4))),groups2906978787729119204at_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5425_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups3539618377306564664at_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,minus_minus_set_nat(A4,B4))),groups3539618377306564664at_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5426_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_real] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups8778361861064173332t_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,minus_minus_set_int(A4,B4))),groups8778361861064173332t_real(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5427_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_rat] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups3906332499630173760nt_rat(G,A4) = aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,minus_minus_set_int(A4,B4))),groups3906332499630173760nt_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5428_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups4541462559716669496nt_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A4,B4))),groups4541462559716669496nt_nat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5429_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups4538972089207619220nt_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,minus_minus_set_int(A4,B4))),groups4538972089207619220nt_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5430_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A4,B4)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5431_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A4,B4)) = aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5432_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A4)),groups5690904116761175830ex_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5433_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
       => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A4,B4)) = aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5434_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
       => ( groups3539618377306564664at_int(F,minus_minus_set_nat(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,A4)),groups3539618377306564664at_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5435_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
       => ( groups8778361861064173332t_real(F,minus_minus_set_int(A4,B4)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5436_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
       => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A4,B4)) = aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A4)),groups3906332499630173760nt_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5437_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
       => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A4)),groups4538972089207619220nt_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5438_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( groups7754918857620584856omplex(F,minus_811609699411566653omplex(A4,B4)) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A4)),groups7754918857620584856omplex(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5439_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),minus_minus_set_nat(A4,B4)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(F),A4)),aa_set_nat_real(groups6591440286371151544t_real(F),B4)) ) ) ) ).

% sum_diff
tff(fact_5440_ln__add__one__self__le__self,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self
tff(fact_5441_ln__mult,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Yb)) ) ) ) ).

% ln_mult
tff(fact_5442_ln__eq__minus__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( ( aa_real_real(ln_ln_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real) )
       => ( Xb = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_5443_ln__div,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Yb)) ) ) ) ).

% ln_div
tff(fact_5444_pos__minus__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5445_pos__minus__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5446_pos__less__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5447_pos__less__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5448_neg__minus__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5449_neg__minus__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5450_neg__less__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5451_neg__less__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5452_minus__divide__less__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% minus_divide_less_eq
tff(fact_5453_minus__divide__less__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_less_eq
tff(fact_5454_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_minus_divide_eq
tff(fact_5455_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_5456_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) )
    <=> $ite(C2 != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2) = B,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5457_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: complex,C2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C2) = B,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5458_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = aa_rat_rat(divide_divide_rat(B),C2) )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2) = B,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5459_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) )
    <=> $ite(C2 != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5460_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C2: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) )
    <=> $ite(C2 != zero_zero_complex,B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5461_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C2) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) )
    <=> $ite(C2 != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5462_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = $ite(Z = zero_zero_real,B,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_5463_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = $ite(Z = zero_zero_complex,B,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_5464_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_5465_minus__divide__add__eq__iff,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z))),Yb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5466_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Yb: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xb),Z))),Yb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(times_times_complex(Yb),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5467_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xb),Z))),Yb) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_rat_rat(times_times_rat(Yb),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5468_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z))),Yb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5469_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Yb: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xb),Z))),Yb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(times_times_complex(Yb),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5470_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xb),Z))),Yb) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_rat_rat(times_times_rat(Yb),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5471_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_5472_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_5473_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_5474_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_5475_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_5476_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_5477_verit__less__mono__div__int2,axiom,
    ! [A4: int,B4: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A4),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Nb))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_5478_div__eq__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_5479_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_real] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_5480_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_5481_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_rat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_5482_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_5483_sum__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_5484_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_nat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B4,A4))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_5485_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B4,A4))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_5486_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_int] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B4,A4))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,B3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,A4)),groups1932886352136224148al_int(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_5487_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B4,A4))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,B3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,A4)),groups5690904116761175830ex_int(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_5488_sum__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B4,A4))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,B3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,A4)),groups3539618377306564664at_int(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_5489_ln__le__minus__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_5490_ln__diff__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Yb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Yb)),Yb)) ) ) ).

% ln_diff_le
tff(fact_5491_pos__minus__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5492_pos__minus__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5493_pos__le__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5494_pos__le__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5495_neg__minus__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5496_neg__minus__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5497_neg__le__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5498_neg__le__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5499_minus__divide__le__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% minus_divide_le_eq
tff(fact_5500_minus__divide__le__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_le_eq
tff(fact_5501_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_5502_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_5503_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5504_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),aa_rat_rat(divide_divide_rat(B),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5505_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5506_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5507_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5508_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5509_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5510_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5511_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5512_realpow__square__minus__le,axiom,
    ! [U: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% realpow_square_minus_le
tff(fact_5513_ln__one__minus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_5514_minus__mod__int__eq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K2)),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,K2),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_5515_zmod__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_5516_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_real] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( member_real(B,minus_minus_set_real(B4,A4))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,B))
           => ( ! [X3: real] :
                  ( member_real(X3,B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5517_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B4,A4))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,B))
           => ( ! [X3: complex] :
                  ( aa_set_complex_o(member_complex(X3),B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5518_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_rat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( member_real(B,minus_minus_set_real(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,B))
           => ( ! [X3: real] :
                  ( member_real(X3,B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5519_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B))
           => ( ! [X3: complex] :
                  ( aa_set_complex_o(member_complex(X3),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5520_sum__strict__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,B: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( aa_set_nat_o(member_nat(B),minus_minus_set_nat(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,B))
           => ( ! [X3: nat] :
                  ( aa_set_nat_o(member_nat(X3),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5521_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_nat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( member_real(B,minus_minus_set_real(B4,A4))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,B))
           => ( ! [X3: real] :
                  ( member_real(X3,B4)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5522_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B4,A4))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,B))
           => ( ! [X3: complex] :
                  ( aa_set_complex_o(member_complex(X3),B4)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5523_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_int] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( member_real(B,minus_minus_set_real(B4,A4))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(F,B))
           => ( ! [X3: real] :
                  ( member_real(X3,B4)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X3)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups1932886352136224148al_int(F,A4)),groups1932886352136224148al_int(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5524_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B4,A4))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_complex_int(F,B))
           => ( ! [X3: complex] :
                  ( aa_set_complex_o(member_complex(X3),B4)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,X3)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A4)),groups5690904116761175830ex_int(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5525_sum__strict__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,B: nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( aa_set_nat_o(member_nat(B),minus_minus_set_nat(B4,A4))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,B))
           => ( ! [X3: nat] :
                  ( aa_set_nat_o(member_nat(X3),B4)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X3)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,A4)),groups3539618377306564664at_int(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5526_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_5527_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_5528_zminus1__lemma,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
     => ( ( B != zero_zero_int )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,
                $ite(R2 = zero_zero_int,aa_int_int(uminus_uminus_int,Q3),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q3)),one_one_int))),
              $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R2)))) ) ) ).

% zminus1_lemma
tff(fact_5529_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5530_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),aa_rat_rat(divide_divide_rat(B),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5531_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5532_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5533_square__le__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_5534_square__le__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Xb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_5535_square__le__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_5536_square__le__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_5537_div__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_5538_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K2)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_5539_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K2) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_5540_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_5541_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_5542_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_5543_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_5544_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_5545_int__bit__induct,axiom,
    ! [P2: fun_int_o,K2: int] :
      ( aa_int_o(P2,zero_zero_int)
     => ( aa_int_o(P2,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K: int] :
              ( aa_int_o(P2,K)
             => ( ( K != zero_zero_int )
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
         => ( ! [K: int] :
                ( aa_int_o(P2,K)
               => ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(P2,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,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(P2,K2) ) ) ) ) ).

% int_bit_induct
tff(fact_5546_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K2) = K2 ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_5547_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K2) = K2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_5548_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_eh(L)),Qr) ).

% divmod_step_nat_def
tff(fact_5549_ln__one__plus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_5550_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_ei(L)),Qr) ).

% divmod_step_int_def
tff(fact_5551_signed__take__bit__int__greater__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K2)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5552_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_5553_tanh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_5554_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_ig),unique5052692396658037445od_int(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_5555_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ih),unique5055182867167087721od_nat(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_5556_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = produc6916734918728496179nteger(aTP_Lamm_ii,unique3479559517661332726nteger(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_5557_divmod__nat__if,axiom,
    ! [Mb: nat,Nb: nat] :
      divmod_nat(Mb,Nb) = $ite(
        ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),Mb),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ij),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_5558_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),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_5559_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_5560_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_5561_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_5562_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_5563_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_5564_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_5565_case__prodI,axiom,
    ! [F: fun_fu3422731660242565741eger_o,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(F,A),B)
     => aa_Pro4652312591602923194eger_o(produc7733317944277263833eger_o(F),produc7264323995540650962nteger(A,B)) ) ).

% case_prodI
tff(fact_5566_case__prodI,axiom,
    ! [F: fun_fu7695273699059480439eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(F,A),B)
     => aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(F),produc565448451787809096nteger(A,B)) ) ).

% case_prodI
tff(fact_5567_case__prodI,axiom,
    ! [F: fun_fu6865642474282033866_int_o,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(F,A),B)
     => aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(F),produc7320969046608631221nt_int(A,B)) ) ).

% case_prodI
tff(fact_5568_case__prodI,axiom,
    ! [F: fun_fu2978539155105838238_int_o,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(F,A),B)
     => aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(F),produc4729913460834754785nt_int(A,B)) ) ).

% case_prodI
tff(fact_5569_case__prodI,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(F,A),B)
     => aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% case_prodI
tff(fact_5570_case__prodI2,axiom,
    ! [P: produc7509114972854644512nteger,C2: fun_fu3422731660242565741eger_o] :
      ( ! [A3: fun_Co3460408610320954864e_term,B3: produc8923325533196201883nteger] :
          ( ( P = produc7264323995540650962nteger(A3,B3) )
         => aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(C2,A3),B3) )
     => aa_Pro4652312591602923194eger_o(produc7733317944277263833eger_o(C2),P) ) ).

% case_prodI2
tff(fact_5571_case__prodI2,axiom,
    ! [P: produc2453742332425399446nteger,C2: fun_fu7695273699059480439eger_o] :
      ( ! [A3: fun_Pr5098669352780829690e_term,B3: produc8923325533196201883nteger] :
          ( ( P = produc565448451787809096nteger(A3,B3) )
         => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(C2,A3),B3) )
     => aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(C2),P) ) ).

% case_prodI2
tff(fact_5572_case__prodI2,axiom,
    ! [P: produc3043004577601930755nt_int,C2: fun_fu6865642474282033866_int_o] :
      ( ! [A3: fun_Pr7763680713840462443e_term,B3: product_prod_int_int] :
          ( ( P = produc7320969046608631221nt_int(A3,B3) )
         => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(C2,A3),B3) )
     => aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(C2),P) ) ).

% case_prodI2
tff(fact_5573_case__prodI2,axiom,
    ! [P: produc4788131091073775919nt_int,C2: fun_fu2978539155105838238_int_o] :
      ( ! [A3: fun_in5730379987311621695e_term,B3: product_prod_int_int] :
          ( ( P = produc4729913460834754785nt_int(A3,B3) )
         => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(C2,A3),B3) )
     => aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(C2),P) ) ).

% case_prodI2
tff(fact_5574_case__prodI2,axiom,
    ! [P: product_prod_int_int,C2: fun_int_fun_int_o] :
      ( ! [A3: int,B3: int] :
          ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_int_o(aa_int_fun_int_o(C2,A3),B3) )
     => aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,C2),P) ) ).

% case_prodI2
tff(fact_5575_mem__case__prodI,axiom,
    ! [Z: complex,C2: fun_in230608924162954550omplex,A: int,B: int] :
      ( aa_set_complex_o(member_complex(Z),aa_int_set_complex(aa_int6119279391169676535omplex(C2,A),B))
     => aa_set_complex_o(member_complex(Z),produc8580519160106071146omplex(C2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5576_mem__case__prodI,axiom,
    ! [Z: real,C2: fun_in6553099857819052212t_real,A: int,B: int] :
      ( member_real(Z,aa_int_set_real(aa_int4847878404602763637t_real(C2,A),B))
     => member_real(Z,produc6452406959799940328t_real(C2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5577_mem__case__prodI,axiom,
    ! [Z: nat,C2: fun_in2630332840903238488et_nat,A: int,B: int] :
      ( aa_set_nat_o(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C2,A),B))
     => aa_set_nat_o(member_nat(Z),produc4251311855443802252et_nat(C2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5578_mem__case__prodI,axiom,
    ! [Z: int,C2: fun_in7040156458970095540et_int,A: int,B: int] :
      ( aa_set_int_o(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C2,A),B))
     => aa_set_int_o(member_int(Z),produc73460835934605544et_int(C2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5579_mem__case__prodI,axiom,
    ! [Z: set_nat,C2: fun_in7207726656141197838et_nat,A: int,B: int] :
      ( member_set_nat(Z,aa_int_set_set_nat(aa_int1843896668418082255et_nat(C2,A),B))
     => member_set_nat(Z,produc5233655623923918146et_nat(C2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5580_mem__case__prodI,axiom,
    ! [Z: complex,C2: fun_fu15390479812311035omplex,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_set_complex_o(member_complex(Z),aa_Pro5605272939097928131omplex(aa_fun5788763187652937060omplex(C2,A),B))
     => aa_set_complex_o(member_complex(Z),produc5772012072983024995omplex(C2,produc7264323995540650962nteger(A,B))) ) ).

% mem_case_prodI
tff(fact_5581_mem__case__prodI,axiom,
    ! [Z: real,C2: fun_fu7234941744988193529t_real,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( member_real(Z,aa_Pro6515153133580421057t_real(aa_fun1966338168372333922t_real(C2,A),B))
     => member_real(Z,produc5067746511659309921t_real(C2,produc7264323995540650962nteger(A,B))) ) ).

% mem_case_prodI
tff(fact_5582_mem__case__prodI,axiom,
    ! [Z: nat,C2: fun_fu6383124537394581533et_nat,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_set_nat_o(member_nat(Z),aa_Pro381016180347496421et_nat(aa_fun3554285010146031238et_nat(C2,A),B))
     => aa_set_nat_o(member_nat(Z),produc7961109426542857093et_nat(C2,produc7264323995540650962nteger(A,B))) ) ).

% mem_case_prodI
tff(fact_5583_mem__case__prodI,axiom,
    ! [Z: int,C2: fun_fu1569576118606662777et_int,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_set_int_o(member_int(Z),aa_Pro5426537197693075521et_int(aa_fun1064201865214304482et_int(C2,A),B))
     => aa_set_int_o(member_int(Z),produc3783258407033660385et_int(C2,produc7264323995540650962nteger(A,B))) ) ).

% mem_case_prodI
tff(fact_5584_mem__case__prodI,axiom,
    ! [Z: complex,C2: fun_fu5523889554619339274omplex,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_set_complex_o(member_complex(Z),aa_Pro4762665599283228129omplex(aa_fun8357827850476113523omplex(C2,A),B))
     => aa_set_complex_o(member_complex(Z),produc4974371134869959794omplex(C2,produc4729913460834754785nt_int(A,B))) ) ).

% mem_case_prodI
tff(fact_5585_mem__case__prodI2,axiom,
    ! [P: product_prod_int_int,Z: complex,C2: fun_in230608924162954550omplex] :
      ( ! [A3: int,B3: int] :
          ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_complex_o(member_complex(Z),aa_int_set_complex(aa_int6119279391169676535omplex(C2,A3),B3)) )
     => aa_set_complex_o(member_complex(Z),produc8580519160106071146omplex(C2,P)) ) ).

% mem_case_prodI2
tff(fact_5586_mem__case__prodI2,axiom,
    ! [P: product_prod_int_int,Z: real,C2: fun_in6553099857819052212t_real] :
      ( ! [A3: int,B3: int] :
          ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => member_real(Z,aa_int_set_real(aa_int4847878404602763637t_real(C2,A3),B3)) )
     => member_real(Z,produc6452406959799940328t_real(C2,P)) ) ).

% mem_case_prodI2
tff(fact_5587_mem__case__prodI2,axiom,
    ! [P: product_prod_int_int,Z: nat,C2: fun_in2630332840903238488et_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_nat_o(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C2,A3),B3)) )
     => aa_set_nat_o(member_nat(Z),produc4251311855443802252et_nat(C2,P)) ) ).

% mem_case_prodI2
tff(fact_5588_mem__case__prodI2,axiom,
    ! [P: product_prod_int_int,Z: int,C2: fun_in7040156458970095540et_int] :
      ( ! [A3: int,B3: int] :
          ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_int_o(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C2,A3),B3)) )
     => aa_set_int_o(member_int(Z),produc73460835934605544et_int(C2,P)) ) ).

% mem_case_prodI2
tff(fact_5589_mem__case__prodI2,axiom,
    ! [P: product_prod_int_int,Z: set_nat,C2: fun_in7207726656141197838et_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => member_set_nat(Z,aa_int_set_set_nat(aa_int1843896668418082255et_nat(C2,A3),B3)) )
     => member_set_nat(Z,produc5233655623923918146et_nat(C2,P)) ) ).

% mem_case_prodI2
tff(fact_5590_mem__case__prodI2,axiom,
    ! [P: produc7509114972854644512nteger,Z: complex,C2: fun_fu15390479812311035omplex] :
      ( ! [A3: fun_Co3460408610320954864e_term,B3: produc8923325533196201883nteger] :
          ( ( P = produc7264323995540650962nteger(A3,B3) )
         => aa_set_complex_o(member_complex(Z),aa_Pro5605272939097928131omplex(aa_fun5788763187652937060omplex(C2,A3),B3)) )
     => aa_set_complex_o(member_complex(Z),produc5772012072983024995omplex(C2,P)) ) ).

% mem_case_prodI2
tff(fact_5591_mem__case__prodI2,axiom,
    ! [P: produc7509114972854644512nteger,Z: real,C2: fun_fu7234941744988193529t_real] :
      ( ! [A3: fun_Co3460408610320954864e_term,B3: produc8923325533196201883nteger] :
          ( ( P = produc7264323995540650962nteger(A3,B3) )
         => member_real(Z,aa_Pro6515153133580421057t_real(aa_fun1966338168372333922t_real(C2,A3),B3)) )
     => member_real(Z,produc5067746511659309921t_real(C2,P)) ) ).

% mem_case_prodI2
tff(fact_5592_mem__case__prodI2,axiom,
    ! [P: produc7509114972854644512nteger,Z: nat,C2: fun_fu6383124537394581533et_nat] :
      ( ! [A3: fun_Co3460408610320954864e_term,B3: produc8923325533196201883nteger] :
          ( ( P = produc7264323995540650962nteger(A3,B3) )
         => aa_set_nat_o(member_nat(Z),aa_Pro381016180347496421et_nat(aa_fun3554285010146031238et_nat(C2,A3),B3)) )
     => aa_set_nat_o(member_nat(Z),produc7961109426542857093et_nat(C2,P)) ) ).

% mem_case_prodI2
tff(fact_5593_mem__case__prodI2,axiom,
    ! [P: produc7509114972854644512nteger,Z: int,C2: fun_fu1569576118606662777et_int] :
      ( ! [A3: fun_Co3460408610320954864e_term,B3: produc8923325533196201883nteger] :
          ( ( P = produc7264323995540650962nteger(A3,B3) )
         => aa_set_int_o(member_int(Z),aa_Pro5426537197693075521et_int(aa_fun1064201865214304482et_int(C2,A3),B3)) )
     => aa_set_int_o(member_int(Z),produc3783258407033660385et_int(C2,P)) ) ).

% mem_case_prodI2
tff(fact_5594_mem__case__prodI2,axiom,
    ! [P: produc4788131091073775919nt_int,Z: complex,C2: fun_fu5523889554619339274omplex] :
      ( ! [A3: fun_in5730379987311621695e_term,B3: product_prod_int_int] :
          ( ( P = produc4729913460834754785nt_int(A3,B3) )
         => aa_set_complex_o(member_complex(Z),aa_Pro4762665599283228129omplex(aa_fun8357827850476113523omplex(C2,A3),B3)) )
     => aa_set_complex_o(member_complex(Z),produc4974371134869959794omplex(C2,P)) ) ).

% mem_case_prodI2
tff(fact_5595_case__prodI2_H,axiom,
    ! [P: product_prod_nat_nat,C2: fun_na5914473681949881652_nat_o,Xb: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) = P )
         => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C2,A3),B3),Xb) )
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,C2),P),Xb) ) ).

% case_prodI2'
tff(fact_5596_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_5597_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_5598_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_5599_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_5600_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_5601_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_5602_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_5603_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_5604_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_5605_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_5606_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_5607_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_5608_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_5609_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_5610_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_5611_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_5612_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_5613_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ).

% abs_add_abs
tff(fact_5614_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_5615_abs__add__abs,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_5616_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_5617_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_5618_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_5619_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_5620_abs__minus__cancel,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus_cancel
tff(fact_5621_tanh__0,axiom,
    aa_complex_complex(tanh_complex,zero_zero_complex) = zero_zero_complex ).

% tanh_0
tff(fact_5622_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_5623_tanh__real__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(tanh_real,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% tanh_real_zero_iff
tff(fact_5624_tanh__real__le__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ).

% tanh_real_le_iff
tff(fact_5625_semiring__norm_I73_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(73)
tff(fact_5626_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_nonneg
tff(fact_5627_abs__of__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_5628_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_nonneg
tff(fact_5629_abs__of__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_5630_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% abs_le_self_iff
tff(fact_5631_abs__le__self__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% abs_le_self_iff
tff(fact_5632_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% abs_le_self_iff
tff(fact_5633_abs__le__self__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% abs_le_self_iff
tff(fact_5634_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_5635_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_5636_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_5637_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_5638_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_5639_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_5640_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_5641_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_5642_semiring__norm_I7_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(Mb),Nb)) ).

% semiring_norm(7)
tff(fact_5643_semiring__norm_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(Mb),Nb)) ).

% semiring_norm(9)
tff(fact_5644_semiring__norm_I16_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(Mb),Nb)),aa_num_num(bit0,aa_num_num(times_times_num(Mb),Nb)))) ).

% semiring_norm(16)
tff(fact_5645_semiring__norm_I79_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(79)
tff(fact_5646_semiring__norm_I74_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(74)
tff(fact_5647_semiring__norm_I72_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(72)
tff(fact_5648_semiring__norm_I70_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),one) ).

% semiring_norm(70)
tff(fact_5649_tanh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% tanh_real_neg_iff
tff(fact_5650_tanh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tanh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% tanh_real_pos_iff
tff(fact_5651_tanh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% tanh_real_nonpos_iff
tff(fact_5652_tanh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% tanh_real_nonneg_iff
tff(fact_5653_sum__abs,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A4))),groups4538972089207619220nt_int(aTP_Lamm_ik(F),A4)) ).

% sum_abs
tff(fact_5654_sum__abs,axiom,
    ! [F: fun_nat_real,A4: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_set_nat_real(groups6591440286371151544t_real(F),A4))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_il(F)),A4)) ).

% sum_abs
tff(fact_5655_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_5656_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_5657_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_5658_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_5659_abs__of__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_nonpos
tff(fact_5660_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_5661_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_nonpos
tff(fact_5662_abs__of__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_5663_semiring__norm_I3_J,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit0,Nb)) = aa_num_num(bit1,Nb) ).

% semiring_norm(3)
tff(fact_5664_semiring__norm_I4_J,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(Nb),one)) ).

% semiring_norm(4)
tff(fact_5665_semiring__norm_I5_J,axiom,
    ! [Mb: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,Mb)),one) = aa_num_num(bit1,Mb) ).

% semiring_norm(5)
tff(fact_5666_semiring__norm_I8_J,axiom,
    ! [Mb: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,Mb)),one) = aa_num_num(bit0,aa_num_num(plus_plus_num(Mb),one)) ).

% semiring_norm(8)
tff(fact_5667_semiring__norm_I10_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(Mb),Nb)),one)) ).

% semiring_norm(10)
tff(fact_5668_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_ik(F),A4)) ).

% sum_abs_ge_zero
tff(fact_5669_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A4: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_il(F)),A4)) ).

% sum_abs_ge_zero
tff(fact_5670_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5671_zero__less__power__abs__iff,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb))
    <=> ( ( A != zero_zero_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5672_zero__less__power__abs__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb))
    <=> ( ( A != zero_zero_rat )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5673_zero__less__power__abs__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb))
    <=> ( ( A != zero_zero_int )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5674_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_complex,aa_complex_complex(plus_plus_complex(aa_set_nat_complex(groups2073611262835488442omplex(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5675_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_rat,aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5676_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5677_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5678_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5679_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(Mb,one) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,Mb)),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_5680_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5055182867167087721od_nat(Mb,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,Mb)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_5681_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique3479559517661332726nteger(Mb,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(Mb),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_5682_sum__zero__power,axiom,
    ! [C2: fun_nat_complex,A4: set_nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_im(C2)),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_nat_complex(C2,zero_zero_nat),
        zero_zero_complex ) ).

% sum_zero_power
tff(fact_5683_sum__zero__power,axiom,
    ! [C2: fun_nat_rat,A4: set_nat] :
      groups2906978787729119204at_rat(aTP_Lamm_in(C2),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_nat_rat(C2,zero_zero_nat),
        zero_zero_rat ) ).

% sum_zero_power
tff(fact_5684_sum__zero__power,axiom,
    ! [C2: fun_nat_real,A4: set_nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_io(C2)),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_nat_real(C2,zero_zero_nat),
        zero_zero_real ) ).

% sum_zero_power
tff(fact_5685_div__Suc__eq__div__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_5686_Suc__div__eq__add3__div__numeral,axiom,
    ! [Mb: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_5687_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_5688_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_5689_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,Nb)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(3)
tff(fact_5690_mod__Suc__eq__mod__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_5691_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Mb: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_5692_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_5693_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_5694_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,Nb)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(4)
tff(fact_5695_sum__zero__power_H,axiom,
    ! [C2: fun_nat_complex,D: fun_nat_complex,A4: set_nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ip(C2),D)),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C2,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)),
        zero_zero_complex ) ).

% sum_zero_power'
tff(fact_5696_sum__zero__power_H,axiom,
    ! [C2: fun_nat_rat,D: fun_nat_rat,A4: set_nat] :
      groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_iq(C2),D),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_rat_rat(divide_divide_rat(aa_nat_rat(C2,zero_zero_nat)),aa_nat_rat(D,zero_zero_nat)),
        zero_zero_rat ) ).

% sum_zero_power'
tff(fact_5697_sum__zero__power_H,axiom,
    ! [C2: fun_nat_real,D: fun_nat_real,A4: set_nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ir(C2),D)),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C2,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)),
        zero_zero_real ) ).

% sum_zero_power'
tff(fact_5698_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_5699_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_5700_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_5701_zmod__numeral__Bit1,axiom,
    ! [V: num,W2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W2)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_5702_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_5703_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_5704_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_5705_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_is),unique5052692396658037445od_int(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_5706_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_it),unique5055182867167087721od_nat(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_5707_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = produc6916734918728496179nteger(aTP_Lamm_iu,unique3479559517661332726nteger(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_5708_abs__ge__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(abs_abs_real,A)) ).

% abs_ge_self
tff(fact_5709_abs__ge__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A)) ).

% abs_ge_self
tff(fact_5710_abs__ge__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_self
tff(fact_5711_abs__ge__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(abs_abs_int,A)) ).

% abs_ge_self
tff(fact_5712_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% abs_le_D1
tff(fact_5713_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% abs_le_D1
tff(fact_5714_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% abs_le_D1
tff(fact_5715_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% abs_le_D1
tff(fact_5716_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_5717_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_5718_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_5719_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_5720_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_5721_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)) ).

% abs_minus_commute
tff(fact_5722_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ).

% abs_minus_commute
tff(fact_5723_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)) ).

% abs_minus_commute
tff(fact_5724_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_5725_mem__case__prodE,axiom,
    ! [Z: complex,C2: fun_in230608924162954550omplex,P: product_prod_int_int] :
      ( aa_set_complex_o(member_complex(Z),produc8580519160106071146omplex(C2,P))
     => ~ ! [X3: int,Y: int] :
            ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) )
           => ~ aa_set_complex_o(member_complex(Z),aa_int_set_complex(aa_int6119279391169676535omplex(C2,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_5726_mem__case__prodE,axiom,
    ! [Z: real,C2: fun_in6553099857819052212t_real,P: product_prod_int_int] :
      ( member_real(Z,produc6452406959799940328t_real(C2,P))
     => ~ ! [X3: int,Y: int] :
            ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) )
           => ~ member_real(Z,aa_int_set_real(aa_int4847878404602763637t_real(C2,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_5727_mem__case__prodE,axiom,
    ! [Z: nat,C2: fun_in2630332840903238488et_nat,P: product_prod_int_int] :
      ( aa_set_nat_o(member_nat(Z),produc4251311855443802252et_nat(C2,P))
     => ~ ! [X3: int,Y: int] :
            ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) )
           => ~ aa_set_nat_o(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C2,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_5728_mem__case__prodE,axiom,
    ! [Z: int,C2: fun_in7040156458970095540et_int,P: product_prod_int_int] :
      ( aa_set_int_o(member_int(Z),produc73460835934605544et_int(C2,P))
     => ~ ! [X3: int,Y: int] :
            ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) )
           => ~ aa_set_int_o(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C2,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_5729_mem__case__prodE,axiom,
    ! [Z: set_nat,C2: fun_in7207726656141197838et_nat,P: product_prod_int_int] :
      ( member_set_nat(Z,produc5233655623923918146et_nat(C2,P))
     => ~ ! [X3: int,Y: int] :
            ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) )
           => ~ member_set_nat(Z,aa_int_set_set_nat(aa_int1843896668418082255et_nat(C2,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_5730_mem__case__prodE,axiom,
    ! [Z: complex,C2: fun_fu15390479812311035omplex,P: produc7509114972854644512nteger] :
      ( aa_set_complex_o(member_complex(Z),produc5772012072983024995omplex(C2,P))
     => ~ ! [X3: fun_Co3460408610320954864e_term,Y: produc8923325533196201883nteger] :
            ( ( P = produc7264323995540650962nteger(X3,Y) )
           => ~ aa_set_complex_o(member_complex(Z),aa_Pro5605272939097928131omplex(aa_fun5788763187652937060omplex(C2,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_5731_mem__case__prodE,axiom,
    ! [Z: real,C2: fun_fu7234941744988193529t_real,P: produc7509114972854644512nteger] :
      ( member_real(Z,produc5067746511659309921t_real(C2,P))
     => ~ ! [X3: fun_Co3460408610320954864e_term,Y: produc8923325533196201883nteger] :
            ( ( P = produc7264323995540650962nteger(X3,Y) )
           => ~ member_real(Z,aa_Pro6515153133580421057t_real(aa_fun1966338168372333922t_real(C2,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_5732_mem__case__prodE,axiom,
    ! [Z: nat,C2: fun_fu6383124537394581533et_nat,P: produc7509114972854644512nteger] :
      ( aa_set_nat_o(member_nat(Z),produc7961109426542857093et_nat(C2,P))
     => ~ ! [X3: fun_Co3460408610320954864e_term,Y: produc8923325533196201883nteger] :
            ( ( P = produc7264323995540650962nteger(X3,Y) )
           => ~ aa_set_nat_o(member_nat(Z),aa_Pro381016180347496421et_nat(aa_fun3554285010146031238et_nat(C2,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_5733_mem__case__prodE,axiom,
    ! [Z: int,C2: fun_fu1569576118606662777et_int,P: produc7509114972854644512nteger] :
      ( aa_set_int_o(member_int(Z),produc3783258407033660385et_int(C2,P))
     => ~ ! [X3: fun_Co3460408610320954864e_term,Y: produc8923325533196201883nteger] :
            ( ( P = produc7264323995540650962nteger(X3,Y) )
           => ~ aa_set_int_o(member_int(Z),aa_Pro5426537197693075521et_int(aa_fun1064201865214304482et_int(C2,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_5734_mem__case__prodE,axiom,
    ! [Z: complex,C2: fun_fu5523889554619339274omplex,P: produc4788131091073775919nt_int] :
      ( aa_set_complex_o(member_complex(Z),produc4974371134869959794omplex(C2,P))
     => ~ ! [X3: fun_in5730379987311621695e_term,Y: product_prod_int_int] :
            ( ( P = produc4729913460834754785nt_int(X3,Y) )
           => ~ aa_set_complex_o(member_complex(Z),aa_Pro4762665599283228129omplex(aa_fun8357827850476113523omplex(C2,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_5735_case__prodD,axiom,
    ! [F: fun_fu3422731660242565741eger_o,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4652312591602923194eger_o(produc7733317944277263833eger_o(F),produc7264323995540650962nteger(A,B))
     => aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(F,A),B) ) ).

% case_prodD
tff(fact_5736_case__prodD,axiom,
    ! [F: fun_fu7695273699059480439eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(F),produc565448451787809096nteger(A,B))
     => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(F,A),B) ) ).

% case_prodD
tff(fact_5737_case__prodD,axiom,
    ! [F: fun_fu6865642474282033866_int_o,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int] :
      ( aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(F),produc7320969046608631221nt_int(A,B))
     => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(F,A),B) ) ).

% case_prodD
tff(fact_5738_case__prodD,axiom,
    ! [F: fun_fu2978539155105838238_int_o,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(F),produc4729913460834754785nt_int(A,B))
     => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(F,A),B) ) ).

% case_prodD
tff(fact_5739_case__prodD,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
     => aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prodD
tff(fact_5740_case__prodE,axiom,
    ! [C2: fun_fu3422731660242565741eger_o,P: produc7509114972854644512nteger] :
      ( aa_Pro4652312591602923194eger_o(produc7733317944277263833eger_o(C2),P)
     => ~ ! [X3: fun_Co3460408610320954864e_term,Y: produc8923325533196201883nteger] :
            ( ( P = produc7264323995540650962nteger(X3,Y) )
           => ~ aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(C2,X3),Y) ) ) ).

% case_prodE
tff(fact_5741_case__prodE,axiom,
    ! [C2: fun_fu7695273699059480439eger_o,P: produc2453742332425399446nteger] :
      ( aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(C2),P)
     => ~ ! [X3: fun_Pr5098669352780829690e_term,Y: produc8923325533196201883nteger] :
            ( ( P = produc565448451787809096nteger(X3,Y) )
           => ~ aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(C2,X3),Y) ) ) ).

% case_prodE
tff(fact_5742_case__prodE,axiom,
    ! [C2: fun_fu6865642474282033866_int_o,P: produc3043004577601930755nt_int] :
      ( aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(C2),P)
     => ~ ! [X3: fun_Pr7763680713840462443e_term,Y: product_prod_int_int] :
            ( ( P = produc7320969046608631221nt_int(X3,Y) )
           => ~ aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(C2,X3),Y) ) ) ).

% case_prodE
tff(fact_5743_case__prodE,axiom,
    ! [C2: fun_fu2978539155105838238_int_o,P: produc4788131091073775919nt_int] :
      ( aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(C2),P)
     => ~ ! [X3: fun_in5730379987311621695e_term,Y: product_prod_int_int] :
            ( ( P = produc4729913460834754785nt_int(X3,Y) )
           => ~ aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(C2,X3),Y) ) ) ).

% case_prodE
tff(fact_5744_case__prodE,axiom,
    ! [C2: fun_int_fun_int_o,P: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,C2),P)
     => ~ ! [X3: int,Y: int] :
            ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) )
           => ~ aa_int_o(aa_int_fun_int_o(C2,X3),Y) ) ) ).

% case_prodE
tff(fact_5745_case__prodD_H,axiom,
    ! [R: fun_na5914473681949881652_nat_o,A: nat,B: nat,C2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,R),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),C2)
     => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(R,A),B),C2) ) ).

% case_prodD'
tff(fact_5746_case__prodE_H,axiom,
    ! [C2: fun_na5914473681949881652_nat_o,P: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,C2),P),Z)
     => ~ ! [X3: nat,Y: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y) )
           => ~ aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C2,X3),Y),Z) ) ) ).

% case_prodE'
tff(fact_5747_abs__ge__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)) ).

% abs_ge_zero
tff(fact_5748_abs__ge__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A)) ).

% abs_ge_zero
tff(fact_5749_abs__ge__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_zero
tff(fact_5750_abs__ge__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A)) ).

% abs_ge_zero
tff(fact_5751_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_pos
tff(fact_5752_abs__of__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_5753_abs__of__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_pos
tff(fact_5754_abs__of__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_5755_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_5756_abs__not__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real) ).

% abs_not_less_zero
tff(fact_5757_abs__not__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_5758_abs__not__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int) ).

% abs_not_less_zero
tff(fact_5759_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq
tff(fact_5760_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq
tff(fact_5761_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq
tff(fact_5762_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(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_5763_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq2
tff(fact_5764_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% abs_triangle_ineq2
tff(fact_5765_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))) ).

% abs_triangle_ineq2
tff(fact_5766_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(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_5767_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq3
tff(fact_5768_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% abs_triangle_ineq3
tff(fact_5769_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))) ).

% abs_triangle_ineq3
tff(fact_5770_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(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_5771_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,A))) ).

% abs_triangle_ineq2_sym
tff(fact_5772_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_5773_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_5774_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(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_5775_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% nonzero_abs_divide
tff(fact_5776_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% nonzero_abs_divide
tff(fact_5777_abs__ge__minus__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A)) ).

% abs_ge_minus_self
tff(fact_5778_abs__ge__minus__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A)) ).

% abs_ge_minus_self
tff(fact_5779_abs__ge__minus__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_minus_self
tff(fact_5780_abs__ge__minus__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A)) ).

% abs_ge_minus_self
tff(fact_5781_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_le_iff
tff(fact_5782_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_le_iff
tff(fact_5783_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_le_iff
tff(fact_5784_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_le_iff
tff(fact_5785_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ).

% abs_le_D2
tff(fact_5786_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).

% abs_le_D2
tff(fact_5787_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ).

% abs_le_D2
tff(fact_5788_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% abs_le_D2
tff(fact_5789_abs__leI,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B) ) ) ).

% abs_leI
tff(fact_5790_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B) ) ) ).

% abs_leI
tff(fact_5791_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B) ) ) ).

% abs_leI
tff(fact_5792_abs__leI,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B) ) ) ).

% abs_leI
tff(fact_5793_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X3)),A4)
           => ( aa_nat_nat(F,aa_nat_nat(suc,X3)) = aa_nat_nat(G,aa_nat_nat(suc,X3)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A4) ) ) ) ).

% sum_cong_Suc
tff(fact_5794_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X3)),A4)
           => ( aa_nat_real(F,aa_nat_nat(suc,X3)) = aa_nat_real(G,aa_nat_nat(suc,X3)) ) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),A4) = aa_set_nat_real(groups6591440286371151544t_real(G),A4) ) ) ) ).

% sum_cong_Suc
tff(fact_5795_abs__real__def,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_real_def
tff(fact_5796_xor__num_Ocases,axiom,
    ! [Xb: product_prod_num_num] :
      ( ( Xb != product_Pair_num_num(one,one) )
     => ( ! [N2: num] : Xb != product_Pair_num_num(one,aa_num_num(bit0,N2))
       => ( ! [N2: num] : Xb != product_Pair_num_num(one,aa_num_num(bit1,N2))
         => ( ! [M: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M),one)
           => ( ! [M: num,N2: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit0,N2))
             => ( ! [M: num,N2: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit1,N2))
               => ( ! [M: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M),one)
                 => ( ! [M: num,N2: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit0,N2))
                   => ~ ! [M: num,N2: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit1,N2)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_5797_sum__subtractf__nat,axiom,
    ! [A4: set_complex,G: fun_complex_nat,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(G,X3)),aa_complex_nat(F,X3)) )
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_iv(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_5798_sum__subtractf__nat,axiom,
    ! [A4: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(G,X3)),aa_real_nat(F,X3)) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_iw(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_5799_sum__subtractf__nat,axiom,
    ! [A4: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(G,X3)),aa_set_nat_nat(F,X3)) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_ix(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A4)),groups8294997508430121362at_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_5800_sum__subtractf__nat,axiom,
    ! [A4: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X3)),aa_int_nat(F,X3)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_iy(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_5801_sum__subtractf__nat,axiom,
    ! [A4: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X3)),aa_nat_nat(F,X3)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_iz(G),F)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A4)) ) ) ).

% sum_subtractf_nat
tff(fact_5802_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ja(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_5803_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jb(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_5804_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jc(G),K2)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_5805_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,K2: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2))) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_jd(G),K2)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_5806_dense__eq0__I,axiom,
    ! [Xb: real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),E) )
     => ( Xb = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_5807_dense__eq0__I,axiom,
    ! [Xb: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),E) )
     => ( Xb = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_5808_abs__mult__pos,axiom,
    ! [Xb: code_integer,Yb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Yb)),Xb) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Yb),Xb)) ) ) ).

% abs_mult_pos
tff(fact_5809_abs__mult__pos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Yb)),Xb) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Xb)) ) ) ).

% abs_mult_pos
tff(fact_5810_abs__mult__pos,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Yb)),Xb) = aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Yb),Xb)) ) ) ).

% abs_mult_pos
tff(fact_5811_abs__mult__pos,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Yb)),Xb) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Xb)) ) ) ).

% abs_mult_pos
tff(fact_5812_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_5813_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_5814_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_5815_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(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_5816_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(abs_abs_real,A) = B )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
        & ( ( A = B )
          | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5817_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer(A) = B )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
        & ( ( A = B )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5818_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
        & ( ( A = B )
          | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5819_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5820_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(abs_abs_real,B) )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        & ( ( B = A )
          | ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5821_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = abs_abs_Code_integer(B) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
        & ( ( B = A )
          | ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5822_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & ( ( B = A )
          | ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5823_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5824_abs__minus__le__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A))),zero_zero_real) ).

% abs_minus_le_zero
tff(fact_5825_abs__minus__le__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_5826_abs__minus__le__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_5827_abs__minus__le__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int) ).

% abs_minus_le_zero
tff(fact_5828_abs__div__pos,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,Xb)),Yb) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)) ) ) ).

% abs_div_pos
tff(fact_5829_abs__div__pos,axiom,
    ! [Yb: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,Xb)),Yb) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% abs_div_pos
tff(fact_5830_zero__le__power__abs,axiom,
    ! [A: code_integer,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb)) ).

% zero_le_power_abs
tff(fact_5831_zero__le__power__abs,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) ).

% zero_le_power_abs
tff(fact_5832_zero__le__power__abs,axiom,
    ! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb)) ).

% zero_le_power_abs
tff(fact_5833_zero__le__power__abs,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb)) ).

% zero_le_power_abs
tff(fact_5834_abs__if,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_if
tff(fact_5835_abs__if,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,A) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int),aa_int_int(uminus_uminus_int,A),A) ).

% abs_if
tff(fact_5836_abs__if,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_if
tff(fact_5837_abs__if,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) ).

% abs_if
tff(fact_5838_abs__if__raw,axiom,
    ! [X: real] :
      aa_real_real(abs_abs_real,X) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real),aa_real_real(uminus_uminus_real,X),X) ).

% abs_if_raw
tff(fact_5839_abs__if__raw,axiom,
    ! [X: int] :
      aa_int_int(abs_abs_int,X) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int),aa_int_int(uminus_uminus_int,X),X) ).

% abs_if_raw
tff(fact_5840_abs__if__raw,axiom,
    ! [X: rat] :
      aa_rat_rat(abs_abs_rat,X) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X),X) ).

% abs_if_raw
tff(fact_5841_abs__if__raw,axiom,
    ! [X: code_integer] :
      abs_abs_Code_integer(X) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X),X) ).

% abs_if_raw
tff(fact_5842_abs__of__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_neg
tff(fact_5843_abs__of__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_5844_abs__of__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_neg
tff(fact_5845_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_5846_abs__diff__le__iff,axiom,
    ! [Xb: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_le_iff
tff(fact_5847_abs__diff__le__iff,axiom,
    ! [Xb: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_5848_abs__diff__le__iff,axiom,
    ! [Xb: rat,A: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),R2)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(plus_plus_rat(A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_5849_abs__diff__le__iff,axiom,
    ! [Xb: int,A: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_5850_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq4
tff(fact_5851_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq4
tff(fact_5852_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq4
tff(fact_5853_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(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_5854_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,B),plus_p5714425477246183910nteger(C2,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C2)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D)))) ).

% abs_diff_triangle_ineq
tff(fact_5855_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C2: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_5856_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(C2),D)))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),C2))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_5857_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C2: int,D: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),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),C2))),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_5858_abs__diff__less__iff,axiom,
    ! [Xb: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_less_iff
tff(fact_5859_abs__diff__less__iff,axiom,
    ! [Xb: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_5860_abs__diff__less__iff,axiom,
    ! [Xb: rat,A: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),R2)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_5861_abs__diff__less__iff,axiom,
    ! [Xb: int,A: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_5862_sum__eq__Suc0__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( groups4541462559716669496nt_nat(F,A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A4)
            & ( aa_int_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A4)
               => ( ( X4 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5863_sum__eq__Suc0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( groups5693394587270226106ex_nat(F,A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A4)
            & ( aa_complex_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: complex] :
                ( aa_set_complex_o(member_complex(Xa3),A4)
               => ( ( X4 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5864_sum__eq__Suc0__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( ( groups977919841031483927at_nat(F,A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
            & ( aa_Pro4243173770044397973at_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa3),A4)
               => ( ( X4 != Xa3 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5865_sum__eq__Suc0__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & ( aa_nat_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A4)
               => ( ( X4 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5866_sum__SucD,axiom,
    ! [F: fun_nat_nat,A4: set_nat,Nb: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = aa_nat_nat(suc,Nb) )
     => ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3)) ) ) ).

% sum_SucD
tff(fact_5867_sum__eq__1__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( groups4541462559716669496nt_nat(F,A4) = one_one_nat )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A4)
            & ( aa_int_nat(F,X4) = one_one_nat )
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A4)
               => ( ( X4 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5868_sum__eq__1__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( groups5693394587270226106ex_nat(F,A4) = one_one_nat )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A4)
            & ( aa_complex_nat(F,X4) = one_one_nat )
            & ! [Xa3: complex] :
                ( aa_set_complex_o(member_complex(Xa3),A4)
               => ( ( X4 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5869_sum__eq__1__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( ( groups977919841031483927at_nat(F,A4) = one_one_nat )
      <=> ? [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
            & ( aa_Pro4243173770044397973at_nat(F,X4) = one_one_nat )
            & ! [Xa3: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa3),A4)
               => ( ( X4 != Xa3 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5870_sum__eq__1__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = one_one_nat )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & ( aa_nat_nat(F,X4) = one_one_nat )
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A4)
               => ( ( X4 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5871_numeral__Bit1,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb))),one_one_complex) ).

% numeral_Bit1
tff(fact_5872_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb))),one_one_real) ).

% numeral_Bit1
tff(fact_5873_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb))),one_one_rat) ).

% numeral_Bit1
tff(fact_5874_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb))),one_one_nat) ).

% numeral_Bit1
tff(fact_5875_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb))),one_one_int) ).

% numeral_Bit1
tff(fact_5876_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% eval_nat_numeral(3)
tff(fact_5877_lemma__interval__lt,axiom,
    ! [A: real,Xb: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [Y3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y3))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),B) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_5878_sum__power__add,axiom,
    ! [Xb: complex,Mb: nat,I6: set_nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_je(Xb),Mb)),I6) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),I6)) ).

% sum_power_add
tff(fact_5879_sum__power__add,axiom,
    ! [Xb: rat,Mb: nat,I6: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_jf(Xb),Mb),I6) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),groups2906978787729119204at_rat(power_power_rat(Xb),I6)) ).

% sum_power_add
tff(fact_5880_sum__power__add,axiom,
    ! [Xb: int,Mb: nat,I6: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_jg(Xb),Mb),I6) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Mb)),groups3539618377306564664at_int(power_power_int(Xb),I6)) ).

% sum_power_add
tff(fact_5881_sum__power__add,axiom,
    ! [Xb: real,Mb: nat,I6: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_jh(Xb),Mb)),I6) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),I6)) ).

% sum_power_add
tff(fact_5882_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Nb,Mb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ji(G),Nb),Mb)),set_or1269000886237332187st_nat(Nb,Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_5883_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,Nb: nat,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Nb,Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_jj(G),Nb),Mb)),set_or1269000886237332187st_nat(Nb,Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_5884_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(m2),m2)),one_one_complex) ) ).

% numeral_code(3)
tff(fact_5885_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),m2)),one_one_real) ) ).

% numeral_code(3)
tff(fact_5886_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(m2),m2)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_5887_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_5888_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2)),one_one_int) ) ).

% numeral_code(3)
tff(fact_5889_sum__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_jk,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_cp(Nb),C2))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_5890_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_jk,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cr,Nb))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_5891_abs__add__one__gt__zero,axiom,
    ! [Xb: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(Xb))) ).

% abs_add_one_gt_zero
tff(fact_5892_abs__add__one__gt__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,Xb))) ).

% abs_add_one_gt_zero
tff(fact_5893_abs__add__one__gt__zero,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(abs_abs_rat,Xb))) ).

% abs_add_one_gt_zero
tff(fact_5894_abs__add__one__gt__zero,axiom,
    ! [Xb: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,Xb))) ).

% abs_add_one_gt_zero
tff(fact_5895_sum__diff__nat,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_5896_sum__diff__nat,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(B4),A4)
       => ( groups977919841031483927at_nat(F,minus_1356011639430497352at_nat(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A4)),groups977919841031483927at_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_5897_sum__diff__nat,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_5898_sum__diff__nat,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B4)) ) ) ) ).

% sum_diff_nat
tff(fact_5899_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_complex,K2: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K2)) = aa_set_nat_complex(groups2073611262835488442omplex(F),set_or1269000886237332187st_nat(zero_zero_nat,K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5900_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_rat,K2: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K2)) = groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(zero_zero_nat,K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5901_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K2: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K2)) = groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(zero_zero_nat,K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5902_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_nat,K2: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K2)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(zero_zero_nat,K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5903_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K2)) = aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(zero_zero_nat,K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5904_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5905_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5906_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5907_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5908_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5909_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5910_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5911_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,Nb))),aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5912_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5913_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5914_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5915_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,Mb)),aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5916_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_5917_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q3: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_5918_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q3: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_5919_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_5920_lemma__interval,axiom,
    ! [A: real,Xb: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [Y3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y3))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),B) ) ) ) ) ) ).

% lemma_interval
tff(fact_5921_Suc3__eq__add__3,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb) ).

% Suc3_eq_add_3
tff(fact_5922_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(aTP_Lamm_jl(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5923_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(aTP_Lamm_jm(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5924_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ja(G)),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5925_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,Mb)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jb(G)),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5926_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(aTP_Lamm_jn(F),set_or1269000886237332187st_nat(Mb,Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,Nb))),aa_nat_rat(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_5927_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(aTP_Lamm_jo(F),set_or1269000886237332187st_nat(Mb,Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,Nb))),aa_nat_int(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_5928_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jp(F)),set_or1269000886237332187st_nat(Mb,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,Nb))),aa_nat_real(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_5929_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : aa_set_nat_complex(groups2073611262835488442omplex(F),set_or1269000886237332187st_nat(A,B)) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_jq(F),A),B),zero_zero_complex) ).

% sum_atLeastAtMost_code
tff(fact_5930_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(A,B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_jr(F),A,B),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_5931_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(A,B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_js(F),A),B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_5932_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_jt(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_5933_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(A,B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_ju(F),A),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_5934_abs__le__square__iff,axiom,
    ! [Xb: code_integer,Yb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),abs_abs_Code_integer(Yb))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_5935_abs__le__square__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_5936_abs__le__square__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),aa_rat_rat(abs_abs_rat,Yb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_5937_abs__le__square__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),aa_int_int(abs_abs_int,Yb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_5938_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_5939_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_5940_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_5941_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_5942_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_5943_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_5944_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5945_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q3: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5946_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q3: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Mb)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5947_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5948_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5949_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5950_Suc__div__eq__add3__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_5951_sum__count__set,axiom,
    ! [Xs: list_complex,X5: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),X5)
     => ( aa_set_complex_o(finite3207457112153483333omplex,X5)
       => ( groups5693394587270226106ex_nat(count_list_complex(Xs),X5) = size_s3451745648224563538omplex(Xs) ) ) ) ).

% sum_count_set
tff(fact_5952_sum__count__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs)),X5)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,X5)
       => ( groups977919841031483927at_nat(count_4203492906077236349at_nat(Xs),X5) = size_s5460976970255530739at_nat(Xs) ) ) ) ).

% sum_count_set
tff(fact_5953_sum__count__set,axiom,
    ! [Xs: list_VEBT_VEBT,X5: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),X5)
     => ( finite5795047828879050333T_VEBT(X5)
       => ( groups771621172384141258BT_nat(count_list_VEBT_VEBT(Xs),X5) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) ) ) ) ).

% sum_count_set
tff(fact_5954_sum__count__set,axiom,
    ! [Xs: list_int,X5: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),X5)
     => ( aa_set_int_o(finite_finite_int,X5)
       => ( groups4541462559716669496nt_nat(count_list_int(Xs),X5) = aa_list_int_nat(size_size_list_int,Xs) ) ) ) ).

% sum_count_set
tff(fact_5955_sum__count__set,axiom,
    ! [Xs: list_nat,X5: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Xs)),X5)
     => ( aa_set_nat_o(finite_finite_nat,X5)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(count_list_nat(Xs)),X5) = aa_list_nat_nat(size_size_list_nat,Xs) ) ) ) ).

% sum_count_set
tff(fact_5956_Suc__mod__eq__add3__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_5957_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_5958_divmod__int__def,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_int_def
tff(fact_5959_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : aa_Pro6799258434465214121nt_int(adjust_div,Qr) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aTP_Lamm_jv),Qr) ).

% Divides.adjust_div_def
tff(fact_5960_abs__sqrt__wlog,axiom,
    ! [P2: fun_Co2081214437782444048eger_o,Xb: code_integer] :
      ( ! [X3: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P2,X3),aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P2,abs_abs_Code_integer(Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_5961_abs__sqrt__wlog,axiom,
    ! [P2: fun_real_fun_real_o,Xb: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
         => aa_real_o(aa_real_fun_real_o(P2,X3),aa_nat_real(aa_real_fun_nat_real(power_power_real,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_real_o(aa_real_fun_real_o(P2,aa_real_real(abs_abs_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_5962_abs__sqrt__wlog,axiom,
    ! [P2: fun_rat_fun_rat_o,Xb: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
         => aa_rat_o(aa_rat_fun_rat_o(P2,X3),aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_rat_o(aa_rat_fun_rat_o(P2,aa_rat_rat(abs_abs_rat,Xb)),aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_5963_abs__sqrt__wlog,axiom,
    ! [P2: fun_int_fun_int_o,Xb: int] :
      ( ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
         => aa_int_o(aa_int_fun_int_o(P2,X3),aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_int_o(aa_int_fun_int_o(P2,aa_int_int(abs_abs_int,Xb)),aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_5964_power2__le__iff__abs__le,axiom,
    ! [Yb: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Yb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),Yb) ) ) ).

% power2_le_iff_abs_le
tff(fact_5965_power2__le__iff__abs__le,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),Yb) ) ) ).

% power2_le_iff_abs_le
tff(fact_5966_power2__le__iff__abs__le,axiom,
    ! [Yb: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),Yb) ) ) ).

% power2_le_iff_abs_le
tff(fact_5967_power2__le__iff__abs__le,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),Yb) ) ) ).

% power2_le_iff_abs_le
tff(fact_5968_abs__square__le__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_5969_abs__square__le__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real) ) ).

% abs_square_le_1
tff(fact_5970_abs__square__le__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_5971_abs__square__le__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),one_one_int) ) ).

% abs_square_le_1
tff(fact_5972_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_def
tff(fact_5973_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod_def
tff(fact_5974_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(Mb,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Nb)),modulo364778990260209775nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Nb))) ).

% divmod_def
tff(fact_5975_power__mono__even,axiom,
    ! [Nb: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% power_mono_even
tff(fact_5976_power__mono__even,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_even
tff(fact_5977_power__mono__even,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_even
tff(fact_5978_power__mono__even,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono_even
tff(fact_5979_divmod_H__nat__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod'_nat_def
tff(fact_5980_convex__sum__bound__le,axiom,
    ! [I6: set_complex,Xb: fun_co8153928105989009288nteger,A: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(Xb,I2)) )
     => ( ( groups6621422865394947399nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_com4217951221296830281nteger(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups6621422865394947399nteger(aa_fun5267417455630836558nteger(aTP_Lamm_jw(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5981_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(Xb,I2)) )
     => ( ( groups7713935264441627589nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_jx(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5982_convex__sum__bound__le,axiom,
    ! [I6: set_nat,Xb: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Xb,I2)) )
     => ( ( groups7501900531339628137nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_jy(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5983_convex__sum__bound__le,axiom,
    ! [I6: set_int,Xb: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Xb,I2)) )
     => ( ( groups7873554091576472773nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_jz(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5984_convex__sum__bound__le,axiom,
    ! [I6: set_complex,Xb: fun_complex_real,A: fun_complex_real,B: real,Delta: real] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(Xb,I2)) )
     => ( ( groups5808333547571424918x_real(Xb,I6) = one_one_real )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ka(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5985_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_real_real,A: fun_real_real,B: real,Delta: real] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Xb,I2)) )
     => ( ( groups8097168146408367636l_real(Xb,I6) = one_one_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_kb(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5986_convex__sum__bound__le,axiom,
    ! [I6: set_int,Xb: fun_int_real,A: fun_int_real,B: real,Delta: real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Xb,I2)) )
     => ( ( groups8778361861064173332t_real(Xb,I6) = one_one_real )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_kc(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5987_convex__sum__bound__le,axiom,
    ! [I6: set_complex,Xb: fun_complex_rat,A: fun_complex_rat,B: rat,Delta: rat] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(Xb,I2)) )
     => ( ( groups5058264527183730370ex_rat(Xb,I6) = one_one_rat )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_complex_rat(A,I2)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_kd(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5988_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_real_rat,A: fun_real_rat,B: rat,Delta: rat] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Xb,I2)) )
     => ( ( groups1300246762558778688al_rat(Xb,I6) = one_one_rat )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_real_rat(A,I2)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_ke(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5989_convex__sum__bound__le,axiom,
    ! [I6: set_nat,Xb: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Xb,I2)) )
     => ( ( groups2906978787729119204at_rat(Xb,I6) = one_one_rat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(A,I2)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kf(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5990_sum__natinterval__diff,axiom,
    ! [F: fun_nat_complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_kg(F)),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_complex_complex(minus_minus_complex(aa_nat_complex(F,Mb)),aa_nat_complex(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_complex) ).

% sum_natinterval_diff
tff(fact_5991_sum__natinterval__diff,axiom,
    ! [F: fun_nat_rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(aTP_Lamm_kh(F),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Mb)),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_rat) ).

% sum_natinterval_diff
tff(fact_5992_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,Mb: nat,Nb: nat] :
      groups3539618377306564664at_int(aTP_Lamm_ki(F),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_5993_sum__natinterval__diff,axiom,
    ! [F: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kj(F)),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Mb)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_real) ).

% sum_natinterval_diff
tff(fact_5994_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(aTP_Lamm_kk(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Nb)),aa_nat_rat(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_5995_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(aTP_Lamm_kl(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_5996_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_km(F)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Nb)),aa_nat_real(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_5997_divmod__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : divmod_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% divmod_nat_def
tff(fact_5998_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),set_or1269000886237332187st_nat(Mb,Nb))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_5999_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(Mb,Nb))) = aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_6000_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(power_power_int(Xb),set_or1269000886237332187st_nat(Mb,Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Mb)),aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_6001_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(Mb,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_6002_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] : groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_kn(G),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.in_pairs
tff(fact_6003_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_ko(G),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.in_pairs
tff(fact_6004_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kp(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.in_pairs
tff(fact_6005_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kq(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.in_pairs
tff(fact_6006_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,Mb)),unique5026877609467782581ep_nat(Nb,unique5055182867167087721od_nat(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6007_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,Mb)),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(Nb),unique5052692396658037445od_int(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6008_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(Mb)),unique4921790084139445826nteger(Nb,unique3479559517661332726nteger(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6009_mask__eq__sum__exp__nat,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bb,Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_6010_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kr),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_6011_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_6012_arith__series__nat,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ks(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_6013_Sum__Icc__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kr),set_or1269000886237332187st_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_6014_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% odd_mod_4_div_2
tff(fact_6015_mod__exhaust__less__4,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_6016_sin__bound__lemma,axiom,
    ! [Xb: real,Yb: real,U: real,V: real] :
      ( ( Xb = Yb )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),U)),Yb))),V) ) ) ).

% sin_bound_lemma
tff(fact_6017_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_6018_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_6019_sum__gp,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_rat,
        $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb)))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ) ).

% sum_gp
tff(fact_6020_sum__gp,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_complex,
        $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb)))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ) ).

% sum_gp
tff(fact_6021_sum__gp,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_real,
        $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ) ).

% sum_gp
tff(fact_6022_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_real(ring_1_of_int_real,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_real,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: real,
            l:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,l),one_one_real)) ) ) ) ).

% of_int_code_if
tff(fact_6023_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_int(ring_1_of_int_int,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_int,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: int,
            l:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_int_int(aa_int_fun_int_int(plus_plus_int,l),one_one_int)) ) ) ) ).

% of_int_code_if
tff(fact_6024_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_complex(ring_17405671764205052669omplex,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_complex,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: complex,
            l:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_complex_complex(plus_plus_complex(l),one_one_complex)) ) ) ) ).

% of_int_code_if
tff(fact_6025_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_rat(ring_1_of_int_rat,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_rat,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: rat,
            l:= aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_rat_rat(plus_plus_rat(l),one_one_rat)) ) ) ) ).

% of_int_code_if
tff(fact_6026_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_Code_integer(ring_18347121197199848620nteger,K2) = $ite(
        K2 = zero_zero_int,
        zero_z3403309356797280102nteger,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% of_int_code_if
tff(fact_6027_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_6028_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_6029_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_6030_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Mb) = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_6031_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Mb) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_6032_of__int__eq__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,W2) = aa_int_real(ring_1_of_int_real,Z) )
    <=> ( W2 = Z ) ) ).

% of_int_eq_iff
tff(fact_6033_of__int__eq__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,W2) = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( W2 = Z ) ) ).

% of_int_eq_iff
tff(fact_6034_of__int__eq__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,W2) = aa_int_complex(ring_17405671764205052669omplex,Z) )
    <=> ( W2 = Z ) ) ).

% of_int_eq_iff
tff(fact_6035_split__part,axiom,
    ! [P2: $o,Q: fun_int_fun_int_o,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun6133342665515662012_int_o(aTP_Lamm_kt((P2)),Q)),X)
    <=> ( (P2)
        & aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Q),X) ) ) ).

% split_part
tff(fact_6036_int__eq__iff__numeral,axiom,
    ! [Mb: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_num_int(numeral_numeral_int,V) )
    <=> ( Mb = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_6037_abs__of__nat,axiom,
    ! [Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)) = aa_nat_rat(semiri681578069525770553at_rat,Nb) ).

% abs_of_nat
tff(fact_6038_abs__of__nat,axiom,
    ! [Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Nb) ).

% abs_of_nat
tff(fact_6039_abs__of__nat,axiom,
    ! [Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(semiri5074537144036343181t_real,Nb) ).

% abs_of_nat
tff(fact_6040_abs__of__nat,axiom,
    ! [Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) ).

% abs_of_nat
tff(fact_6041_negative__eq__positive,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Mb) )
    <=> ( ( Nb = zero_zero_nat )
        & ( Mb = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_6042_of__int__of__nat__eq,axiom,
    ! [Nb: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_rat(semiri681578069525770553at_rat,Nb) ).

% of_int_of_nat_eq
tff(fact_6043_of__int__of__nat__eq,axiom,
    ! [Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Nb) ).

% of_int_of_nat_eq
tff(fact_6044_of__int__of__nat__eq,axiom,
    ! [Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_real(semiri5074537144036343181t_real,Nb) ).

% of_int_of_nat_eq
tff(fact_6045_of__int__of__nat__eq,axiom,
    ! [Nb: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_complex(semiri8010041392384452111omplex,Nb) ).

% of_int_of_nat_eq
tff(fact_6046_of__int__of__nat__eq,axiom,
    ! [Nb: nat] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) ).

% of_int_of_nat_eq
tff(fact_6047_of__int__abs,axiom,
    ! [Xb: int] : aa_int_int(ring_1_of_int_int,aa_int_int(abs_abs_int,Xb)) = aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Xb)) ).

% of_int_abs
tff(fact_6048_of__int__abs,axiom,
    ! [Xb: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(abs_abs_int,Xb)) = abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Xb)) ).

% of_int_abs
tff(fact_6049_of__int__abs,axiom,
    ! [Xb: int] : aa_int_real(ring_1_of_int_real,aa_int_int(abs_abs_int,Xb)) = aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Xb)) ).

% of_int_abs
tff(fact_6050_of__int__abs,axiom,
    ! [Xb: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(abs_abs_int,Xb)) = aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Xb)) ).

% of_int_abs
tff(fact_6051_negative__zle,axiom,
    ! [Nb: nat,Mb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zle
tff(fact_6052_zdvd1__eq,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
    <=> ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_6053_int__dvd__int__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% int_dvd_int_iff
tff(fact_6054_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Mb) = zero_zero_rat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6055_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = zero_zero_int )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6056_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = zero_zero_real )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6057_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = zero_zero_nat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6058_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Mb) = zero_zero_complex )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6059_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Mb) = zero_z3403309356797280102nteger )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6060_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6061_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6062_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6063_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6064_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6065_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6066_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_6067_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_6068_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_6069_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_6070_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_6071_of__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% of_nat_0
tff(fact_6072_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_6073_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_6074_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_6075_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_6076_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_6077_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_6078_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_6079_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_6080_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_6081_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_6082_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_add
tff(fact_6083_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_add
tff(fact_6084_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_add
tff(fact_6085_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_add
tff(fact_6086_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% of_nat_add
tff(fact_6087_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Mb),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_add
tff(fact_6088_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_mult
tff(fact_6089_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mult
tff(fact_6090_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_mult
tff(fact_6091_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mult
tff(fact_6092_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% of_nat_mult
tff(fact_6093_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_mult
tff(fact_6094_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_6095_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_6096_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_6097_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_6098_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int = aa_int_int(ring_1_of_int_int,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_6099_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_real = aa_int_real(ring_1_of_int_real,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_6100_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_6101_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_complex = aa_int_complex(ring_17405671764205052669omplex,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_6102_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = zero_zero_int )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_6103_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = zero_zero_real )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_6104_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = zero_zero_rat )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_6105_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_6106_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Nb) = one_one_rat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6107_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Nb) = one_one_int )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6108_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Nb) = one_one_real )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6109_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Nb) = one_one_nat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6110_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Nb) = one_one_complex )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6111_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) = one_one_Code_integer )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6112_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6113_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6114_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6115_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6116_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6117_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6118_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_6119_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_6120_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_6121_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_6122_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_6123_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_6124_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_6125_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_6126_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_6127_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = numera6690914467698888265omplex(Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6128_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6129_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6130_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6131_of__int__numeral,axiom,
    ! [K2: num] : aa_int_complex(ring_17405671764205052669omplex,aa_num_int(numeral_numeral_int,K2)) = numera6690914467698888265omplex(K2) ).

% of_int_numeral
tff(fact_6132_of__int__numeral,axiom,
    ! [K2: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K2)) = aa_num_real(numeral_numeral_real,K2) ).

% of_int_numeral
tff(fact_6133_of__int__numeral,axiom,
    ! [K2: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K2)) = aa_num_rat(numeral_numeral_rat,K2) ).

% of_int_numeral
tff(fact_6134_of__int__numeral,axiom,
    ! [K2: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,K2) ).

% of_int_numeral
tff(fact_6135_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_6136_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_6137_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_6138_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = one_one_int )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_6139_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = one_one_real )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_6140_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = one_one_rat )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_6141_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = one_one_complex )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_6142_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_6143_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_6144_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_6145_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_6146_of__int__mult,axiom,
    ! [W2: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)) = aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,W2)),aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_mult
tff(fact_6147_of__int__mult,axiom,
    ! [W2: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_mult
tff(fact_6148_of__int__mult,axiom,
    ! [W2: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)) = aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_mult
tff(fact_6149_of__int__mult,axiom,
    ! [W2: int,Z: int] : aa_int_int(ring_1_of_int_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,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_mult
tff(fact_6150_of__int__add,axiom,
    ! [W2: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_add
tff(fact_6151_of__int__add,axiom,
    ! [W2: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_add
tff(fact_6152_of__int__add,axiom,
    ! [W2: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z)) = aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_add
tff(fact_6153_of__int__add,axiom,
    ! [W2: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z)) = aa_complex_complex(plus_plus_complex(aa_int_complex(ring_17405671764205052669omplex,W2)),aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_add
tff(fact_6154_negative__zless,axiom,
    ! [Nb: nat,Mb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zless
tff(fact_6155_of__int__minus,axiom,
    ! [Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,Z)) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_minus
tff(fact_6156_of__int__minus,axiom,
    ! [Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,Z)) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_minus
tff(fact_6157_of__int__minus,axiom,
    ! [Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,Z)) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_minus
tff(fact_6158_of__int__minus,axiom,
    ! [Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,Z)) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_minus
tff(fact_6159_of__int__minus,axiom,
    ! [Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,Z)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).

% of_int_minus
tff(fact_6160_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int)
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_6161_of__int__diff,axiom,
    ! [W2: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z)) = aa_complex_complex(minus_minus_complex(aa_int_complex(ring_17405671764205052669omplex,W2)),aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_diff
tff(fact_6162_of__int__diff,axiom,
    ! [W2: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_diff
tff(fact_6163_of__int__diff,axiom,
    ! [W2: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z)) = aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_diff
tff(fact_6164_of__int__diff,axiom,
    ! [W2: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_diff
tff(fact_6165_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_6166_eq__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K2) = aa_nat_nat(suc,Nb) )
    <=> ( aa_num_nat(pred_numeral,K2) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_6167_Suc__eq__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( ( aa_nat_nat(suc,Nb) = aa_num_nat(numeral_numeral_nat,K2) )
    <=> ( Nb = aa_num_nat(pred_numeral,K2) ) ) ).

% Suc_eq_numeral
tff(fact_6168_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Xb) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6169_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Xb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6170_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Xb) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6171_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Xb) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W2) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6172_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2) = aa_int_rat(ring_1_of_int_rat,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W2) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_6173_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2) = aa_int_real(ring_1_of_int_real,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W2) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_6174_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2) = aa_int_int(ring_1_of_int_int,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W2) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_6175_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W2) = aa_int_complex(ring_17405671764205052669omplex,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W2) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_6176_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),Nb) ).

% of_int_power
tff(fact_6177_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),Nb) ).

% of_int_power
tff(fact_6178_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),Nb) ).

% of_int_power
tff(fact_6179_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),Nb) ).

% of_int_power
tff(fact_6180_of__int__of__bool,axiom,
    ! [P2: $o] : aa_int_real(ring_1_of_int_real,aa_o_int(zero_n2684676970156552555ol_int,(P2))) = zero_n3304061248610475627l_real((P2)) ).

% of_int_of_bool
tff(fact_6181_of__int__of__bool,axiom,
    ! [P2: $o] : aa_int_rat(ring_1_of_int_rat,aa_o_int(zero_n2684676970156552555ol_int,(P2))) = zero_n2052037380579107095ol_rat((P2)) ).

% of_int_of_bool
tff(fact_6182_of__int__of__bool,axiom,
    ! [P2: $o] : aa_int_complex(ring_17405671764205052669omplex,aa_o_int(zero_n2684676970156552555ol_int,(P2))) = zero_n1201886186963655149omplex((P2)) ).

% of_int_of_bool
tff(fact_6183_of__int__of__bool,axiom,
    ! [P2: $o] : aa_int_int(ring_1_of_int_int,aa_o_int(zero_n2684676970156552555ol_int,(P2))) = aa_o_int(zero_n2684676970156552555ol_int,(P2)) ).

% of_int_of_bool
tff(fact_6184_of__nat__of__bool,axiom,
    ! [P2: $o] : aa_nat_real(semiri5074537144036343181t_real,zero_n2687167440665602831ol_nat((P2))) = zero_n3304061248610475627l_real((P2)) ).

% of_nat_of_bool
tff(fact_6185_of__nat__of__bool,axiom,
    ! [P2: $o] : aa_nat_complex(semiri8010041392384452111omplex,zero_n2687167440665602831ol_nat((P2))) = zero_n1201886186963655149omplex((P2)) ).

% of_nat_of_bool
tff(fact_6186_of__nat__of__bool,axiom,
    ! [P2: $o] : aa_nat_Code_integer(semiri4939895301339042750nteger,zero_n2687167440665602831ol_nat((P2))) = zero_n356916108424825756nteger((P2)) ).

% of_nat_of_bool
tff(fact_6187_of__nat__of__bool,axiom,
    ! [P2: $o] : aa_nat_nat(semiri1316708129612266289at_nat,zero_n2687167440665602831ol_nat((P2))) = zero_n2687167440665602831ol_nat((P2)) ).

% of_nat_of_bool
tff(fact_6188_of__nat__of__bool,axiom,
    ! [P2: $o] : aa_nat_int(semiri1314217659103216013at_int,zero_n2687167440665602831ol_nat((P2))) = aa_o_int(zero_n2684676970156552555ol_int,(P2)) ).

% of_nat_of_bool
tff(fact_6189_of__nat__sum,axiom,
    ! [F: fun_int_nat,A4: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A4)) = groups4538972089207619220nt_int(aTP_Lamm_ku(F),A4) ).

% of_nat_sum
tff(fact_6190_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A4: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A4)) = groups7754918857620584856omplex(aTP_Lamm_kv(F),A4) ).

% of_nat_sum
tff(fact_6191_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = groups3539618377306564664at_int(aTP_Lamm_kw(F),A4) ).

% of_nat_sum
tff(fact_6192_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_kx(F)),A4) ).

% of_nat_sum
tff(fact_6193_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = groups7501900531339628137nteger(aTP_Lamm_ky(F),A4) ).

% of_nat_sum
tff(fact_6194_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kz(F)),A4) ).

% of_nat_sum
tff(fact_6195_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_la(F)),A4) ).

% of_nat_sum
tff(fact_6196_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6197_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),zero_z3403309356797280102nteger)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6198_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6199_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6200_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6201_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb)) ).

% of_nat_Suc
tff(fact_6202_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% of_nat_Suc
tff(fact_6203_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb)) ).

% of_nat_Suc
tff(fact_6204_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,Mb)) ).

% of_nat_Suc
tff(fact_6205_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,Mb)) ).

% of_nat_Suc
tff(fact_6206_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Mb)) = plus_p5714425477246183910nteger(one_one_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)) ).

% of_nat_Suc
tff(fact_6207_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),Mb) ) ).

% numeral_le_real_of_nat_iff
tff(fact_6208_less__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K2)),Nb) ) ).

% less_numeral_Suc
tff(fact_6209_less__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(pred_numeral,K2)) ) ).

% less_Suc_numeral
tff(fact_6210_le__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K2)),Nb) ) ).

% le_numeral_Suc
tff(fact_6211_le__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_num_nat(pred_numeral,K2)) ) ).

% le_Suc_numeral
tff(fact_6212_of__int__sum,axiom,
    ! [F: fun_complex_int,A4: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A4)) = groups7754918857620584856omplex(aTP_Lamm_lb(F),A4) ).

% of_int_sum
tff(fact_6213_of__int__sum,axiom,
    ! [F: fun_nat_int,A4: set_nat] : aa_int_real(ring_1_of_int_real,groups3539618377306564664at_int(F,A4)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lc(F)),A4) ).

% of_int_sum
tff(fact_6214_of__int__sum,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(F,A4)) = groups8778361861064173332t_real(aTP_Lamm_ld(F),A4) ).

% of_int_sum
tff(fact_6215_of__int__sum,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A4)) = groups3906332499630173760nt_rat(aTP_Lamm_le(F),A4) ).

% of_int_sum
tff(fact_6216_of__int__sum,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_complex(ring_17405671764205052669omplex,groups4538972089207619220nt_int(F,A4)) = groups3049146728041665814omplex(aTP_Lamm_lf(F),A4) ).

% of_int_sum
tff(fact_6217_of__int__sum,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A4)) = groups4538972089207619220nt_int(aTP_Lamm_lg(F),A4) ).

% of_int_sum
tff(fact_6218_diff__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(pred_numeral,K2)) ).

% diff_Suc_numeral
tff(fact_6219_diff__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K2)),Nb) ).

% diff_numeral_Suc
tff(fact_6220_max__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),aa_num_nat(pred_numeral,K2))) ).

% max_Suc_numeral
tff(fact_6221_max__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K2)),Nb)) ).

% max_numeral_Suc
tff(fact_6222_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_6223_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_6224_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_6225_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_6226_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_6227_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_6228_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_6229_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_6230_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_6231_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_6232_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_6233_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_6234_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_6235_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_6236_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_6237_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_6238_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_6239_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_6240_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_6241_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_6242_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6243_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6244_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6245_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6246_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6247_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6248_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6249_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6250_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6251_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W2)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6252_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6253_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6254_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6255_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_6256_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_6257_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_6258_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6259_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6260_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6261_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_6262_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_6263_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_6264_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_6265_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_6266_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_6267_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_6268_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_6269_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_6270_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_6271_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_6272_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_6273_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6274_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6275_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6276_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6277_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Xb)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6278_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Xb)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6279_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Yb) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6280_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Yb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6281_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Yb) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6282_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Yb) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6283_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) = aa_int_complex(ring_17405671764205052669omplex,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6284_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) = aa_int_real(ring_1_of_int_real,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6285_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) = aa_int_rat(ring_1_of_int_rat,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6286_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = aa_int_int(ring_1_of_int_int,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6287_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W2)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6288_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W2)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6289_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W2)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6290_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6291_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Xb)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6292_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Xb)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6293_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6294_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6295_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6296_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6297_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6298_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6299_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),Nb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6300_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6301_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6302_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6303_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6304_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6305_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6306_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6307_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6308_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6309_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6310_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6311_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_6312_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_6313_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_6314_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6315_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6316_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6317_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_6318_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_6319_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_6320_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Yb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6321_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Yb) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6322_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Yb) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6323_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Yb) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6324_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Yb) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6325_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb) = aa_int_real(ring_1_of_int_real,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Yb ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6326_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = aa_int_int(ring_1_of_int_int,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Yb ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6327_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),Nb) = aa_int_complex(ring_17405671764205052669omplex,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Yb ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6328_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb) = aa_int_rat(ring_1_of_int_rat,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Yb ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6329_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb) = aa_int_Code_integer(ring_18347121197199848620nteger,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Yb ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6330_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6331_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6332_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6333_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6334_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6335_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6336_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6337_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6338_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6339_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6340_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6341_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6342_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6343_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6344_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6345_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6346_Collect__case__prod__mono,axiom,
    ! [A4: fun_int_fun_int_o,B4: fun_int_fun_int_o] :
      ( ord_le77980682512674322_int_o(A4,B4)
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,A4)),collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,B4))) ) ).

% Collect_case_prod_mono
tff(fact_6347_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_lh),Prod) ).

% prod.disc_eq_case
tff(fact_6348_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_6349_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_6350_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_6351_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),aa_int_Code_integer(ring_18347121197199848620nteger,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_6352_ex__le__of__int,axiom,
    ! [Xb: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_le_of_int
tff(fact_6353_ex__le__of__int,axiom,
    ! [Xb: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_le_of_int
tff(fact_6354_mult__of__int__commute,axiom,
    ! [Xb: int,Yb: complex] : aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,Xb)),Yb) = aa_complex_complex(times_times_complex(Yb),aa_int_complex(ring_17405671764205052669omplex,Xb)) ).

% mult_of_int_commute
tff(fact_6355_mult__of__int__commute,axiom,
    ! [Xb: int,Yb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Xb)),Yb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),aa_int_real(ring_1_of_int_real,Xb)) ).

% mult_of_int_commute
tff(fact_6356_mult__of__int__commute,axiom,
    ! [Xb: int,Yb: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,Xb)),Yb) = aa_rat_rat(times_times_rat(Yb),aa_int_rat(ring_1_of_int_rat,Xb)) ).

% mult_of_int_commute
tff(fact_6357_mult__of__int__commute,axiom,
    ! [Xb: int,Yb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,Xb)),Yb) = aa_int_int(aa_int_fun_int_int(times_times_int,Yb),aa_int_int(ring_1_of_int_int,Xb)) ).

% mult_of_int_commute
tff(fact_6358_real__arch__simple,axiom,
    ! [Xb: real] :
    ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N2)) ).

% real_arch_simple
tff(fact_6359_real__arch__simple,axiom,
    ! [Xb: rat] :
    ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% real_arch_simple
tff(fact_6360_mult__of__nat__commute,axiom,
    ! [Xb: nat,Yb: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Yb) = aa_rat_rat(times_times_rat(Yb),aa_nat_rat(semiri681578069525770553at_rat,Xb)) ).

% mult_of_nat_commute
tff(fact_6361_mult__of__nat__commute,axiom,
    ! [Xb: nat,Yb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),Yb) = aa_int_int(aa_int_fun_int_int(times_times_int,Yb),aa_nat_int(semiri1314217659103216013at_int,Xb)) ).

% mult_of_nat_commute
tff(fact_6362_mult__of__nat__commute,axiom,
    ! [Xb: nat,Yb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Yb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),aa_nat_real(semiri5074537144036343181t_real,Xb)) ).

% mult_of_nat_commute
tff(fact_6363_mult__of__nat__commute,axiom,
    ! [Xb: nat,Yb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Yb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Yb),aa_nat_nat(semiri1316708129612266289at_nat,Xb)) ).

% mult_of_nat_commute
tff(fact_6364_mult__of__nat__commute,axiom,
    ! [Xb: nat,Yb: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Xb)),Yb) = aa_complex_complex(times_times_complex(Yb),aa_nat_complex(semiri8010041392384452111omplex,Xb)) ).

% mult_of_nat_commute
tff(fact_6365_mult__of__nat__commute,axiom,
    ! [Xb: nat,Yb: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),Yb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Yb),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)) ).

% mult_of_nat_commute
tff(fact_6366_of__int__max,axiom,
    ! [Xb: int,Yb: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_int_real(ring_1_of_int_real,Xb)),aa_int_real(ring_1_of_int_real,Yb)) ).

% of_int_max
tff(fact_6367_of__int__max,axiom,
    ! [Xb: int,Yb: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb)) = aa_rat_rat(ord_max_rat(aa_int_rat(ring_1_of_int_rat,Xb)),aa_int_rat(ring_1_of_int_rat,Yb)) ).

% of_int_max
tff(fact_6368_of__int__max,axiom,
    ! [Xb: int,Yb: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(ring_1_of_int_int,Xb)),aa_int_int(ring_1_of_int_int,Yb)) ).

% of_int_max
tff(fact_6369_of__int__max,axiom,
    ! [Xb: int,Yb: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb)) = ord_max_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Xb),aa_int_Code_integer(ring_18347121197199848620nteger,Yb)) ).

% of_int_max
tff(fact_6370_int__cases2,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,N2)) ) ).

% int_cases2
tff(fact_6371_zdvd__antisym__abs,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B) ) ) ) ).

% zdvd_antisym_abs
tff(fact_6372_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M: nat,N2: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_diff_cases
tff(fact_6373_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_0_le_iff
tff(fact_6374_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_0_le_iff
tff(fact_6375_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_0_le_iff
tff(fact_6376_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_0_le_iff
tff(fact_6377_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_0_le_iff
tff(fact_6378_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_6379_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_6380_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_6381_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_6382_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),zero_z3403309356797280102nteger) ).

% of_nat_less_0_iff
tff(fact_6383_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_6384_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_6385_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_6386_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_6387_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Nb)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_6388_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Nb)) != zero_z3403309356797280102nteger ).

% of_nat_neq_0
tff(fact_6389_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_6390_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_6391_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_6392_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_6393_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_6394_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_6395_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_6396_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_6397_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_6398_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_6399_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J)) ) ).

% of_nat_mono
tff(fact_6400_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,I)),aa_nat_Code_integer(semiri4939895301339042750nteger,J)) ) ).

% of_nat_mono
tff(fact_6401_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J)) ) ).

% of_nat_mono
tff(fact_6402_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_6403_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_6404_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_6405_abs__zmult__eq__1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)) = one_one_int )
     => ( aa_int_int(abs_abs_int,Mb) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_6406_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_6407_int__of__nat__induct,axiom,
    ! [P2: fun_int_o,Z: int] :
      ( ! [N2: nat] : aa_int_o(P2,aa_nat_int(semiri1314217659103216013at_int,N2))
     => ( ! [N2: nat] : aa_int_o(P2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))))
       => aa_int_o(P2,Z) ) ) ).

% int_of_nat_induct
tff(fact_6408_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(3)
tff(fact_6409_zle__int,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% zle_int
tff(fact_6410_zero__le__imp__eq__int,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ? [N2: nat] : K2 = aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% zero_le_imp_eq_int
tff(fact_6411_nonneg__int__cases,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ~ ! [N2: nat] : K2 != aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% nonneg_int_cases
tff(fact_6412_int__plus,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% int_plus
tff(fact_6413_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(5)
tff(fact_6414_zadd__int__left,axiom,
    ! [Mb: nat,Nb: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))),Z) ).

% zadd_int_left
tff(fact_6415_zle__iff__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z)
    <=> ? [N3: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,N3)) ) ).

% zle_iff_zadd
tff(fact_6416_not__int__zless__negative,axiom,
    ! [Nb: nat,Mb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Mb))) ).

% not_int_zless_negative
tff(fact_6417_int__sum,axiom,
    ! [F: fun_int_nat,A4: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A4)) = groups4538972089207619220nt_int(aTP_Lamm_ku(F),A4) ).

% int_sum
tff(fact_6418_int__sum,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = groups3539618377306564664at_int(aTP_Lamm_kw(F),A4) ).

% int_sum
tff(fact_6419_numeral__eq__Suc,axiom,
    ! [K2: num] : aa_num_nat(numeral_numeral_nat,K2) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K2)) ).

% numeral_eq_Suc
tff(fact_6420_of__nat__max,axiom,
    ! [Xb: nat,Yb: nat] : semiri4216267220026989637d_enat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb)) = ord_ma741700101516333627d_enat(semiri4216267220026989637d_enat(Xb),semiri4216267220026989637d_enat(Yb)) ).

% of_nat_max
tff(fact_6421_of__nat__max,axiom,
    ! [Xb: nat,Yb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Yb)) ).

% of_nat_max
tff(fact_6422_of__nat__max,axiom,
    ! [Xb: nat,Yb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,Yb)) ).

% of_nat_max
tff(fact_6423_of__nat__max,axiom,
    ! [Xb: nat,Yb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(semiri1316708129612266289at_nat,Yb)) ).

% of_nat_max
tff(fact_6424_of__nat__max,axiom,
    ! [Xb: nat,Yb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb)) = ord_max_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Xb),aa_nat_Code_integer(semiri4939895301339042750nteger,Yb)) ).

% of_nat_max
tff(fact_6425_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
    <=> ! [M4: int] :
        ? [N3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M4),aa_int_int(abs_abs_int,N3))
          & aa_set_int_o(member_int(N3),S2) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_6426_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
    <=> ! [M4: int] :
        ? [N3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,M4),aa_int_int(abs_abs_int,N3))
          & aa_set_int_o(member_int(N3),S2) ) ) ).

% infinite_int_iff_unbounded
tff(fact_6427_nat__leq__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_leq_as_int
tff(fact_6428_ex__less__of__nat__mult,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_6429_ex__less__of__nat__mult,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_6430_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ) ).

% of_nat_diff
tff(fact_6431_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ) ).

% of_nat_diff
tff(fact_6432_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% of_nat_diff
tff(fact_6433_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ) ).

% of_nat_diff
tff(fact_6434_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ) ) ).

% of_nat_diff
tff(fact_6435_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = minus_8373710615458151222nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Mb),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ) ) ).

% of_nat_diff
tff(fact_6436_zabs__def,axiom,
    ! [I: int] :
      aa_int_int(abs_abs_int,I) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int),aa_int_int(uminus_uminus_int,I),I) ).

% zabs_def
tff(fact_6437_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ! [Y3: real] :
        ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),Xb)) ) ).

% reals_Archimedean3
tff(fact_6438_real__of__int__div4,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))) ).

% real_of_int_div4
tff(fact_6439_int__cases4,axiom,
    ! [Mb: int] :
      ( ! [N2: nat] : Mb != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( Mb != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ) ).

% int_cases4
tff(fact_6440_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),I)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I)) ) ) ).

% dvd_imp_le_int
tff(fact_6441_real__of__nat__div4,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))) ).

% real_of_nat_div4
tff(fact_6442_int__zle__neg,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)))
    <=> ( ( Nb = zero_zero_nat )
        & ( Mb = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_6443_int__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ).

% int_Suc
tff(fact_6444_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_6445_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
    <=> ? [N3: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% zless_iff_Suc_zadd
tff(fact_6446_abs__mod__less,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_6447_nonpos__int__cases,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),zero_zero_int)
     => ~ ! [N2: nat] : K2 != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% nonpos_int_cases
tff(fact_6448_negative__zle__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),zero_zero_int) ).

% negative_zle_0
tff(fact_6449_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_nonneg
tff(fact_6450_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_nonneg
tff(fact_6451_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_nonneg
tff(fact_6452_of__int__leD,axiom,
    ! [Nb: int,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),Xb) ) ) ).

% of_int_leD
tff(fact_6453_of__int__leD,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% of_int_leD
tff(fact_6454_of__int__leD,axiom,
    ! [Nb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ) ).

% of_int_leD
tff(fact_6455_of__int__leD,axiom,
    ! [Nb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xb) ) ) ).

% of_int_leD
tff(fact_6456_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_pos
tff(fact_6457_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_pos
tff(fact_6458_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_pos
tff(fact_6459_of__int__lessD,axiom,
    ! [Nb: int,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Xb) ) ) ).

% of_int_lessD
tff(fact_6460_of__int__lessD,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% of_int_lessD
tff(fact_6461_of__int__lessD,axiom,
    ! [Nb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ) ).

% of_int_lessD
tff(fact_6462_of__int__lessD,axiom,
    ! [Nb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xb) ) ) ).

% of_int_lessD
tff(fact_6463_floor__exists,axiom,
    ! [Xb: real] :
    ? [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int))) ) ).

% floor_exists
tff(fact_6464_floor__exists,axiom,
    ! [Xb: rat] :
    ? [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int))) ) ).

% floor_exists
tff(fact_6465_floor__exists1,axiom,
    ! [Xb: real] :
    ? [X3: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X3)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),one_one_int)))
      & ! [Y3: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y3)),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),one_one_int))) )
         => ( Y3 = X3 ) ) ) ).

% floor_exists1
tff(fact_6466_floor__exists1,axiom,
    ! [Xb: rat] :
    ? [X3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X3)),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),one_one_int)))
      & ! [Y3: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y3)),Xb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),one_one_int))) )
         => ( Y3 = X3 ) ) ) ).

% floor_exists1
tff(fact_6467_mod__mult2__eq_H,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,Mb))) ).

% mod_mult2_eq'
tff(fact_6468_mod__mult2__eq_H,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))) ).

% mod_mult2_eq'
tff(fact_6469_mod__mult2__eq_H,axiom,
    ! [A: code_integer,Mb: nat,Nb: nat] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),modulo364778990260209775nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb))) ).

% mod_mult2_eq'
tff(fact_6470_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K2)) ).

% of_int_neg_numeral
tff(fact_6471_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)) ).

% of_int_neg_numeral
tff(fact_6472_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2)) ).

% of_int_neg_numeral
tff(fact_6473_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2)) ).

% of_int_neg_numeral
tff(fact_6474_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K2)) ).

% of_int_neg_numeral
tff(fact_6475_int__le__real__less,axiom,
    ! [Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Mb)),one_one_real)) ) ).

% int_le_real_less
tff(fact_6476_int__less__real__le,axiom,
    ! [Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real)),aa_int_real(ring_1_of_int_real,Mb)) ) ).

% int_less_real_le
tff(fact_6477_pos__int__cases,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ~ ! [N2: nat] :
            ( ( K2 = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% pos_int_cases
tff(fact_6478_zero__less__imp__eq__int,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
          & ( K2 = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_6479_int__cases3,axiom,
    ! [K2: int] :
      ( ( K2 != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K2 = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) )
       => ~ ! [N2: nat] :
              ( ( K2 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ) ).

% int_cases3
tff(fact_6480_nat__less__real__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ).

% nat_less_real_le
tff(fact_6481_nat__le__real__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_6482_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K2)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K2)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_6483_zdvd__mult__cancel1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( Mb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)),Mb)
      <=> ( aa_int_int(abs_abs_int,Nb) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_6484_not__zle__0__negative,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))) ).

% not_zle_0_negative
tff(fact_6485_negative__zless__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),zero_zero_int) ).

% negative_zless_0
tff(fact_6486_negD,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
     => ? [N2: nat] : Xb = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% negD
tff(fact_6487_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ).

% int_ops(6)
tff(fact_6488_real__of__int__div__aux,axiom,
    ! [Xb: int,D: int] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Xb)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_6489_real__of__nat__div__aux,axiom,
    ! [Xb: nat,D: nat] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_6490_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
     => ~ ! [N2: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),E2) ) ).

% nat_approx_posE
tff(fact_6491_nat__approx__posE,axiom,
    ! [E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
     => ~ ! [N2: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))),E2) ) ).

% nat_approx_posE
tff(fact_6492_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_6493_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_6494_even__add__abs__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(abs_abs_int,L)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_add_abs_iff
tff(fact_6495_even__abs__add__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,K2)),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_abs_add_iff
tff(fact_6496_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),Xb)),C2) )
         => ( Xb = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_6497_real__of__int__div2,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xb)))) ).

% real_of_int_div2
tff(fact_6498_real__of__int__div3,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xb)))),one_one_real) ).

% real_of_int_div3
tff(fact_6499_neg__int__cases,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
     => ~ ! [N2: nat] :
            ( ( K2 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% neg_int_cases
tff(fact_6500_zdiff__int__split,axiom,
    ! [P2: fun_int_o,Xb: nat,Yb: nat] :
      ( aa_int_o(P2,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Yb)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
         => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Yb))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
         => aa_int_o(P2,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_6501_real__of__nat__div2,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb)))) ).

% real_of_nat_div2
tff(fact_6502_ln__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_realpow
tff(fact_6503_real__of__nat__div3,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb)))),one_one_real) ).

% real_of_nat_div3
tff(fact_6504_nat__intermed__int__val,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int,K2: int] :
      ( ! [I2: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Mb)),K2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(F,Nb))
           => ? [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
                & ( aa_nat_int(F,I2) = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_6505_decr__lemma,axiom,
    ! [D: int,Xb: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Z))),one_one_int)),D))),Z) ) ).

% decr_lemma
tff(fact_6506_incr__lemma,axiom,
    ! [D: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Z))),one_one_int)),D))) ) ).

% incr_lemma
tff(fact_6507_linear__plus__1__le__power,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_6508_Bernoulli__inequality,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_6509_nat__ivt__aux,axiom,
    ! [Nb: nat,F: fun_nat_int,K2: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(F,Nb))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
              & ( aa_nat_int(F,I2) = K2 ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_6510_eq__diff__eq_H,axiom,
    ! [Xb: real,Yb: real,Z: real] :
      ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Yb),Z) )
    <=> ( Yb = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Z) ) ) ).

% eq_diff_eq'
tff(fact_6511_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F: fun_nat_int,K2: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(F,Nb))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
              & ( aa_nat_int(F,I2) = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_6512_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).

% double_gauss_sum
tff(fact_6513_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).

% double_gauss_sum
tff(fact_6514_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(semiri8010041392384452111omplex),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).

% double_gauss_sum
tff(fact_6515_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)) ).

% double_gauss_sum
tff(fact_6516_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).

% double_gauss_sum
tff(fact_6517_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum
tff(fact_6518_double__arith__series,axiom,
    ! [A: rat,D: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_li(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),D))) ).

% double_arith_series
tff(fact_6519_double__arith__series,axiom,
    ! [A: int,D: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_lj(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),D))) ).

% double_arith_series
tff(fact_6520_double__arith__series,axiom,
    ! [A: complex,D: complex,Nb: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_lk(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),D))) ).

% double_arith_series
tff(fact_6521_double__arith__series,axiom,
    ! [A: code_integer,D: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_ll(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),D))) ).

% double_arith_series
tff(fact_6522_double__arith__series,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lm(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),D))) ).

% double_arith_series
tff(fact_6523_double__arith__series,axiom,
    ! [A: real,D: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ln(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),D))) ).

% double_arith_series
tff(fact_6524_gauss__sum,axiom,
    ! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_6525_gauss__sum,axiom,
    ! [Nb: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_6526_gauss__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_6527_arith__series,axiom,
    ! [A: int,D: int,Nb: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_lj(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6528_arith__series,axiom,
    ! [A: code_integer,D: code_integer,Nb: nat] : groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_ll(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),D))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6529_arith__series,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lm(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6530_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6531_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6532_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(semiri8010041392384452111omplex),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6533_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6534_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6535_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6536_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_6537_sum__gp__offset,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ).

% sum_gp_offset
tff(fact_6538_sum__gp__offset,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_complex,aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ).

% sum_gp_offset
tff(fact_6539_sum__gp__offset,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).

% sum_gp_offset
tff(fact_6540_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_6541_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_6542_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_6543_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_rat(semiri681578069525770553at_rat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_rat,produc6207742614233964070at_rat(aTP_Lamm_lo,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_6544_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_int(semiri1314217659103216013at_int,Nb) = $ite(Nb = zero_zero_nat,zero_zero_int,aa_Pro4240683299535347697at_int(produc6840382203811409530at_int(aTP_Lamm_lp),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_6545_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_real(semiri5074537144036343181t_real,Nb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_Pro6168199777849092593t_real(produc1703576794950452218t_real(aTP_Lamm_lq),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_6546_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_nat(semiri1316708129612266289at_nat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_lr),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_6547_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_complex(semiri8010041392384452111omplex,Nb) = $ite(Nb = zero_zero_nat,zero_zero_complex,produc1917071388513777916omplex(aTP_Lamm_ls,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_6548_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) = $ite(Nb = zero_zero_nat,zero_z3403309356797280102nteger,aa_Pro4764841431850000930nteger(produc1830744345554046123nteger(aTP_Lamm_lt),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_6549_monoseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_lu,Xb)) ) ).

% monoseq_arctan_series
tff(fact_6550_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_6551_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),Nb)),aa_nat_complex(power_power_complex(Z),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_6552_round__unique,axiom,
    ! [Xb: real,Yb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Yb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Yb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_int(archim8280529875227126926d_real,Xb) = Yb ) ) ) ).

% round_unique
tff(fact_6553_round__unique,axiom,
    ! [Xb: rat,Yb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Yb))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Yb)),aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,Xb) = Yb ) ) ) ).

% round_unique
tff(fact_6554_ln__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_lv,Xb)) ) ) ) ).

% ln_series
tff(fact_6555_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,Nb: nat] :
      ( ( H != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Z),H)),Nb)),aa_nat_rat(power_power_rat(Z),Nb))),H)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(times_times_rat(H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_lx(H),Z),Nb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6556_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),Nb)),aa_nat_complex(power_power_complex(Z),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(times_times_complex(H),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_lz(H),Z),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6557_lemma__termdiff2,axiom,
    ! [H: real,Z: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,H),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mb(H),Z),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6558_lessThan__eq__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Xb) = aa_nat_set_nat(set_ord_lessThan_nat,Yb) )
    <=> ( Xb = Yb ) ) ).

% lessThan_eq_iff
tff(fact_6559_lessThan__eq__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( ( set_ord_lessThan_int(Xb) = set_ord_lessThan_int(Yb) )
    <=> ( Xb = Yb ) ) ).

% lessThan_eq_iff
tff(fact_6560_lessThan__eq__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( ( set_or5984915006950818249n_real(Xb) = set_or5984915006950818249n_real(Yb) )
    <=> ( Xb = Yb ) ) ).

% lessThan_eq_iff
tff(fact_6561_lessThan__iff,axiom,
    ! [I: set_nat,K2: set_nat] :
      ( member_set_nat(I,set_or890127255671739683et_nat(K2))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,I),K2) ) ).

% lessThan_iff
tff(fact_6562_lessThan__iff,axiom,
    ! [I: rat,K2: rat] :
      ( member_rat(I,set_ord_lessThan_rat(K2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),K2) ) ).

% lessThan_iff
tff(fact_6563_lessThan__iff,axiom,
    ! [I: num,K2: num] :
      ( member_num(I,set_ord_lessThan_num(K2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,I),K2) ) ).

% lessThan_iff
tff(fact_6564_lessThan__iff,axiom,
    ! [I: nat,K2: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2) ) ).

% lessThan_iff
tff(fact_6565_lessThan__iff,axiom,
    ! [I: int,K2: int] :
      ( aa_set_int_o(member_int(I),set_ord_lessThan_int(K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,I),K2) ) ).

% lessThan_iff
tff(fact_6566_lessThan__iff,axiom,
    ! [I: real,K2: real] :
      ( member_real(I,set_or5984915006950818249n_real(K2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,I),K2) ) ).

% lessThan_iff
tff(fact_6567_finite__lessThan,axiom,
    ! [K2: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_ord_lessThan_nat,K2)) ).

% finite_lessThan
tff(fact_6568_lessThan__subset__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ord_less_eq_set_rat(set_ord_lessThan_rat(Xb),set_ord_lessThan_rat(Yb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ).

% lessThan_subset_iff
tff(fact_6569_lessThan__subset__iff,axiom,
    ! [Xb: num,Yb: num] :
      ( ord_less_eq_set_num(set_ord_lessThan_num(Xb),set_ord_lessThan_num(Yb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb) ) ).

% lessThan_subset_iff
tff(fact_6570_lessThan__subset__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Yb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ).

% lessThan_subset_iff
tff(fact_6571_lessThan__subset__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_lessThan_int(Xb)),set_ord_lessThan_int(Yb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ).

% lessThan_subset_iff
tff(fact_6572_lessThan__subset__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( ord_less_eq_set_real(set_or5984915006950818249n_real(Xb),set_or5984915006950818249n_real(Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ).

% lessThan_subset_iff
tff(fact_6573_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_6574_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_6575_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_6576_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_rat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_6577_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_int(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_6578_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_nat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_6579_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_real(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_6580_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_im(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_6581_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_io(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_6582_int__int__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% int_int_eq
tff(fact_6583_less__by__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( A4 = bot_bo2099793752762293965at_nat )
     => aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A4),B4) ) ).

% less_by_empty
tff(fact_6584_lessThan__non__empty,axiom,
    ! [Xb: int] : set_ord_lessThan_int(Xb) != bot_bot_set_int ).

% lessThan_non_empty
tff(fact_6585_lessThan__non__empty,axiom,
    ! [Xb: real] : set_or5984915006950818249n_real(Xb) != bot_bot_set_real ).

% lessThan_non_empty
tff(fact_6586_infinite__Iio,axiom,
    ! [A: int] : ~ aa_set_int_o(finite_finite_int,set_ord_lessThan_int(A)) ).

% infinite_Iio
tff(fact_6587_infinite__Iio,axiom,
    ! [A: real] : ~ finite_finite_real(set_or5984915006950818249n_real(A)) ).

% infinite_Iio
tff(fact_6588_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_mc,U)) ).

% lessThan_def
tff(fact_6589_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_md,U)) ).

% lessThan_def
tff(fact_6590_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_me,U)) ).

% lessThan_def
tff(fact_6591_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bb,U)) ).

% lessThan_def
tff(fact_6592_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_mf,U)) ).

% lessThan_def
tff(fact_6593_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_mg,U)) ).

% lessThan_def
tff(fact_6594_Iio__eq__empty__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Nb) = bot_bot_set_nat )
    <=> ( Nb = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
tff(fact_6595_lessThan__strict__subset__iff,axiom,
    ! [Mb: rat,Nb: rat] :
      ( ord_less_set_rat(set_ord_lessThan_rat(Mb),set_ord_lessThan_rat(Nb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_6596_lessThan__strict__subset__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ord_less_set_num(set_ord_lessThan_num(Mb),set_ord_lessThan_num(Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_6597_lessThan__strict__subset__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_6598_lessThan__strict__subset__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_set_int_o(ord_less_set_int(set_ord_lessThan_int(Mb)),set_ord_lessThan_int(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_6599_lessThan__strict__subset__iff,axiom,
    ! [Mb: real,Nb: real] :
      ( ord_less_set_real(set_or5984915006950818249n_real(Mb),set_or5984915006950818249n_real(Nb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_6600_complex__mod__minus__le__complex__mod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Xb))),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% complex_mod_minus_le_complex_mod
tff(fact_6601_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(B),A))),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% complex_mod_triangle_ineq2
tff(fact_6602_lessThan__empty__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Nb) = bot_bot_set_nat )
    <=> ( Nb = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_6603_finite__nat__iff__bounded,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
    <=> ? [K3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_lessThan_nat,K3)) ) ).

% finite_nat_iff_bounded
tff(fact_6604_finite__nat__bounded,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ? [K: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_lessThan_nat,K)) ) ).

% finite_nat_bounded
tff(fact_6605_round__mono,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,Xb)),aa_rat_int(archim7778729529865785530nd_rat,Yb)) ) ).

% round_mono
tff(fact_6606_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mh(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nat_diff_reindex
tff(fact_6607_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_mi(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nat_diff_reindex
tff(fact_6608_sum__diff__distrib,axiom,
    ! [Q: fun_int_nat,P2: fun_int_nat,Nb: int] :
      ( ! [X3: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Q,X3)),aa_int_nat(P2,X3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(P2,set_ord_lessThan_int(Nb))),groups4541462559716669496nt_nat(Q,set_ord_lessThan_int(Nb))) = groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_iy(Q),P2),set_ord_lessThan_int(Nb)) ) ) ).

% sum_diff_distrib
tff(fact_6609_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P2: fun_real_nat,Nb: real] :
      ( ! [X3: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Q,X3)),aa_real_nat(P2,X3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P2,set_or5984915006950818249n_real(Nb))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(Nb))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_iw(Q),P2),set_or5984915006950818249n_real(Nb)) ) ) ).

% sum_diff_distrib
tff(fact_6610_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P2: fun_nat_nat,Nb: nat] :
      ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Q,X3)),aa_nat_nat(P2,X3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P2),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_iz(Q),P2)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ) ) ).

% sum_diff_distrib
tff(fact_6611_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_jl(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_6612_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_jm(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_6613_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ja(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_6614_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jb(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_6615_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_jn(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Mb)),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6616_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_jo(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6617_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jp(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Mb)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6618_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_mj(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_6619_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_mk(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_6620_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ml(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_6621_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ja(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_6622_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jb(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_6623_power__diff__1__eq,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Nb)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),one_one_complex)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_6624_power__diff__1__eq,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),one_one_rat)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_6625_power__diff__1__eq,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Nb)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),one_one_int)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_6626_power__diff__1__eq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_6627_one__diff__power__eq,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_6628_one__diff__power__eq,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_6629_one__diff__power__eq,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_6630_one__diff__power__eq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_6631_geometric__sum,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( Xb != one_one_complex )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Nb)),one_one_complex)),aa_complex_complex(minus_minus_complex(Xb),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_6632_geometric__sum,axiom,
    ! [Xb: rat,Nb: nat] :
      ( ( Xb != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),one_one_rat)),aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_6633_geometric__sum,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != one_one_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ) ) ).

% geometric_sum
tff(fact_6634_monoseq__realpow,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,Xb)) ) ) ).

% monoseq_realpow
tff(fact_6635_round__diff__minimal,axiom,
    ! [Z: real,Mb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z))))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,Mb)))) ).

% round_diff_minimal
tff(fact_6636_round__diff__minimal,axiom,
    ! [Z: rat,Mb: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,Mb)))) ).

% round_diff_minimal
tff(fact_6637_sum__gp__strict,axiom,
    ! [Xb: rat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ).

% sum_gp_strict
tff(fact_6638_sum__gp__strict,axiom,
    ! [Xb: complex,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Nb))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ).

% sum_gp_strict
tff(fact_6639_sum__gp__strict,axiom,
    ! [Xb: real,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).

% sum_gp_strict
tff(fact_6640_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mm(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mn(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_6641_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,Mb: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mo(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mp(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_6642_lemma__termdiff1,axiom,
    ! [Z: int,H: int,Mb: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_mq(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_mr(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_6643_lemma__termdiff1,axiom,
    ! [Z: real,H: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ms(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mt(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_6644_diff__power__eq__sum,axiom,
    ! [Xb: complex,Nb: nat,Yb: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))),aa_nat_complex(power_power_complex(Yb),aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Yb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_mu(Xb),Nb),Yb)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_6645_diff__power__eq__sum,axiom,
    ! [Xb: rat,Nb: nat,Yb: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(Yb),aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Yb)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_mv(Xb),Nb),Yb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_6646_diff__power__eq__sum,axiom,
    ! [Xb: int,Nb: nat,Yb: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Nb))),aa_nat_int(power_power_int(Yb),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Yb)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_mw(Xb),Nb),Yb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_6647_diff__power__eq__sum,axiom,
    ! [Xb: real,Nb: nat,Yb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Yb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_mx(Xb),Nb),Yb)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_6648_power__diff__sumr2,axiom,
    ! [Xb: complex,Nb: nat,Yb: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Yb),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Yb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_my(Xb),Nb),Yb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_6649_power__diff__sumr2,axiom,
    ! [Xb: rat,Nb: nat,Yb: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(Yb),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Yb)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_mz(Xb),Nb),Yb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_6650_power__diff__sumr2,axiom,
    ! [Xb: int,Nb: nat,Yb: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Yb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Yb)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_na(Xb),Nb),Yb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_6651_power__diff__sumr2,axiom,
    ! [Xb: real,Nb: nat,Yb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Yb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_nb(Xb),Nb),Yb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_6652_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_Code_integer,K5: code_integer,K2: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,P4)),K5) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),K5)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7501900531339628137nteger(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6653_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_rat,K5: rat,K2: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,P4)),K5) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),K5)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6654_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_int,K5: int,K2: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,P4)),K5) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K5)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6655_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_nat,K5: nat,K2: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,P4)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),K5)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6656_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_real,K5: real,K2: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,P4)),K5) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),K5)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6657_one__diff__power__eq_H,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_nc(Xb),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_6658_one__diff__power__eq_H,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_nd(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_6659_one__diff__power__eq_H,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ne(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_6660_one__diff__power__eq_H,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_nf(Xb),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_6661_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ng(F),G)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nh(F)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ni(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum_split_even_odd
tff(fact_6662_of__int__round__le,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_6663_of__int__round__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))),aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_6664_of__int__round__ge,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_ge
tff(fact_6665_of__int__round__ge,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))) ).

% of_int_round_ge
tff(fact_6666_of__int__round__abs__le,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_6667_of__int__round__abs__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))),Xb))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_6668_norm__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xb)),zero_zero_real)
    <=> ( Xb = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_6669_norm__le__zero__iff,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),zero_zero_real)
    <=> ( Xb = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_6670_zero__less__norm__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,Xb))
    <=> ( Xb != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_6671_zero__less__norm__iff,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xb))
    <=> ( Xb != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_6672_norm__eq__zero,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_6673_norm__eq__zero,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_6674_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_6675_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_6676_norm__power__diff,axiom,
    ! [Z: real,W2: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W2)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),Mb)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W2)))) ) ) ).

% norm_power_diff
tff(fact_6677_norm__power__diff,axiom,
    ! [Z: complex,W2: complex,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W2)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Z),Mb)),aa_nat_complex(power_power_complex(W2),Mb)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z),W2)))) ) ) ).

% norm_power_diff
tff(fact_6678_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_nj) = zero_zero_complex ).

% suminf_zero
tff(fact_6679_suminf__zero,axiom,
    suminf_real(aTP_Lamm_ek) = zero_zero_real ).

% suminf_zero
tff(fact_6680_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_bx) = zero_zero_nat ).

% suminf_zero
tff(fact_6681_suminf__zero,axiom,
    suminf_int(aTP_Lamm_nk) = zero_zero_int ).

% suminf_zero
tff(fact_6682_norm__not__less__zero,axiom,
    ! [Xb: complex] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),zero_zero_real) ).

% norm_not_less_zero
tff(fact_6683_norm__ge__zero,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% norm_ge_zero
tff(fact_6684_sum__norm__le,axiom,
    ! [S2: set_real,F: fun_real_complex,G: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(F,X3))),aa_real_real(G,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups5754745047067104278omplex(F,S2))),groups8097168146408367636l_real(G,S2)) ) ).

% sum_norm_le
tff(fact_6685_sum__norm__le,axiom,
    ! [S2: set_set_nat,F: fun_set_nat_complex,G: fun_set_nat_real] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(F,X3))),aa_set_nat_real(G,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups8255218700646806128omplex(F,S2))),groups5107569545109728110t_real(G,S2)) ) ).

% sum_norm_le
tff(fact_6686_sum__norm__le,axiom,
    ! [S2: set_int,F: fun_int_complex,G: fun_int_real] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(F,X3))),aa_int_real(G,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups3049146728041665814omplex(F,S2))),groups8778361861064173332t_real(G,S2)) ) ).

% sum_norm_le
tff(fact_6687_sum__norm__le,axiom,
    ! [S2: set_nat,F: fun_nat_complex,G: fun_nat_real] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,X3))),aa_nat_real(G,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),S2))),aa_set_nat_real(groups6591440286371151544t_real(G),S2)) ) ).

% sum_norm_le
tff(fact_6688_sum__norm__le,axiom,
    ! [S2: set_complex,F: fun_complex_complex,G: fun_complex_real] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(F,X3))),aa_complex_real(G,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,S2))),groups5808333547571424918x_real(G,S2)) ) ).

% sum_norm_le
tff(fact_6689_sum__norm__le,axiom,
    ! [S2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,X3))),aa_nat_real(G,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),S2))),aa_set_nat_real(groups6591440286371151544t_real(G),S2)) ) ).

% sum_norm_le
tff(fact_6690_norm__sum,axiom,
    ! [F: fun_nat_complex,A4: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),A4))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nl(F)),A4)) ).

% norm_sum
tff(fact_6691_norm__sum,axiom,
    ! [F: fun_complex_complex,A4: set_complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,A4))),groups5808333547571424918x_real(aTP_Lamm_nm(F),A4)) ).

% norm_sum
tff(fact_6692_norm__sum,axiom,
    ! [F: fun_nat_real,A4: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),A4))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nn(F)),A4)) ).

% norm_sum
tff(fact_6693_norm__uminus__minus,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xb)),Yb)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)) ).

% norm_uminus_minus
tff(fact_6694_norm__uminus__minus,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),Yb)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Yb)) ).

% norm_uminus_minus
tff(fact_6695_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).

% nonzero_norm_divide
tff(fact_6696_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).

% nonzero_norm_divide
tff(fact_6697_power__eq__imp__eq__norm,axiom,
    ! [W2: real,Nb: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_real_real(real_V7735802525324610683m_real,W2) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_6698_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,Nb: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),Nb) = aa_nat_complex(power_power_complex(Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_complex_real(real_V1022390504157884413omplex,W2) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_6699_norm__mult__ineq,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Yb))) ).

% norm_mult_ineq
tff(fact_6700_norm__mult__ineq,axiom,
    ! [Xb: complex,Yb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xb),Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Yb))) ).

% norm_mult_ineq
tff(fact_6701_norm__add__less,axiom,
    ! [Xb: real,R2: real,Yb: real,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Yb)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_add_less
tff(fact_6702_norm__add__less,axiom,
    ! [Xb: complex,R2: real,Yb: complex,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Yb)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Yb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_add_less
tff(fact_6703_norm__triangle__lt,axiom,
    ! [Xb: real,Yb: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Yb))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb))),E2) ) ).

% norm_triangle_lt
tff(fact_6704_norm__triangle__lt,axiom,
    ! [Xb: complex,Yb: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Yb))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Yb))),E2) ) ).

% norm_triangle_lt
tff(fact_6705_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_triangle_mono
tff(fact_6706_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_triangle_mono
tff(fact_6707_norm__triangle__ineq,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Yb))) ).

% norm_triangle_ineq
tff(fact_6708_norm__triangle__ineq,axiom,
    ! [Xb: complex,Yb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Yb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Yb))) ).

% norm_triangle_ineq
tff(fact_6709_norm__triangle__le,axiom,
    ! [Xb: real,Yb: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Yb))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb))),E2) ) ).

% norm_triangle_le
tff(fact_6710_norm__triangle__le,axiom,
    ! [Xb: complex,Yb: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Yb))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Yb))),E2) ) ).

% norm_triangle_le
tff(fact_6711_norm__add__leD,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),C2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),C2)) ) ).

% norm_add_leD
tff(fact_6712_norm__add__leD,axiom,
    ! [A: complex,B: complex,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),C2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),C2)) ) ).

% norm_add_leD
tff(fact_6713_norm__power__ineq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),Nb)) ).

% norm_power_ineq
tff(fact_6714_norm__power__ineq,axiom,
    ! [Xb: complex,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(Xb),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),Nb)) ).

% norm_power_ineq
tff(fact_6715_norm__diff__triangle__less,axiom,
    ! [Xb: real,Yb: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Yb))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Yb),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_6716_norm__diff__triangle__less,axiom,
    ! [Xb: complex,Yb: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Yb))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Yb),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_6717_norm__triangle__le__diff,axiom,
    ! [Xb: real,Yb: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Yb))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Yb))),E2) ) ).

% norm_triangle_le_diff
tff(fact_6718_norm__triangle__le__diff,axiom,
    ! [Xb: complex,Yb: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Yb))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Yb))),E2) ) ).

% norm_triangle_le_diff
tff(fact_6719_norm__diff__triangle__le,axiom,
    ! [Xb: real,Yb: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Yb))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Yb),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_6720_norm__diff__triangle__le,axiom,
    ! [Xb: complex,Yb: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Yb))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Yb),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_6721_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))) ).

% norm_triangle_ineq4
tff(fact_6722_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))) ).

% norm_triangle_ineq4
tff(fact_6723_norm__triangle__sub,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Yb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Yb)))) ).

% norm_triangle_sub
tff(fact_6724_norm__triangle__sub,axiom,
    ! [Xb: complex,Yb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Yb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Yb)))) ).

% norm_triangle_sub
tff(fact_6725_norm__diff__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ).

% norm_diff_ineq
tff(fact_6726_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))) ).

% norm_diff_ineq
tff(fact_6727_norm__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% norm_triangle_ineq2
tff(fact_6728_norm__triangle__ineq2,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B))) ).

% norm_triangle_ineq2
tff(fact_6729_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N4)
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => ( suminf_complex(F) = aa_set_nat_complex(groups2073611262835488442omplex(F),N4) ) ) ) ).

% suminf_finite
tff(fact_6730_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N4)
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => ( suminf_int(F) = groups3539618377306564664at_int(F,N4) ) ) ) ).

% suminf_finite
tff(fact_6731_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N4)
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => ( suminf_nat(F) = aa_set_nat_nat(groups3542108847815614940at_nat(F),N4) ) ) ) ).

% suminf_finite
tff(fact_6732_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N4)
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => ( suminf_real(F) = aa_set_nat_real(groups6591440286371151544t_real(F),N4) ) ) ) ).

% suminf_finite
tff(fact_6733_power__eq__1__iff,axiom,
    ! [W2: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),Nb) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W2) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_6734_power__eq__1__iff,axiom,
    ! [W2: complex,Nb: nat] :
      ( ( aa_nat_complex(power_power_complex(W2),Nb) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W2) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_6735_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C2: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_6736_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C2: complex,D: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(plus_plus_complex(C2),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C2))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_6737_norm__triangle__ineq3,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% norm_triangle_ineq3
tff(fact_6738_norm__triangle__ineq3,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B))) ).

% norm_triangle_ineq3
tff(fact_6739_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ja(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_6740_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jb(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_6741_arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(arctan,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_no,Xb)) ) ) ).

% arctan_series
tff(fact_6742_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_np),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_6743_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_nq) ).

% pi_series
tff(fact_6744_summable__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_no,Xb)) ) ).

% summable_arctan_series
tff(fact_6745_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aTP_Lamm_nr(R),aTP_Lamm_nr(S2))
    <=> ord_le2843351958646193337nt_int(R,S2) ) ).

% pred_subset_eq2
tff(fact_6746_pred__subset__eq2,axiom,
    ! [R: set_Pr1200682188532975830nteger,S2: set_Pr1200682188532975830nteger] :
      ( ord_le7899862361585801933eger_o(aTP_Lamm_ns(R),aTP_Lamm_ns(S2))
    <=> ord_le2645283422117218614nteger(R,S2) ) ).

% pred_subset_eq2
tff(fact_6747_pred__subset__eq2,axiom,
    ! [R: set_Pr3486312109254157132nteger,S2: set_Pr3486312109254157132nteger] :
      ( ord_le4825817444062779351eger_o(aTP_Lamm_nt(R),aTP_Lamm_nt(S2))
    <=> ord_le2932375828722672556nteger(R,S2) ) ).

% pred_subset_eq2
tff(fact_6748_pred__subset__eq2,axiom,
    ! [R: set_Pr6700895561065273913nt_int,S2: set_Pr6700895561065273913nt_int] :
      ( ord_le4806730579060245290_int_o(aTP_Lamm_nu(R),aTP_Lamm_nu(S2))
    <=> ord_le3990897393281686169nt_int(R,S2) ) ).

% pred_subset_eq2
tff(fact_6749_pred__subset__eq2,axiom,
    ! [R: set_Pr4166680992137330533nt_int,S2: set_Pr4166680992137330533nt_int] :
      ( ord_le2876595303158420734_int_o(aTP_Lamm_nv(R),aTP_Lamm_nv(S2))
    <=> ord_le3197784583289274309nt_int(R,S2) ) ).

% pred_subset_eq2
tff(fact_6750_arctan__eq__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(arctan,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% arctan_eq_zero_iff
tff(fact_6751_arctan__zero__zero,axiom,
    aa_real_real(arctan,zero_zero_real) = zero_zero_real ).

% arctan_zero_zero
tff(fact_6752_summable__zero,axiom,
    summable_complex(aTP_Lamm_nj) ).

% summable_zero
tff(fact_6753_summable__zero,axiom,
    summable_real(aTP_Lamm_ek) ).

% summable_zero
tff(fact_6754_summable__zero,axiom,
    summable_nat(aTP_Lamm_bx) ).

% summable_zero
tff(fact_6755_summable__zero,axiom,
    summable_int(aTP_Lamm_nk) ).

% summable_zero
tff(fact_6756_summable__single,axiom,
    ! [I: nat,F: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_em(I),F)) ).

% summable_single
tff(fact_6757_summable__single,axiom,
    ! [I: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_nw(I),F)) ).

% summable_single
tff(fact_6758_summable__single,axiom,
    ! [I: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nx(I),F)) ).

% summable_single
tff(fact_6759_summable__single,axiom,
    ! [I: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ny(I),F)) ).

% summable_single
tff(fact_6760_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_jd(F),K2))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_6761_arctan__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% arctan_less_zero_iff
tff(fact_6762_zero__less__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arctan,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% zero_less_arctan_iff
tff(fact_6763_zero__le__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_arctan_iff
tff(fact_6764_arctan__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% arctan_le_zero_iff
tff(fact_6765_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_6766_summable__cmult__iff,axiom,
    ! [C2: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_nz(C2),F))
    <=> ( ( C2 = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_6767_summable__cmult__iff,axiom,
    ! [C2: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_oa(C2),F))
    <=> ( ( C2 = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_6768_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ob(F),C2))
    <=> ( ( C2 = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_6769_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_oc(F),C2))
    <=> ( ( C2 = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_6770_summable__If__finite,axiom,
    ! [P2: fun_nat_o,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P2))
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_od(P2),F)) ) ).

% summable_If_finite
tff(fact_6771_summable__If__finite,axiom,
    ! [P2: fun_nat_o,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P2))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_oe(P2),F)) ) ).

% summable_If_finite
tff(fact_6772_summable__If__finite,axiom,
    ! [P2: fun_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P2))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_of(P2),F)) ) ).

% summable_If_finite
tff(fact_6773_summable__If__finite,axiom,
    ! [P2: fun_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P2))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_og(P2),F)) ) ).

% summable_If_finite
tff(fact_6774_summable__If__finite__set,axiom,
    ! [A4: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_oh(A4),F)) ) ).

% summable_If_finite_set
tff(fact_6775_summable__If__finite__set,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_oi(A4),F)) ) ).

% summable_If_finite_set
tff(fact_6776_summable__If__finite__set,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_oj(A4),F)) ) ).

% summable_If_finite_set
tff(fact_6777_summable__If__finite__set,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ok(A4),F)) ) ).

% summable_If_finite_set
tff(fact_6778_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
tff(fact_6779_arctan__monotone_H,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Yb)) ) ).

% arctan_monotone'
tff(fact_6780_arctan__le__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ).

% arctan_le_iff
tff(fact_6781_summable__const__iff,axiom,
    ! [C2: complex] :
      ( summable_complex(aTP_Lamm_ol(C2))
    <=> ( C2 = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_6782_summable__const__iff,axiom,
    ! [C2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_om,C2))
    <=> ( C2 = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_6783_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N4: nat,F: fun_nat_real] :
      ( summable_real(G)
     => ( ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))),aa_nat_real(G,N2)) )
       => summable_real(F) ) ) ).

% summable_comparison_test'
tff(fact_6784_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N4: nat,F: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2)) )
       => summable_complex(F) ) ) ).

% summable_comparison_test'
tff(fact_6785_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))),aa_nat_real(G,N2)) )
     => ( summable_real(G)
       => summable_real(F) ) ) ).

% summable_comparison_test
tff(fact_6786_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2)) )
     => ( summable_real(G)
       => summable_complex(F) ) ) ).

% summable_comparison_test
tff(fact_6787_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_fs(F),G)) ) ) ).

% summable_add
tff(fact_6788_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fr(F),G)) ) ) ).

% summable_add
tff(fact_6789_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_on(F),G)) ) ) ).

% summable_add
tff(fact_6790_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_jb(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_6791_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_jd(F),K2)) ) ).

% summable_ignore_initial_segment
tff(fact_6792_suminf__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2))
     => ( summable_real(F)
       => ( summable_real(G)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),suminf_real(G)) ) ) ) ).

% suminf_le
tff(fact_6793_suminf__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G,N2))
     => ( summable_nat(F)
       => ( summable_nat(G)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),suminf_nat(G)) ) ) ) ).

% suminf_le
tff(fact_6794_suminf__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(G,N2))
     => ( summable_int(F)
       => ( summable_int(G)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),suminf_int(G)) ) ) ) ).

% suminf_le
tff(fact_6795_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N4)
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => summable_complex(F) ) ) ).

% summable_finite
tff(fact_6796_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N4)
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => summable_real(F) ) ) ).

% summable_finite
tff(fact_6797_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N4)
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => summable_nat(F) ) ) ).

% summable_finite
tff(fact_6798_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N4)
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => summable_int(F) ) ) ).

% summable_finite
tff(fact_6799_summable__mult__D,axiom,
    ! [C2: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_nz(C2),F))
     => ( ( C2 != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_6800_summable__mult__D,axiom,
    ! [C2: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_oa(C2),F))
     => ( ( C2 != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_6801_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_6802_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_6803_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_6804_pi__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),pi) ).

% pi_gt_zero
tff(fact_6805_pi__not__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,pi),zero_zero_real) ).

% pi_not_less_zero
tff(fact_6806_pi__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),pi) ).

% pi_ge_zero
tff(fact_6807_suminf__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_fs(F),G)) ) ) ) ).

% suminf_add
tff(fact_6808_suminf__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,suminf_nat(F)),suminf_nat(G)) = suminf_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fr(F),G)) ) ) ) ).

% suminf_add
tff(fact_6809_suminf__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,suminf_int(F)),suminf_int(G)) = suminf_int(aa_fun5565043122120104496at_int(aTP_Lamm_on(F),G)) ) ) ) ).

% suminf_add
tff(fact_6810_suminf__nonneg,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),suminf_real(F)) ) ) ).

% suminf_nonneg
tff(fact_6811_suminf__nonneg,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),suminf_nat(F)) ) ) ).

% suminf_nonneg
tff(fact_6812_suminf__nonneg,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),suminf_int(F)) ) ) ).

% suminf_nonneg
tff(fact_6813_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2))
       => ( ( suminf_real(F) = zero_zero_real )
        <=> ! [N3: nat] : aa_nat_real(F,N3) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_6814_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2))
       => ( ( suminf_nat(F) = zero_zero_nat )
        <=> ! [N3: nat] : aa_nat_nat(F,N3) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_6815_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2))
       => ( ( suminf_int(F) = zero_zero_int )
        <=> ! [N3: nat] : aa_nat_int(F,N3) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_6816_suminf__pos,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,N2))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ).

% suminf_pos
tff(fact_6817_suminf__pos,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ).

% suminf_pos
tff(fact_6818_suminf__pos,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,N2))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ).

% suminf_pos
tff(fact_6819_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_im(F)) ).

% summable_0_powser
tff(fact_6820_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_io(F)) ).

% summable_0_powser
tff(fact_6821_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_im(F)) ).

% summable_zero_power'
tff(fact_6822_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_io(F)) ).

% summable_zero_power'
tff(fact_6823_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_oo(F)) ).

% summable_zero_power'
tff(fact_6824_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_op(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_oq(F),Z)) ) ).

% summable_powser_split_head
tff(fact_6825_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_or(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_os(F),Z)) ) ).

% summable_powser_split_head
tff(fact_6826_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_oq(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_op(F),Z)) ) ).

% powser_split_head(3)
tff(fact_6827_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_os(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_or(F),Z)) ) ).

% powser_split_head(3)
tff(fact_6828_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,Mb: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ot(F),Mb),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_oq(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_6829_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,Mb: nat,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ou(F),Mb),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_os(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_6830_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_nl(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_6831_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N2))),aa_nat_real(G,N2)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_il(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_6832_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_il(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_il(F))) ) ).

% summable_rabs
tff(fact_6833_suminf__pos2,axiom,
    ! [F: fun_nat_real,I: nat] :
      ( summable_real(F)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ) ).

% suminf_pos2
tff(fact_6834_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I: nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ) ).

% suminf_pos2
tff(fact_6835_suminf__pos2,axiom,
    ! [F: fun_nat_int,I: nat] :
      ( summable_int(F)
     => ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ) ).

% suminf_pos2
tff(fact_6836_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F))
        <=> ? [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I3)) ) ) ) ).

% suminf_pos_iff
tff(fact_6837_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F))
        <=> ? [I3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I3)) ) ) ) ).

% suminf_pos_iff
tff(fact_6838_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F))
        <=> ? [I3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I3)) ) ) ) ).

% suminf_pos_iff
tff(fact_6839_suminf__le__const,axiom,
    ! [F: fun_nat_int,Xb: int] :
      ( summable_int(F)
     => ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N2))),Xb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_6840_suminf__le__const,axiom,
    ! [F: fun_nat_nat,Xb: nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),Xb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_6841_suminf__le__const,axiom,
    ! [F: fun_nat_real,Xb: real] :
      ( summable_real(F)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_6842_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_int,Xb: int] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2))
     => ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N2))),Xb)
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_6843_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_nat,Xb: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2))
     => ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),Xb)
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_6844_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_real,Xb: real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2))
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),Xb)
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_6845_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_jb(F)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_6846_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_6847_pi__ge__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi) ).

% pi_ge_two
tff(fact_6848_summable__norm,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_nn(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(F))),suminf_real(aTP_Lamm_nn(F))) ) ).

% summable_norm
tff(fact_6849_summable__norm,axiom,
    ! [F: fun_nat_complex] :
      ( summable_real(aTP_Lamm_nl(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(F))),suminf_real(aTP_Lamm_nl(F))) ) ).

% summable_norm
tff(fact_6850_sum__le__suminf,axiom,
    ! [F: fun_nat_int,I6: set_nat] :
      ( summable_int(F)
     => ( aa_set_nat_o(finite_finite_nat,I6)
       => ( ! [N2: nat] :
              ( aa_set_nat_o(member_nat(N2),uminus5710092332889474511et_nat(I6))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,I6)),suminf_int(F)) ) ) ) ).

% sum_le_suminf
tff(fact_6851_sum__le__suminf,axiom,
    ! [F: fun_nat_nat,I6: set_nat] :
      ( summable_nat(F)
     => ( aa_set_nat_o(finite_finite_nat,I6)
       => ( ! [N2: nat] :
              ( aa_set_nat_o(member_nat(N2),uminus5710092332889474511et_nat(I6))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),I6)),suminf_nat(F)) ) ) ) ).

% sum_le_suminf
tff(fact_6852_sum__le__suminf,axiom,
    ! [F: fun_nat_real,I6: set_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(finite_finite_nat,I6)
       => ( ! [N2: nat] :
              ( aa_set_nat_o(member_nat(N2),uminus5710092332889474511et_nat(I6))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),I6)),suminf_real(F)) ) ) ) ).

% sum_le_suminf
tff(fact_6853_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Xa3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa3),R)
        <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_6854_pred__equals__eq2,axiom,
    ! [R: set_Pr1200682188532975830nteger,S2: set_Pr1200682188532975830nteger] :
      ( ! [X4: fun_Co3460408610320954864e_term,Xa3: produc8923325533196201883nteger] :
          ( member6901806533512420791nteger(produc7264323995540650962nteger(X4,Xa3),R)
        <=> member6901806533512420791nteger(produc7264323995540650962nteger(X4,Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_6855_pred__equals__eq2,axiom,
    ! [R: set_Pr3486312109254157132nteger,S2: set_Pr3486312109254157132nteger] :
      ( ! [X4: fun_Pr5098669352780829690e_term,Xa3: produc8923325533196201883nteger] :
          ( member8978266922497061677nteger(produc565448451787809096nteger(X4,Xa3),R)
        <=> member8978266922497061677nteger(produc565448451787809096nteger(X4,Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_6856_pred__equals__eq2,axiom,
    ! [R: set_Pr6700895561065273913nt_int,S2: set_Pr6700895561065273913nt_int] :
      ( ! [X4: fun_Pr7763680713840462443e_term,Xa3: product_prod_int_int] :
          ( member753276645541486746nt_int(produc7320969046608631221nt_int(X4,Xa3),R)
        <=> member753276645541486746nt_int(produc7320969046608631221nt_int(X4,Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_6857_pred__equals__eq2,axiom,
    ! [R: set_Pr4166680992137330533nt_int,S2: set_Pr4166680992137330533nt_int] :
      ( ! [X4: fun_in5730379987311621695e_term,Xa3: product_prod_int_int] :
          ( member5890090881338784710nt_int(produc4729913460834754785nt_int(X4,Xa3),R)
        <=> member5890090881338784710nt_int(produc4729913460834754785nt_int(X4,Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_6858_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => ( suminf_real(F) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_jd(F),K2))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K2))) ) ) ).

% suminf_split_initial_segment
tff(fact_6859_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_jd(F),K2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K2))) ) ) ).

% suminf_minus_initial_segment
tff(fact_6860_pi__half__neq__zero,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_6861_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_le_two
tff(fact_6862_sum__less__suminf,axiom,
    ! [F: fun_nat_int,Nb: nat] :
      ( summable_int(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,M)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ).

% sum_less_suminf
tff(fact_6863_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,Nb: nat] :
      ( summable_nat(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M)) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_nat(F)) ) ) ).

% sum_less_suminf
tff(fact_6864_sum__less__suminf,axiom,
    ! [F: fun_nat_real,Nb: nat] :
      ( summable_real(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,M)) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_real(F)) ) ) ).

% sum_less_suminf
tff(fact_6865_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_oq(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_oq(F),Z)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(F,zero_zero_nat)),aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_op(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_6866_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_os(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_os(F),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,zero_zero_nat)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_or(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_6867_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_oq(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_op(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_oq(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_6868_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_os(F),Z))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_or(F),Z))),Z) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_os(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_6869_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E2: real] :
      ( summable_complex(F)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
       => ~ ! [N7: nat] :
              ~ ! [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M2)
                 => ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),set_or1269000886237332187st_nat(M2,N5)))),E2) ) ) ) ).

% summable_partial_sum_bound
tff(fact_6870_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E2: real] :
      ( summable_real(F)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
       => ~ ! [N7: nat] :
              ~ ! [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M2)
                 => ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(M2,N5)))),E2) ) ) ) ).

% summable_partial_sum_bound
tff(fact_6871_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
     => ( summable_real(F)
       => ? [N7: nat] :
          ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N5)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_jd(F),N5)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_6872_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
     => ( summable_complex(F)
       => ? [N7: nat] :
          ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N5)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ov(F),N5)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_6873_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I2)),one_one_real)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_os(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_6874_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: fun_nat_complex,M5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,R2),R0)
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A,N2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N2))),M5)
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_ow(R2),A)) ) ) ) ).

% Abel_lemma
tff(fact_6875_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_6876_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_ge_zero
tff(fact_6877_summable__ratio__test,axiom,
    ! [C2: real,N4: nat,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),one_one_real)
     => ( ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2)))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_6878_summable__ratio__test,axiom,
    ! [C2: real,N4: nat,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),one_one_real)
     => ( ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2)))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_6879_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,Nb: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I))
           => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_6880_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,Nb: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_nat(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_6881_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,Nb: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I))
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_real(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_6882_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y),R2)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y),S3) )
     => ord_le2843351958646193337nt_int(R2,S3) ) ).

% subrelI
tff(fact_6883_subrelI,axiom,
    ! [R2: set_Pr1200682188532975830nteger,S3: set_Pr1200682188532975830nteger] :
      ( ! [X3: fun_Co3460408610320954864e_term,Y: produc8923325533196201883nteger] :
          ( member6901806533512420791nteger(produc7264323995540650962nteger(X3,Y),R2)
         => member6901806533512420791nteger(produc7264323995540650962nteger(X3,Y),S3) )
     => ord_le2645283422117218614nteger(R2,S3) ) ).

% subrelI
tff(fact_6884_subrelI,axiom,
    ! [R2: set_Pr3486312109254157132nteger,S3: set_Pr3486312109254157132nteger] :
      ( ! [X3: fun_Pr5098669352780829690e_term,Y: produc8923325533196201883nteger] :
          ( member8978266922497061677nteger(produc565448451787809096nteger(X3,Y),R2)
         => member8978266922497061677nteger(produc565448451787809096nteger(X3,Y),S3) )
     => ord_le2932375828722672556nteger(R2,S3) ) ).

% subrelI
tff(fact_6885_subrelI,axiom,
    ! [R2: set_Pr6700895561065273913nt_int,S3: set_Pr6700895561065273913nt_int] :
      ( ! [X3: fun_Pr7763680713840462443e_term,Y: product_prod_int_int] :
          ( member753276645541486746nt_int(produc7320969046608631221nt_int(X3,Y),R2)
         => member753276645541486746nt_int(produc7320969046608631221nt_int(X3,Y),S3) )
     => ord_le3990897393281686169nt_int(R2,S3) ) ).

% subrelI
tff(fact_6886_subrelI,axiom,
    ! [R2: set_Pr4166680992137330533nt_int,S3: set_Pr4166680992137330533nt_int] :
      ( ! [X3: fun_in5730379987311621695e_term,Y: product_prod_int_int] :
          ( member5890090881338784710nt_int(produc4729913460834754785nt_int(X3,Y),R2)
         => member5890090881338784710nt_int(produc4729913460834754785nt_int(X3,Y),S3) )
     => ord_le3197784583289274309nt_int(R2,S3) ) ).

% subrelI
tff(fact_6887_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_6888_arctan__add,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Yb)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Yb)))) ) ) ) ).

% arctan_add
tff(fact_6889_bot__empty__eq2,axiom,
    ! [X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(bot_bo428638681919004382_int_o,X),Xa)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_6890_bot__empty__eq2,axiom,
    ! [X: fun_Co3460408610320954864e_term,Xa: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(bot_bo2238103298444869889eger_o,X),Xa)
    <=> member6901806533512420791nteger(produc7264323995540650962nteger(X,Xa),bot_bo5621876822565464938nteger) ) ).

% bot_empty_eq2
tff(fact_6891_bot__empty__eq2,axiom,
    ! [X: fun_Pr5098669352780829690e_term,Xa: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(bot_bo4567088895781559819eger_o,X),Xa)
    <=> member8978266922497061677nteger(produc565448451787809096nteger(X,Xa),bot_bo9050247954524366304nteger) ) ).

% bot_empty_eq2
tff(fact_6892_bot__empty__eq2,axiom,
    ! [X: fun_Pr7763680713840462443e_term,Xa: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(bot_bo1117279750869432670_int_o,X),Xa)
    <=> member753276645541486746nt_int(produc7320969046608631221nt_int(X,Xa),bot_bo2501116328598523085nt_int) ) ).

% bot_empty_eq2
tff(fact_6893_bot__empty__eq2,axiom,
    ! [X: fun_in5730379987311621695e_term,Xa: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(bot_bo6524246659364023090_int_o,X),Xa)
    <=> member5890090881338784710nt_int(produc4729913460834754785nt_int(X,Xa),bot_bo5189594446080201209nt_int) ) ).

% bot_empty_eq2
tff(fact_6894_bot__empty__eq2,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X),Xa)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_6895_pred__subset__eq,axiom,
    ! [R: set_complex,S2: set_complex] :
      ( ord_le7644322562001619171plex_o(aTP_Lamm_a(R),aTP_Lamm_a(S2))
    <=> aa_set_complex_o(ord_le211207098394363844omplex(R),S2) ) ).

% pred_subset_eq
tff(fact_6896_pred__subset__eq,axiom,
    ! [R: set_real,S2: set_real] :
      ( ord_le3605727316730466277real_o(aTP_Lamm_aa(R),aTP_Lamm_aa(S2))
    <=> ord_less_eq_set_real(R,S2) ) ).

% pred_subset_eq
tff(fact_6897_pred__subset__eq,axiom,
    ! [R: set_set_nat,S2: set_set_nat] :
      ( ord_le6468474429079055883_nat_o(aTP_Lamm_ac(R),aTP_Lamm_ac(S2))
    <=> ord_le6893508408891458716et_nat(R,S2) ) ).

% pred_subset_eq
tff(fact_6898_pred__subset__eq,axiom,
    ! [R: set_nat,S2: set_nat] :
      ( ord_le7240017405068453697_nat_o(aTP_Lamm_ad(R),aTP_Lamm_ad(S2))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,R),S2) ) ).

% pred_subset_eq
tff(fact_6899_pred__subset__eq,axiom,
    ! [R: set_int,S2: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_ae(R),aTP_Lamm_ae(S2))
    <=> aa_set_int_o(ord_less_eq_set_int(R),S2) ) ).

% pred_subset_eq
tff(fact_6900_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => ( ! [D2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),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,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K2))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_6901_sin__cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% sin_cos_npi
tff(fact_6902_cos__pi__eq__zero,axiom,
    ! [Mb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_6903_accp__subset,axiom,
    ! [R1: fun_Pr4799464687642144302_num_o,R22: fun_Pr4799464687642144302_num_o] :
      ( ord_le6421466325430216142_num_o(R1,R22)
     => ord_le3107050023622517222_num_o(accp_P3113834385874906142um_num(R22),accp_P3113834385874906142um_num(R1)) ) ).

% accp_subset
tff(fact_6904_accp__subset,axiom,
    ! [R1: fun_Pr2028404973800837934_nat_o,R22: fun_Pr2028404973800837934_nat_o] :
      ( ord_le3650406611588909774_nat_o(R1,R22)
     => aa_fun8408151826566693824at_o_o(ord_le1572679713340831354_nat_o(accp_P4275260045618599050at_nat(R22)),accp_P4275260045618599050at_nat(R1)) ) ).

% accp_subset
tff(fact_6905_accp__subset,axiom,
    ! [R1: fun_Pr9018117161326028078_int_o,R22: fun_Pr9018117161326028078_int_o] :
      ( ord_le1416746762259324110_int_o(R1,R22)
     => ord_le14110778710936002_int_o(accp_P1096762738010456898nt_int(R22),accp_P1096762738010456898nt_int(R1)) ) ).

% accp_subset
tff(fact_6906_accp__subset,axiom,
    ! [R1: fun_li1025265955262268170_nat_o,R22: fun_li1025265955262268170_nat_o] :
      ( ord_le4470944457981725354_nat_o(R1,R22)
     => ord_le4972187363741583025_nat_o(accp_list_nat(R22),accp_list_nat(R1)) ) ).

% accp_subset
tff(fact_6907_accp__subset,axiom,
    ! [R1: fun_nat_fun_nat_o,R22: fun_nat_fun_nat_o] :
      ( ord_le768339454706049738_nat_o(R1,R22)
     => ord_le7240017405068453697_nat_o(accp_nat(R22),accp_nat(R1)) ) ).

% accp_subset
tff(fact_6908_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_ox,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_6909_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => aa_complex_o(sums_complex(aTP_Lamm_oy(Z)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_6910_monoseq__def,axiom,
    ! [X5: fun_nat_real] :
      ( topolo6980174941875973593q_real(X5)
    <=> ( ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,M4)),aa_nat_real(X5,N3)) )
        | ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,N3)),aa_nat_real(X5,M4)) ) ) ) ).

% monoseq_def
tff(fact_6911_monoseq__def,axiom,
    ! [X5: fun_nat_set_int] :
      ( topolo3100542954746470799et_int(X5)
    <=> ( ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,M4)),aa_nat_set_int(X5,N3)) )
        | ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,N3)),aa_nat_set_int(X5,M4)) ) ) ) ).

% monoseq_def
tff(fact_6912_monoseq__def,axiom,
    ! [X5: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X5)
    <=> ( ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,M4)),aa_nat_rat(X5,N3)) )
        | ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N3)),aa_nat_rat(X5,M4)) ) ) ) ).

% monoseq_def
tff(fact_6913_monoseq__def,axiom,
    ! [X5: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X5)
    <=> ( ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,M4)),aa_nat_num(X5,N3)) )
        | ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,N3)),aa_nat_num(X5,M4)) ) ) ) ).

% monoseq_def
tff(fact_6914_monoseq__def,axiom,
    ! [X5: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X5)
    <=> ( ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,M4)),aa_nat_nat(X5,N3)) )
        | ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,N3)),aa_nat_nat(X5,M4)) ) ) ) ).

% monoseq_def
tff(fact_6915_monoseq__def,axiom,
    ! [X5: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X5)
    <=> ( ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,M4)),aa_nat_int(X5,N3)) )
        | ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,N3)),aa_nat_int(X5,M4)) ) ) ) ).

% monoseq_def
tff(fact_6916_monoI2,axiom,
    ! [X5: fun_nat_real] :
      ( ! [M: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,N2)),aa_nat_real(X5,M)) )
     => topolo6980174941875973593q_real(X5) ) ).

% monoI2
tff(fact_6917_monoI2,axiom,
    ! [X5: fun_nat_set_int] :
      ( ! [M: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
         => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,N2)),aa_nat_set_int(X5,M)) )
     => topolo3100542954746470799et_int(X5) ) ).

% monoI2
tff(fact_6918_monoI2,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [M: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N2)),aa_nat_rat(X5,M)) )
     => topolo4267028734544971653eq_rat(X5) ) ).

% monoI2
tff(fact_6919_monoI2,axiom,
    ! [X5: fun_nat_num] :
      ( ! [M: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,N2)),aa_nat_num(X5,M)) )
     => topolo1459490580787246023eq_num(X5) ) ).

% monoI2
tff(fact_6920_monoI2,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [M: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,N2)),aa_nat_nat(X5,M)) )
     => topolo4902158794631467389eq_nat(X5) ) ).

% monoI2
tff(fact_6921_monoI2,axiom,
    ! [X5: fun_nat_int] :
      ( ! [M: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,N2)),aa_nat_int(X5,M)) )
     => topolo4899668324122417113eq_int(X5) ) ).

% monoI2
tff(fact_6922_sin__zero,axiom,
    aa_complex_complex(sin_complex,zero_zero_complex) = zero_zero_complex ).

% sin_zero
tff(fact_6923_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_6924_sums__zero,axiom,
    aa_complex_o(sums_complex(aTP_Lamm_nj),zero_zero_complex) ).

% sums_zero
tff(fact_6925_sums__zero,axiom,
    aa_real_o(sums_real(aTP_Lamm_ek),zero_zero_real) ).

% sums_zero
tff(fact_6926_sums__zero,axiom,
    aa_nat_o(sums_nat(aTP_Lamm_bx),zero_zero_nat) ).

% sums_zero
tff(fact_6927_sums__zero,axiom,
    aa_int_o(sums_int(aTP_Lamm_nk),zero_zero_int) ).

% sums_zero
tff(fact_6928_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_6929_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_6930_sin__pi,axiom,
    aa_real_real(sin_real,pi) = zero_zero_real ).

% sin_pi
tff(fact_6931_cos__periodic__pi2,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_periodic_pi2
tff(fact_6932_cos__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_periodic_pi
tff(fact_6933_sin__periodic__pi2,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_periodic_pi2
tff(fact_6934_sin__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_periodic_pi
tff(fact_6935_sin__cos__squared__add3,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Xb))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,Xb)),aa_complex_complex(sin_complex,Xb))) = one_one_complex ).

% sin_cos_squared_add3
tff(fact_6936_sin__cos__squared__add3,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Xb))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_6937_sin__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_6938_sin__npi2,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = zero_zero_real ).

% sin_npi2
tff(fact_6939_sin__npi__int,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_npi_int
tff(fact_6940_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,Xb: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_im(A)),Xb)
    <=> ( aa_nat_complex(A,zero_zero_nat) = Xb ) ) ).

% powser_sums_zero_iff
tff(fact_6941_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(aTP_Lamm_io(A)),Xb)
    <=> ( aa_nat_real(A,zero_zero_nat) = Xb ) ) ).

% powser_sums_zero_iff
tff(fact_6942_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_6943_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_6944_cos__periodic,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,Xb) ).

% cos_periodic
tff(fact_6945_sin__periodic,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,Xb) ).

% sin_periodic
tff(fact_6946_sin__cos__squared__add,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_6947_sin__cos__squared__add,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_6948_sin__cos__squared__add2,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_6949_sin__cos__squared__add2,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_6950_sin__2npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_6951_sin__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_int_2pin
tff(fact_6952_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_6953_cos__one__sin__zero,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) = one_one_complex )
     => ( aa_complex_complex(sin_complex,Xb) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_6954_cos__one__sin__zero,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
     => ( aa_real_real(sin_real,Xb) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_6955_sums__0,axiom,
    ! [F: fun_nat_complex] :
      ( ! [N2: nat] : aa_nat_complex(F,N2) = zero_zero_complex
     => aa_complex_o(sums_complex(F),zero_zero_complex) ) ).

% sums_0
tff(fact_6956_sums__0,axiom,
    ! [F: fun_nat_real] :
      ( ! [N2: nat] : aa_nat_real(F,N2) = zero_zero_real
     => aa_real_o(sums_real(F),zero_zero_real) ) ).

% sums_0
tff(fact_6957_sums__0,axiom,
    ! [F: fun_nat_nat] :
      ( ! [N2: nat] : aa_nat_nat(F,N2) = zero_zero_nat
     => aa_nat_o(sums_nat(F),zero_zero_nat) ) ).

% sums_0
tff(fact_6958_sums__0,axiom,
    ! [F: fun_nat_int] :
      ( ! [N2: nat] : aa_nat_int(F,N2) = zero_zero_int
     => aa_int_o(sums_int(F),zero_zero_int) ) ).

% sums_0
tff(fact_6959_sin__add,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(cos_real,Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(sin_real,Yb))) ).

% sin_add
tff(fact_6960_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S3: real,Ta: real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2))
     => ( aa_real_o(sums_real(F),S3)
       => ( aa_real_o(sums_real(G),Ta)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,S3),Ta) ) ) ) ).

% sums_le
tff(fact_6961_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S3: nat,Ta: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G,N2))
     => ( aa_nat_o(sums_nat(F),S3)
       => ( aa_nat_o(sums_nat(G),Ta)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S3),Ta) ) ) ) ).

% sums_le
tff(fact_6962_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S3: int,Ta: int] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(G,N2))
     => ( aa_int_o(sums_int(F),S3)
       => ( aa_int_o(sums_int(G),Ta)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,S3),Ta) ) ) ) ).

% sums_le
tff(fact_6963_sums__single,axiom,
    ! [I: nat,F: fun_nat_complex] : aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_em(I),F)),aa_nat_complex(F,I)) ).

% sums_single
tff(fact_6964_sums__single,axiom,
    ! [I: nat,F: fun_nat_real] : aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_nw(I),F)),aa_nat_real(F,I)) ).

% sums_single
tff(fact_6965_sums__single,axiom,
    ! [I: nat,F: fun_nat_nat] : aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nx(I),F)),aa_nat_nat(F,I)) ).

% sums_single
tff(fact_6966_sums__single,axiom,
    ! [I: nat,F: fun_nat_int] : aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ny(I),F)),aa_nat_int(F,I)) ).

% sums_single
tff(fact_6967_sums__add,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( aa_real_o(sums_real(F),A)
     => ( aa_real_o(sums_real(G),B)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_fs(F),G)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% sums_add
tff(fact_6968_sums__add,axiom,
    ! [F: fun_nat_nat,A: nat,G: fun_nat_nat,B: nat] :
      ( aa_nat_o(sums_nat(F),A)
     => ( aa_nat_o(sums_nat(G),B)
       => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fr(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% sums_add
tff(fact_6969_sums__add,axiom,
    ! [F: fun_nat_int,A: int,G: fun_nat_int,B: int] :
      ( aa_int_o(sums_int(F),A)
     => ( aa_int_o(sums_int(G),B)
       => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_on(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% sums_add
tff(fact_6970_cos__add,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Yb))) ).

% cos_add
tff(fact_6971_cos__diff,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Yb))) ).

% cos_diff
tff(fact_6972_sin__zero__norm__cos__one,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,Xb)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_6973_sin__zero__norm__cos__one,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(sin_complex,Xb) = zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,Xb)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_6974_sin__zero__abs__cos__one,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,Xb)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_6975_sincos__principal__value,axiom,
    ! [Xb: real] :
    ? [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
      & ( aa_real_real(sin_real,Y) = aa_real_real(sin_real,Xb) )
      & ( aa_real_real(cos_real,Y) = aa_real_real(cos_real,Xb) ) ) ).

% sincos_principal_value
tff(fact_6976_sin__x__le__x,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),Xb) ) ).

% sin_x_le_x
tff(fact_6977_sin__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),one_one_real) ).

% sin_le_one
tff(fact_6978_cos__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),one_one_real) ).

% cos_le_one
tff(fact_6979_abs__sin__x__le__abs__x,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xb))),aa_real_real(abs_abs_real,Xb)) ).

% abs_sin_x_le_abs_x
tff(fact_6980_sums__mult2__iff,axiom,
    ! [C2: complex,F: fun_nat_complex,D: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_oz(C2),F)),aa_complex_complex(times_times_complex(D),C2))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult2_iff
tff(fact_6981_sums__mult2__iff,axiom,
    ! [C2: real,F: fun_nat_real,D: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_pa(C2),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C2))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult2_iff
tff(fact_6982_sums__mult__iff,axiom,
    ! [C2: complex,F: fun_nat_complex,D: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_nz(C2),F)),aa_complex_complex(times_times_complex(C2),D))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult_iff
tff(fact_6983_sums__mult__iff,axiom,
    ! [C2: real,F: fun_nat_real,D: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_oa(C2),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),D))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult_iff
tff(fact_6984_cos__arctan__not__zero,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(arctan,Xb)) != zero_zero_real ).

% cos_arctan_not_zero
tff(fact_6985_sin__cos__le1,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Yb))))),one_one_real) ).

% sin_cos_le1
tff(fact_6986_sums__mult__D,axiom,
    ! [C2: complex,F: fun_nat_complex,A: complex] :
      ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_nz(C2),F)),A)
     => ( ( C2 != zero_zero_complex )
       => aa_complex_o(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C2)) ) ) ).

% sums_mult_D
tff(fact_6987_sums__mult__D,axiom,
    ! [C2: real,F: fun_nat_real,A: real] :
      ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_oa(C2),F)),A)
     => ( ( C2 != zero_zero_real )
       => aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)) ) ) ).

% sums_mult_D
tff(fact_6988_sums__Suc__imp,axiom,
    ! [F: fun_nat_complex,S3: complex] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( aa_complex_o(sums_complex(aTP_Lamm_pb(F)),S3)
       => aa_complex_o(sums_complex(F),S3) ) ) ).

% sums_Suc_imp
tff(fact_6989_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S3: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_real_o(sums_real(aTP_Lamm_jb(F)),S3)
       => aa_real_o(sums_real(F),S3) ) ) ).

% sums_Suc_imp
tff(fact_6990_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S3: real] :
      ( aa_real_o(sums_real(aTP_Lamm_jb(F)),S3)
    <=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S3),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc_iff
tff(fact_6991_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( aa_real_o(sums_real(aTP_Lamm_jb(F)),L)
     => aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,L),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_6992_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( aa_nat_o(sums_nat(aTP_Lamm_ja(F)),L)
     => aa_nat_o(sums_nat(F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),aa_nat_nat(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_6993_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( aa_int_o(sums_int(aTP_Lamm_jm(F)),L)
     => aa_int_o(sums_int(F),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),aa_nat_int(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_6994_sin__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero
tff(fact_6995_sin__x__ge__neg__x,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(sin_real,Xb)) ) ).

% sin_x_ge_neg_x
tff(fact_6996_sin__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_ge_zero
tff(fact_6997_sums__zero__iff__shift,axiom,
    ! [Nb: nat,F: fun_nat_complex,S3: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => ( aa_nat_complex(F,I2) = zero_zero_complex ) )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_pc(Nb),F)),S3)
      <=> aa_complex_o(sums_complex(F),S3) ) ) ).

% sums_zero_iff_shift
tff(fact_6998_sums__zero__iff__shift,axiom,
    ! [Nb: nat,F: fun_nat_real,S3: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => ( aa_nat_real(F,I2) = zero_zero_real ) )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_pd(Nb),F)),S3)
      <=> aa_real_o(sums_real(F),S3) ) ) ).

% sums_zero_iff_shift
tff(fact_6999_sin__ge__minus__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,Xb)) ).

% sin_ge_minus_one
tff(fact_7000_cos__monotone__0__pi__le,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Yb)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_7001_cos__mono__le__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Yb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_7002_cos__inj__pi,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),pi)
           => ( ( aa_real_real(cos_real,Xb) = aa_real_real(cos_real,Yb) )
             => ( Xb = Yb ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_7003_cos__ge__minus__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,Xb)) ).

% cos_ge_minus_one
tff(fact_7004_abs__sin__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xb))),one_one_real) ).

% abs_sin_le_one
tff(fact_7005_abs__cos__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,Xb))),one_one_real) ).

% abs_cos_le_one
tff(fact_7006_sin__times__sin,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W2)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W2),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W2),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_7007_sin__times__sin,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_7008_sin__times__cos,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W2)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W2),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W2),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_7009_sin__times__cos,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W2)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_7010_cos__times__sin,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W2)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W2),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W2),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_7011_cos__times__sin,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W2)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_7012_sin__plus__sin,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,W2)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W2),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W2),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_7013_sin__plus__sin,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_7014_sin__diff__sin,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,W2)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W2),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W2),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_7015_sin__diff__sin,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_7016_cos__diff__cos,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,W2)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W2),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Z),W2)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_7017_cos__diff__cos,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_7018_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N4)
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => aa_complex_o(sums_complex(F),aa_set_nat_complex(groups2073611262835488442omplex(F),N4)) ) ) ).

% sums_finite
tff(fact_7019_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N4)
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => aa_int_o(sums_int(F),groups3539618377306564664at_int(F,N4)) ) ) ).

% sums_finite
tff(fact_7020_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N4)
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => aa_nat_o(sums_nat(F),aa_set_nat_nat(groups3542108847815614940at_nat(F),N4)) ) ) ).

% sums_finite
tff(fact_7021_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N2: nat] :
            ( ~ aa_set_nat_o(member_nat(N2),N4)
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => aa_real_o(sums_real(F),aa_set_nat_real(groups6591440286371151544t_real(F),N4)) ) ) ).

% sums_finite
tff(fact_7022_sums__If__finite,axiom,
    ! [P2: fun_nat_o,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P2))
     => aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_od(P2),F)),aa_set_nat_complex(groups2073611262835488442omplex(F),collect_nat(P2))) ) ).

% sums_If_finite
tff(fact_7023_sums__If__finite,axiom,
    ! [P2: fun_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P2))
     => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_og(P2),F)),groups3539618377306564664at_int(F,collect_nat(P2))) ) ).

% sums_If_finite
tff(fact_7024_sums__If__finite,axiom,
    ! [P2: fun_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P2))
     => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_of(P2),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),collect_nat(P2))) ) ).

% sums_If_finite
tff(fact_7025_sums__If__finite,axiom,
    ! [P2: fun_nat_o,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P2))
     => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_oe(P2),F)),aa_set_nat_real(groups6591440286371151544t_real(F),collect_nat(P2))) ) ).

% sums_If_finite
tff(fact_7026_sums__If__finite__set,axiom,
    ! [A4: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_oh(A4),F)),aa_set_nat_complex(groups2073611262835488442omplex(F),A4)) ) ).

% sums_If_finite_set
tff(fact_7027_sums__If__finite__set,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ok(A4),F)),groups3539618377306564664at_int(F,A4)) ) ).

% sums_If_finite_set
tff(fact_7028_sums__If__finite__set,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_oj(A4),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) ) ).

% sums_If_finite_set
tff(fact_7029_sums__If__finite__set,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_oi(A4),F)),aa_set_nat_real(groups6591440286371151544t_real(F),A4)) ) ).

% sums_If_finite_set
tff(fact_7030_powser__sums__if,axiom,
    ! [Mb: nat,Z: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_pe(Mb),Z)),aa_nat_complex(power_power_complex(Z),Mb)) ).

% powser_sums_if
tff(fact_7031_powser__sums__if,axiom,
    ! [Mb: nat,Z: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_pf(Mb),Z)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Mb)) ).

% powser_sums_if
tff(fact_7032_powser__sums__if,axiom,
    ! [Mb: nat,Z: int] : aa_int_o(sums_int(aa_int_fun_nat_int(aTP_Lamm_pg(Mb),Z)),aa_nat_int(power_power_int(Z),Mb)) ).

% powser_sums_if
tff(fact_7033_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : aa_complex_o(sums_complex(aTP_Lamm_im(A)),aa_nat_complex(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_7034_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : aa_real_o(sums_real(aTP_Lamm_io(A)),aa_nat_real(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_7035_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_7036_cos__mono__less__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Yb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_7037_cos__monotone__0__pi,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Yb)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_7038_sin__eq__0__pi,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),pi)
       => ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
         => ( Xb = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_7039_sums__iff__shift,axiom,
    ! [F: fun_nat_real,Nb: nat,S3: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_jd(F),Nb)),S3)
    <=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S3),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).

% sums_iff_shift
tff(fact_7040_sums__iff__shift_H,axiom,
    ! [F: fun_nat_real,Nb: nat,S3: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_jd(F),Nb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S3),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))
    <=> aa_real_o(sums_real(F),S3) ) ).

% sums_iff_shift'
tff(fact_7041_sums__split__initial__segment,axiom,
    ! [F: fun_nat_real,S3: real,Nb: nat] :
      ( aa_real_o(sums_real(F),S3)
     => aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_jd(F),Nb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S3),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).

% sums_split_initial_segment
tff(fact_7042_sin__zero__pi__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),pi)
     => ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_7043_cos__monotone__minus__pi__0_H,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Yb)),aa_real_real(cos_real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_7044_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S2: real,A4: set_nat,S5: real,F: fun_nat_real] :
      ( aa_real_o(sums_real(G),S2)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( ( S5 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ph(G),F)),A4)) )
         => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_pi(G),A4),F)),S5) ) ) ) ).

% sums_If_finite_set'
tff(fact_7045_sin__zero__iff__int2,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I3: int] : Xb = 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_7046_sincos__total__pi,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),pi)
            & ( Xb = aa_real_real(cos_real,T4) )
            & ( Yb = aa_real_real(sin_real,T4) ) ) ) ) ).

% sincos_total_pi
tff(fact_7047_sin__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_7048_cos__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_7049_sin__gt__zero__02,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero_02
tff(fact_7050_cos__two__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_less_zero
tff(fact_7051_cos__two__le__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_le_zero
tff(fact_7052_cos__is__zero,axiom,
    ? [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X3) = zero_zero_real )
      & ! [Y3: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y3) = zero_zero_real ) )
         => ( Y3 = X3 ) ) ) ).

% cos_is_zero
tff(fact_7053_cos__monotone__minus__pi__0,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Yb)),aa_real_real(cos_real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_7054_cos__total,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ? [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),pi)
            & ( aa_real_real(cos_real,X3) = Yb )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),pi)
                  & ( aa_real_real(cos_real,Y3) = Yb ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% cos_total
tff(fact_7055_sincos__total__pi__half,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
              & ( Xb = aa_real_real(cos_real,T4) )
              & ( Yb = aa_real_real(sin_real,T4) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_7056_sincos__total__2pi__le,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
          & ( Xb = aa_real_real(cos_real,T4) )
          & ( Yb = aa_real_real(sin_real,T4) ) ) ) ).

% sincos_total_2pi_le
tff(fact_7057_sincos__total__2pi,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( ( Xb = aa_real_real(cos_real,T4) )
               => ( Yb != aa_real_real(sin_real,T4) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_7058_sin__pi__divide__n__ge__0,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_7059_cos__plus__cos,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,W2)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W2),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W2),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_7060_cos__plus__cos,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_7061_cos__times__cos,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W2)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W2),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W2),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_7062_cos__times__cos,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_7063_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_pj),one_one_real) ).

% power_half_series
tff(fact_7064_sin__gt__zero2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero2
tff(fact_7065_sin__lt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_7066_cos__double__less__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_7067_cos__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_gt_zero
tff(fact_7068_sin__inj__pi,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,Xb) = aa_real_real(sin_real,Yb) )
             => ( Xb = Yb ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_7069_sin__mono__le__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Yb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_7070_sin__monotone__2pi__le,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Yb)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_7071_sums__if_H,axiom,
    ! [G: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(G),Xb)
     => aa_real_o(sums_real(aTP_Lamm_pk(G)),Xb) ) ).

% sums_if'
tff(fact_7072_sums__if,axiom,
    ! [G: fun_nat_real,Xb: real,F: fun_nat_real,Yb: real] :
      ( aa_real_o(sums_real(G),Xb)
     => ( aa_real_o(sums_real(F),Yb)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_pl(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)) ) ) ).

% sums_if
tff(fact_7073_sin__le__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_7074_sin__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_7075_sin__monotone__2pi,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Yb)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi
tff(fact_7076_sin__mono__less__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Yb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_7077_sin__total,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ? [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(sin_real,X3) = Yb )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
                  & ( aa_real_real(sin_real,Y3) = Yb ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% sin_total
tff(fact_7078_cos__gt__zero__pi,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_gt_zero_pi
tff(fact_7079_cos__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_ge_zero
tff(fact_7080_accp__subset__induct,axiom,
    ! [D4: fun_Pr6331313853612757382_num_o,R: fun_Pr4799464687642144302_num_o,Xb: product_prod_num_num,P2: fun_Pr6331313853612757382_num_o] :
      ( ord_le3107050023622517222_num_o(D4,accp_P3113834385874906142um_num(R))
     => ( ! [X3: product_prod_num_num,Z2: product_prod_num_num] :
            ( aa_Pro7685007914727650815_num_o(D4,X3)
           => ( aa_Pro7685007914727650815_num_o(aa_Pro4218333719626769503_num_o(R,Z2),X3)
             => aa_Pro7685007914727650815_num_o(D4,Z2) ) )
       => ( aa_Pro7685007914727650815_num_o(D4,Xb)
         => ( ! [X3: product_prod_num_num] :
                ( aa_Pro7685007914727650815_num_o(D4,X3)
               => ( ! [Z3: product_prod_num_num] :
                      ( aa_Pro7685007914727650815_num_o(aa_Pro4218333719626769503_num_o(R,Z3),X3)
                     => aa_Pro7685007914727650815_num_o(P2,Z3) )
                 => aa_Pro7685007914727650815_num_o(P2,X3) ) )
           => aa_Pro7685007914727650815_num_o(P2,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7081_accp__subset__induct,axiom,
    ! [D4: fun_Pr4796943543331071514_nat_o,R: fun_Pr2028404973800837934_nat_o,Xb: product_prod_nat_nat,P2: fun_Pr4796943543331071514_nat_o] :
      ( aa_fun8408151826566693824at_o_o(ord_le1572679713340831354_nat_o(D4),accp_P4275260045618599050at_nat(R))
     => ( ! [X3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
            ( aa_Pro6150637604445964947_nat_o(D4,X3)
           => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,Z2),X3)
             => aa_Pro6150637604445964947_nat_o(D4,Z2) ) )
       => ( aa_Pro6150637604445964947_nat_o(D4,Xb)
         => ( ! [X3: product_prod_nat_nat] :
                ( aa_Pro6150637604445964947_nat_o(D4,X3)
               => ( ! [Z3: product_prod_nat_nat] :
                      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,Z3),X3)
                     => aa_Pro6150637604445964947_nat_o(P2,Z3) )
                 => aa_Pro6150637604445964947_nat_o(P2,X3) ) )
           => aa_Pro6150637604445964947_nat_o(P2,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7082_accp__subset__induct,axiom,
    ! [D4: fun_Pr3238374608701176162_int_o,R: fun_Pr9018117161326028078_int_o,Xb: product_prod_int_int,P2: fun_Pr3238374608701176162_int_o] :
      ( ord_le14110778710936002_int_o(D4,accp_P1096762738010456898nt_int(R))
     => ( ! [X3: product_prod_int_int,Z2: product_prod_int_int] :
            ( aa_Pro4592068669816069595_int_o(D4,X3)
           => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(R,Z2),X3)
             => aa_Pro4592068669816069595_int_o(D4,Z2) ) )
       => ( aa_Pro4592068669816069595_int_o(D4,Xb)
         => ( ! [X3: product_prod_int_int] :
                ( aa_Pro4592068669816069595_int_o(D4,X3)
               => ( ! [Z3: product_prod_int_int] :
                      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(R,Z3),X3)
                     => aa_Pro4592068669816069595_int_o(P2,Z3) )
                 => aa_Pro4592068669816069595_int_o(P2,X3) ) )
           => aa_Pro4592068669816069595_int_o(P2,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7083_accp__subset__induct,axiom,
    ! [D4: fun_list_nat_o,R: fun_li1025265955262268170_nat_o,Xb: list_nat,P2: fun_list_nat_o] :
      ( ord_le4972187363741583025_nat_o(D4,accp_list_nat(R))
     => ( ! [X3: list_nat,Z2: list_nat] :
            ( aa_list_nat_o(D4,X3)
           => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(R,Z2),X3)
             => aa_list_nat_o(D4,Z2) ) )
       => ( aa_list_nat_o(D4,Xb)
         => ( ! [X3: list_nat] :
                ( aa_list_nat_o(D4,X3)
               => ( ! [Z3: list_nat] :
                      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(R,Z3),X3)
                     => aa_list_nat_o(P2,Z3) )
                 => aa_list_nat_o(P2,X3) ) )
           => aa_list_nat_o(P2,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7084_accp__subset__induct,axiom,
    ! [D4: fun_nat_o,R: fun_nat_fun_nat_o,Xb: nat,P2: fun_nat_o] :
      ( ord_le7240017405068453697_nat_o(D4,accp_nat(R))
     => ( ! [X3: nat,Z2: nat] :
            ( aa_nat_o(D4,X3)
           => ( aa_nat_o(aa_nat_fun_nat_o(R,Z2),X3)
             => aa_nat_o(D4,Z2) ) )
       => ( aa_nat_o(D4,Xb)
         => ( ! [X3: nat] :
                ( aa_nat_o(D4,X3)
               => ( ! [Z3: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(R,Z3),X3)
                     => aa_nat_o(P2,Z3) )
                 => aa_nat_o(P2,X3) ) )
           => aa_nat_o(P2,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7085_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_7086_sin__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I3: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I3)
          & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_7087_cos__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ? [I3: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I3)
          & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_7088_sin__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_7089_sin__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ( ? [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_7090_cos__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
       => ? [N2: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_7091_cos__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ( ? [N3: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N3: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_7092_monoseq__Suc,axiom,
    ! [X5: fun_nat_real] :
      ( topolo6980174941875973593q_real(X5)
    <=> ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,N3)),aa_nat_real(X5,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,aa_nat_nat(suc,N3))),aa_nat_real(X5,N3)) ) ) ).

% monoseq_Suc
tff(fact_7093_monoseq__Suc,axiom,
    ! [X5: fun_nat_set_int] :
      ( topolo3100542954746470799et_int(X5)
    <=> ( ! [N3: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,N3)),aa_nat_set_int(X5,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,aa_nat_nat(suc,N3))),aa_nat_set_int(X5,N3)) ) ) ).

% monoseq_Suc
tff(fact_7094_monoseq__Suc,axiom,
    ! [X5: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X5)
    <=> ( ! [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N3)),aa_nat_rat(X5,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,aa_nat_nat(suc,N3))),aa_nat_rat(X5,N3)) ) ) ).

% monoseq_Suc
tff(fact_7095_monoseq__Suc,axiom,
    ! [X5: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X5)
    <=> ( ! [N3: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,N3)),aa_nat_num(X5,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,aa_nat_nat(suc,N3))),aa_nat_num(X5,N3)) ) ) ).

% monoseq_Suc
tff(fact_7096_monoseq__Suc,axiom,
    ! [X5: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X5)
    <=> ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,N3)),aa_nat_nat(X5,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,aa_nat_nat(suc,N3))),aa_nat_nat(X5,N3)) ) ) ).

% monoseq_Suc
tff(fact_7097_monoseq__Suc,axiom,
    ! [X5: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X5)
    <=> ( ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,N3)),aa_nat_int(X5,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,aa_nat_nat(suc,N3))),aa_nat_int(X5,N3)) ) ) ).

% monoseq_Suc
tff(fact_7098_mono__SucI2,axiom,
    ! [X5: fun_nat_real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,aa_nat_nat(suc,N2))),aa_nat_real(X5,N2))
     => topolo6980174941875973593q_real(X5) ) ).

% mono_SucI2
tff(fact_7099_mono__SucI2,axiom,
    ! [X5: fun_nat_set_int] :
      ( ! [N2: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,aa_nat_nat(suc,N2))),aa_nat_set_int(X5,N2))
     => topolo3100542954746470799et_int(X5) ) ).

% mono_SucI2
tff(fact_7100_mono__SucI2,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,aa_nat_nat(suc,N2))),aa_nat_rat(X5,N2))
     => topolo4267028734544971653eq_rat(X5) ) ).

% mono_SucI2
tff(fact_7101_mono__SucI2,axiom,
    ! [X5: fun_nat_num] :
      ( ! [N2: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,aa_nat_nat(suc,N2))),aa_nat_num(X5,N2))
     => topolo1459490580787246023eq_num(X5) ) ).

% mono_SucI2
tff(fact_7102_mono__SucI2,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,aa_nat_nat(suc,N2))),aa_nat_nat(X5,N2))
     => topolo4902158794631467389eq_nat(X5) ) ).

% mono_SucI2
tff(fact_7103_mono__SucI2,axiom,
    ! [X5: fun_nat_int] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,aa_nat_nat(suc,N2))),aa_nat_int(X5,N2))
     => topolo4899668324122417113eq_int(X5) ) ).

% mono_SucI2
tff(fact_7104_mono__SucI1,axiom,
    ! [X5: fun_nat_real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,N2)),aa_nat_real(X5,aa_nat_nat(suc,N2)))
     => topolo6980174941875973593q_real(X5) ) ).

% mono_SucI1
tff(fact_7105_mono__SucI1,axiom,
    ! [X5: fun_nat_set_int] :
      ( ! [N2: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,N2)),aa_nat_set_int(X5,aa_nat_nat(suc,N2)))
     => topolo3100542954746470799et_int(X5) ) ).

% mono_SucI1
tff(fact_7106_mono__SucI1,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N2)),aa_nat_rat(X5,aa_nat_nat(suc,N2)))
     => topolo4267028734544971653eq_rat(X5) ) ).

% mono_SucI1
tff(fact_7107_mono__SucI1,axiom,
    ! [X5: fun_nat_num] :
      ( ! [N2: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,N2)),aa_nat_num(X5,aa_nat_nat(suc,N2)))
     => topolo1459490580787246023eq_num(X5) ) ).

% mono_SucI1
tff(fact_7108_mono__SucI1,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,N2)),aa_nat_nat(X5,aa_nat_nat(suc,N2)))
     => topolo4902158794631467389eq_nat(X5) ) ).

% mono_SucI1
tff(fact_7109_mono__SucI1,axiom,
    ! [X5: fun_nat_int] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,N2)),aa_nat_int(X5,aa_nat_nat(suc,N2)))
     => topolo4899668324122417113eq_int(X5) ) ).

% mono_SucI1
tff(fact_7110_monoI1,axiom,
    ! [X5: fun_nat_real] :
      ( ! [M: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,M)),aa_nat_real(X5,N2)) )
     => topolo6980174941875973593q_real(X5) ) ).

% monoI1
tff(fact_7111_monoI1,axiom,
    ! [X5: fun_nat_set_int] :
      ( ! [M: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
         => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,M)),aa_nat_set_int(X5,N2)) )
     => topolo3100542954746470799et_int(X5) ) ).

% monoI1
tff(fact_7112_monoI1,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [M: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,M)),aa_nat_rat(X5,N2)) )
     => topolo4267028734544971653eq_rat(X5) ) ).

% monoI1
tff(fact_7113_monoI1,axiom,
    ! [X5: fun_nat_num] :
      ( ! [M: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,M)),aa_nat_num(X5,N2)) )
     => topolo1459490580787246023eq_num(X5) ) ).

% monoI1
tff(fact_7114_monoI1,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [M: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,M)),aa_nat_nat(X5,N2)) )
     => topolo4902158794631467389eq_nat(X5) ) ).

% monoI1
tff(fact_7115_monoI1,axiom,
    ! [X5: fun_nat_int] :
      ( ! [M: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,M)),aa_nat_int(X5,N2)) )
     => topolo4899668324122417113eq_int(X5) ) ).

% monoI1
tff(fact_7116_Maclaurin__cos__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xb)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pm,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_7117_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),zero_zero_real)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pm,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_7118_Maclaurin__cos__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pm,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_7119_diffs__equiv,axiom,
    ! [C2: fun_nat_real,Xb: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_pn(C2),Xb))
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_po(C2),Xb)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_pn(C2),Xb))) ) ).

% diffs_equiv
tff(fact_7120_diffs__equiv,axiom,
    ! [C2: fun_nat_complex,Xb: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_pp(C2),Xb))
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_pq(C2),Xb)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_pp(C2),Xb))) ) ).

% diffs_equiv
tff(fact_7121_tan__double,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,Xb))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_7122_tan__double,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(tan_real,Xb))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_7123_in__measure,axiom,
    ! [Xb: int,Yb: int,F: fun_int_nat] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Yb),measure_int(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,Xb)),aa_int_nat(F,Yb)) ) ).

% in_measure
tff(fact_7124_tan__pi,axiom,
    aa_real_real(tan_real,pi) = zero_zero_real ).

% tan_pi
tff(fact_7125_tan__zero,axiom,
    aa_complex_complex(tan_complex,zero_zero_complex) = zero_zero_complex ).

% tan_zero
tff(fact_7126_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_7127_tan__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),pi)) = aa_real_real(tan_real,Xb) ).

% tan_periodic_pi
tff(fact_7128_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_7129_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_7130_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_7131_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_7132_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_7133_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_7134_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_7135_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_7136_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_7137_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_7138_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_Suc
tff(fact_7139_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_Suc
tff(fact_7140_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Nb))),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ).

% fact_Suc
tff(fact_7141_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_Suc
tff(fact_7142_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_Suc
tff(fact_7143_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Nb))),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% fact_Suc
tff(fact_7144_tan__npi,axiom,
    ! [Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_7145_tan__periodic__n,axiom,
    ! [Xb: real,Nb: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,Nb)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_n
tff(fact_7146_tan__periodic__nat,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_nat
tff(fact_7147_tan__periodic__int,axiom,
    ! [Xb: real,I: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_int
tff(fact_7148_tan__periodic,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic
tff(fact_7149_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) != zero_zero_rat ).

% fact_nonzero
tff(fact_7150_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) != zero_zero_int ).

% fact_nonzero
tff(fact_7151_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) != zero_zero_real ).

% fact_nonzero
tff(fact_7152_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) != zero_zero_nat ).

% fact_nonzero
tff(fact_7153_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,Nb) != zero_zero_complex ).

% fact_nonzero
tff(fact_7154_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_zero
tff(fact_7155_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_zero
tff(fact_7156_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_zero
tff(fact_7157_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_zero
tff(fact_7158_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),zero_zero_rat) ).

% fact_not_neg
tff(fact_7159_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),zero_zero_int) ).

% fact_not_neg
tff(fact_7160_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),zero_zero_real) ).

% fact_not_neg
tff(fact_7161_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),zero_zero_nat) ).

% fact_not_neg
tff(fact_7162_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_gt_zero
tff(fact_7163_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_gt_zero
tff(fact_7164_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_gt_zero
tff(fact_7165_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_gt_zero
tff(fact_7166_fact__ge__1,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_1
tff(fact_7167_fact__ge__1,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_1
tff(fact_7168_fact__ge__1,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_1
tff(fact_7169_fact__ge__1,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_1
tff(fact_7170_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% fact_mono
tff(fact_7171_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% fact_mono
tff(fact_7172_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% fact_mono
tff(fact_7173_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono
tff(fact_7174_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) ) ).

% fact_dvd
tff(fact_7175_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Mb)) ) ).

% fact_dvd
tff(fact_7176_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Mb)) ) ).

% fact_dvd
tff(fact_7177_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) ) ).

% fact_dvd
tff(fact_7178_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ) ).

% fact_less_mono
tff(fact_7179_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ) ).

% fact_less_mono
tff(fact_7180_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ) ).

% fact_less_mono
tff(fact_7181_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono
tff(fact_7182_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K2)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_7183_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_7184_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K2)),aa_nat_int(semiri1406184849735516958ct_int,Nb))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_7185_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_7186_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_7187_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_7188_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( modulo364778990260209775nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,Nb),aa_nat_Code_integer(semiri3624122377584611663nteger,Mb)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_7189_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_7190_fact__le__power,axiom,
    ! [Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_7191_fact__le__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_7192_fact__le__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_7193_fact__le__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_7194_fact__le__power,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_7195_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K2)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ) ).

% choose_dvd
tff(fact_7196_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% choose_dvd
tff(fact_7197_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K2)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% choose_dvd
tff(fact_7198_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% choose_dvd
tff(fact_7199_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% choose_dvd
tff(fact_7200_diffs__def,axiom,
    ! [C2: fun_nat_rat,X: nat] : aa_nat_rat(diffs_rat(C2),X) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,X))),aa_nat_rat(C2,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_7201_diffs__def,axiom,
    ! [C2: fun_nat_int,X: nat] : aa_nat_int(diffs_int(C2),X) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,X))),aa_nat_int(C2,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_7202_diffs__def,axiom,
    ! [C2: fun_nat_real,X: nat] : aa_nat_real(diffs_real(C2),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,X))),aa_nat_real(C2,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_7203_diffs__def,axiom,
    ! [C2: fun_nat_complex,X: nat] : aa_nat_complex(diffs_complex(C2),X) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,X))),aa_nat_complex(C2,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_7204_diffs__def,axiom,
    ! [C2: fun_nat_Code_integer,X: nat] : aa_nat_Code_integer(diffs_Code_integer(C2),X) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,X))),aa_nat_Code_integer(C2,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_7205_square__fact__le__2__fact,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% square_fact_le_2_fact
tff(fact_7206_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_rat(semiri773545260158071498ct_rat,Mb) = $ite(Mb = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7207_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_int(semiri1406184849735516958ct_int,Mb) = $ite(Mb = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7208_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_Code_integer(semiri3624122377584611663nteger,Mb) = $ite(Mb = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7209_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_real(semiri2265585572941072030t_real,Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7210_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = $ite(Mb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7211_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_complex(semiri5044797733671781792omplex,Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7212_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7213_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7214_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_Code_integer(semiri3624122377584611663nteger,Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7215_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7216_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7217_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(semiri5044797733671781792omplex,Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7218_tan__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,Xb)) ) ) ).

% tan_gt_zero
tff(fact_7219_lemma__tan__total,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(tan_real,X3)) ) ) ).

% lemma_tan_total
tff(fact_7220_add__tan__eq,axiom,
    ! [Xb: complex,Yb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Yb) != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Yb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(Xb),Yb))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Yb))) ) ) ) ).

% add_tan_eq
tff(fact_7221_add__tan__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Yb) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Yb))) ) ) ) ).

% add_tan_eq
tff(fact_7222_tan__pos__pi2__le,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,Xb)) ) ) ).

% tan_pos_pi2_le
tff(fact_7223_tan__total__pos,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
     => ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X3) = Yb ) ) ) ).

% tan_total_pos
tff(fact_7224_tan__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xb)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_7225_tan__mono__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb)) ) ) ) ).

% tan_mono_le
tff(fact_7226_tan__mono__le__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_7227_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_na5723442673449079972x_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun5746040925223427547t_real(aTP_Lamm_pr(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).

% Maclaurin_zero
tff(fact_7228_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ps(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_7229_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_rat_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_pt(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).

% Maclaurin_zero
tff(fact_7230_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_nat_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_pu(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_7231_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_int_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_pv(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_7232_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B7: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_pw(H),J)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B7),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_7233_tan__add,axiom,
    ! [Xb: complex,Yb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Yb) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Yb)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(Xb),Yb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Yb))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Yb)))) ) ) ) ) ).

% tan_add
tff(fact_7234_tan__add,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Yb) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb)))) ) ) ) ) ).

% tan_add
tff(fact_7235_tan__diff,axiom,
    ! [Xb: complex,Yb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Yb) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(Xb),Yb)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(Xb),Yb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Yb))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Yb)))) ) ) ) ) ).

% tan_diff
tff(fact_7236_tan__diff,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Yb) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Yb)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb)))) ) ) ) ) ).

% tan_diff
tff(fact_7237_lemma__tan__add1,axiom,
    ! [Xb: complex,Yb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Yb) != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Yb))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Yb))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Yb))) ) ) ) ).

% lemma_tan_add1
tff(fact_7238_lemma__tan__add1,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Yb) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Yb))) ) ) ) ).

% lemma_tan_add1
tff(fact_7239_tan__half,axiom,
    ! [Xb: complex] : aa_complex_complex(tan_complex,Xb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb))),one_one_complex)) ).

% tan_half
tff(fact_7240_tan__half,axiom,
    ! [Xb: real] : aa_real_real(tan_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),one_one_real)) ).

% tan_half
tff(fact_7241_cos__coeff__def,axiom,
    ! [X: nat] :
      aa_nat_real(cos_coeff,X) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X)),zero_zero_real) ).

% cos_coeff_def
tff(fact_7242_sin__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_px,Xb)),aa_real_real(sin_real,Xb)) ).

% sin_paired
tff(fact_7243_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xb)
            & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_py,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_7244_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),Xb)
          & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_py,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_7245_Maclaurin__sin__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_py,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_7246_Maclaurin__sin__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] : aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_py,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_7247_sin__coeff__def,axiom,
    ! [X: nat] :
      aa_nat_real(sin_coeff,X) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X))) ).

% sin_coeff_def
tff(fact_7248_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T4),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,T4)),aa_real_real(sin_real,T4)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_7249_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_7250_fact__ge__self,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_self
tff(fact_7251_fact__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono_nat
tff(fact_7252_fact__less__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_7253_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_7254_zero__complex_Ocode,axiom,
    zero_zero_complex = aa_real_complex(complex2(zero_zero_real),zero_zero_real) ).

% zero_complex.code
tff(fact_7255_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_7256_complex__add,axiom,
    ! [A: real,B: real,C2: real,D: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C2),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ).

% complex_add
tff(fact_7257_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_7258_dvd__fact,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% dvd_fact
tff(fact_7259_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_7260_complex__mult,axiom,
    ! [A: real,B: real,C2: real,D: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C2),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))) ).

% complex_mult
tff(fact_7261_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_7262_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_7263_fact__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_7264_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_7265_Complex__sum_H,axiom,
    ! [F: fun_nat_real,S3: set_nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_pz(F)),S3) = aa_real_complex(complex2(aa_set_nat_real(groups6591440286371151544t_real(F),S3)),zero_zero_real) ).

% Complex_sum'
tff(fact_7266_Complex__sum_H,axiom,
    ! [F: fun_complex_real,S3: set_complex] : groups7754918857620584856omplex(aTP_Lamm_qa(F),S3) = aa_real_complex(complex2(groups5808333547571424918x_real(F,S3)),zero_zero_real) ).

% Complex_sum'
tff(fact_7267_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_7268_sin__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% sin_coeff_Suc
tff(fact_7269_cos__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Nb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% cos_coeff_Suc
tff(fact_7270_sin__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,Xb)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_7271_cos__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_7272_Maclaurin__exp__lt,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T4))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qb,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_7273_in__finite__psubset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(A4,B4),finite_psubset_nat)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),B4)
        & aa_set_nat_o(finite_finite_nat,B4) ) ) ).

% in_finite_psubset
tff(fact_7274_in__finite__psubset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( member2572552093476627150et_int(produc6363374080413544029et_int(A4,B4),finite_psubset_int)
    <=> ( aa_set_int_o(ord_less_set_int(A4),B4)
        & aa_set_int_o(finite_finite_int,B4) ) ) ).

% in_finite_psubset
tff(fact_7275_in__finite__psubset,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( member351165363924911826omplex(produc3790773574474814305omplex(A4,B4),finite8643634255014194347omplex)
    <=> ( aa_set_complex_o(ord_less_set_complex(A4),B4)
        & aa_set_complex_o(finite3207457112153483333omplex,B4) ) ) ).

% in_finite_psubset
tff(fact_7276_in__finite__psubset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A4,B4),finite469560695537375940at_nat)
    <=> ( aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(A4),B4)
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4) ) ) ).

% in_finite_psubset
tff(fact_7277_pochhammer__double,axiom,
    ! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_7278_pochhammer__double,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_7279_pochhammer__double,axiom,
    ! [Z: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_7280_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,Nb) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_qc,Nb),zero_zero_rat) ).

% of_nat_code
tff(fact_7281_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_int_int(aa_nat_fun_int_int(semiri8420488043553186161ux_int(aTP_Lamm_qd),Nb),zero_zero_int) ).

% of_nat_code
tff(fact_7282_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_qe),Nb),zero_zero_real) ).

% of_nat_code
tff(fact_7283_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_qf),Nb),zero_zero_nat) ).

% of_nat_code
tff(fact_7284_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,Nb) = aa_complex_complex(aa_nat3829984526574172007omplex(semiri2816024913162550771omplex(aTP_Lamm_qg),Nb),zero_zero_complex) ).

% of_nat_code
tff(fact_7285_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) = semiri4055485073559036834nteger(aTP_Lamm_qh,Nb,zero_z3403309356797280102nteger) ).

% of_nat_code
tff(fact_7286_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sqrt,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_7287_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_7288_real__sqrt__le__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ).

% real_sqrt_le_iff
tff(fact_7289_exp__le__cancel__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ).

% exp_le_cancel_iff
tff(fact_7290_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_7291_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_7292_real__sqrt__gt__0__iff,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb) ) ).

% real_sqrt_gt_0_iff
tff(fact_7293_real__sqrt__lt__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_7294_real__sqrt__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_7295_real__sqrt__ge__0__iff,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb) ) ).

% real_sqrt_ge_0_iff
tff(fact_7296_real__sqrt__ge__1__iff,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Yb) ) ).

% real_sqrt_ge_1_iff
tff(fact_7297_real__sqrt__le__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ).

% real_sqrt_le_1_iff
tff(fact_7298_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_7299_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_7300_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_7301_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_7302_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_7303_exp__eq__one__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(exp_real,Xb) = one_one_real )
    <=> ( Xb = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_7304_exp__less__one__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% exp_less_one_iff
tff(fact_7305_one__less__exp__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% one_less_exp_iff
tff(fact_7306_one__le__exp__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% one_le_exp_iff
tff(fact_7307_exp__le__one__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_7308_exp__ln__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xb)) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% exp_ln_iff
tff(fact_7309_exp__ln,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xb)) = Xb ) ) ).

% exp_ln
tff(fact_7310_real__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb ) ) ).

% real_sqrt_pow2
tff(fact_7311_real__sqrt__pow2__iff,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% real_sqrt_pow2_iff
tff(fact_7312_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xb: real,Yb: real,Xaa: real,Yaa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_7313_norm__exp,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Xb))),aa_real_real(exp_real,aa_real_real(real_V7735802525324610683m_real,Xb))) ).

% norm_exp
tff(fact_7314_norm__exp,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Xb))),aa_real_real(exp_real,aa_complex_real(real_V1022390504157884413omplex,Xb))) ).

% norm_exp
tff(fact_7315_real__sqrt__le__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Yb)) ) ).

% real_sqrt_le_mono
tff(fact_7316_exp__not__eq__zero,axiom,
    ! [Xb: complex] : aa_complex_complex(exp_complex,Xb) != zero_zero_complex ).

% exp_not_eq_zero
tff(fact_7317_exp__not__eq__zero,axiom,
    ! [Xb: real] : aa_real_real(exp_real,Xb) != zero_zero_real ).

% exp_not_eq_zero
tff(fact_7318_real__sqrt__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_gt_zero
tff(fact_7319_exp__total,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ? [X3: real] : aa_real_real(exp_real,X3) = Yb ) ).

% exp_total
tff(fact_7320_exp__gt__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(exp_real,Xb)) ).

% exp_gt_zero
tff(fact_7321_not__exp__less__zero,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),zero_zero_real) ).

% not_exp_less_zero
tff(fact_7322_real__sqrt__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_ge_zero
tff(fact_7323_real__sqrt__eq__zero__cancel,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(sqrt,Xb) = zero_zero_real )
       => ( Xb = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_7324_exp__ge__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,Xb)) ).

% exp_ge_zero
tff(fact_7325_not__exp__le__zero,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),zero_zero_real) ).

% not_exp_le_zero
tff(fact_7326_real__sqrt__ge__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_ge_one
tff(fact_7327_mult__exp__exp,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Yb)) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)) ).

% mult_exp_exp
tff(fact_7328_exp__add__commuting,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Yb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Xb) )
     => ( aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Yb)) ) ) ).

% exp_add_commuting
tff(fact_7329_pochhammer__pos,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_7330_pochhammer__pos,axiom,
    ! [Xb: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_7331_pochhammer__pos,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_7332_pochhammer__pos,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_7333_pochhammer__eq__0__mono,axiom,
    ! [A: complex,Nb: nat,Mb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),Mb) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7334_pochhammer__eq__0__mono,axiom,
    ! [A: real,Nb: nat,Mb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Mb) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7335_pochhammer__eq__0__mono,axiom,
    ! [A: rat,Nb: nat,Mb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7336_pochhammer__neq__0__mono,axiom,
    ! [A: complex,Mb: nat,Nb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Mb) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7337_pochhammer__neq__0__mono,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Mb) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7338_pochhammer__neq__0__mono,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7339_exp__gt__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xb)) ) ).

% exp_gt_one
tff(fact_7340_real__div__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,Xb)) = aa_real_real(sqrt,Xb) ) ) ).

% real_div_sqrt
tff(fact_7341_sqrt__add__le__add__sqrt,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Yb))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_7342_exp__ge__add__one__self,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(exp_real,Xb)) ).

% exp_ge_add_one_self
tff(fact_7343_le__real__sqrt__sumsq,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),Yb)))) ).

% le_real_sqrt_sumsq
tff(fact_7344_pochhammer__nonneg,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_7345_pochhammer__nonneg,axiom,
    ! [Xb: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_7346_pochhammer__nonneg,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_7347_pochhammer__nonneg,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_7348_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% pochhammer_0_left
tff(fact_7349_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% pochhammer_0_left
tff(fact_7350_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% pochhammer_0_left
tff(fact_7351_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% pochhammer_0_left
tff(fact_7352_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% pochhammer_0_left
tff(fact_7353_exp__ge__add__one__self__aux,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(exp_real,Xb)) ) ).

% exp_ge_add_one_self_aux
tff(fact_7354_lemma__exp__total,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Yb)
     => ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Yb),one_one_real))
          & ( aa_real_real(exp_real,X3) = Yb ) ) ) ).

% lemma_exp_total
tff(fact_7355_ln__ge__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Yb)),Xb) ) ) ).

% ln_ge_iff
tff(fact_7356_ln__x__over__x__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Yb)),Yb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xb)),Xb)) ) ) ).

% ln_x_over_x_mono
tff(fact_7357_pochhammer__rec,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),Nb)) ).

% pochhammer_rec
tff(fact_7358_pochhammer__rec,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),Nb)) ).

% pochhammer_rec
tff(fact_7359_pochhammer__rec,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),Nb)) ).

% pochhammer_rec
tff(fact_7360_pochhammer__rec,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),Nb)) ).

% pochhammer_rec
tff(fact_7361_pochhammer__rec,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),Nb)) ).

% pochhammer_rec
tff(fact_7362_pochhammer__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb)),aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ).

% pochhammer_Suc
tff(fact_7363_pochhammer__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% pochhammer_Suc
tff(fact_7364_pochhammer__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,Nb))) ).

% pochhammer_Suc
tff(fact_7365_pochhammer__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) ).

% pochhammer_Suc
tff(fact_7366_pochhammer__Suc,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(A),Nb)),aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,Nb))) ).

% pochhammer_Suc
tff(fact_7367_pochhammer__Suc,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(A),Nb)),plus_p5714425477246183910nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) ).

% pochhammer_Suc
tff(fact_7368_pochhammer__rec_H,axiom,
    ! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)) ).

% pochhammer_rec'
tff(fact_7369_pochhammer__rec_H,axiom,
    ! [Z: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),aa_nat_int(comm_s4660882817536571857er_int(Z),Nb)) ).

% pochhammer_rec'
tff(fact_7370_pochhammer__rec_H,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb)) ).

% pochhammer_rec'
tff(fact_7371_pochhammer__rec_H,axiom,
    ! [Z: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)) ).

% pochhammer_rec'
tff(fact_7372_pochhammer__rec_H,axiom,
    ! [Z: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,Nb))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb)) ).

% pochhammer_rec'
tff(fact_7373_pochhammer__rec_H,axiom,
    ! [Z: code_integer,Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(Z,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Nb)) ).

% pochhammer_rec'
tff(fact_7374_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K2) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7375_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K2) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7376_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K2) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7377_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K2) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7378_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K2) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7379_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K2) = zero_zero_rat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7380_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K2) = zero_zero_int )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7381_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K2) = zero_zero_real )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7382_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K2) = zero_zero_complex )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7383_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K2) = zero_z3403309356797280102nteger )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7384_pochhammer__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7385_pochhammer__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7386_pochhammer__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = zero_zero_complex )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7387_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K2) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7388_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K2) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7389_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K2) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7390_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K2) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7391_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K2) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7392_pochhammer__product_H,axiom,
    ! [Z: rat,Nb: nat,Mb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_7393_pochhammer__product_H,axiom,
    ! [Z: int,Nb: nat,Mb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Nb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_7394_pochhammer__product_H,axiom,
    ! [Z: real,Nb: nat,Mb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_7395_pochhammer__product_H,axiom,
    ! [Z: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_7396_pochhammer__product_H,axiom,
    ! [Z: complex,Nb: nat,Mb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_7397_pochhammer__product_H,axiom,
    ! [Z: code_integer,Nb: nat,Mb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Nb)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(Z,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_7398_real__le__rsqrt,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Yb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,Yb)) ) ).

% real_le_rsqrt
tff(fact_7399_sqrt__le__D,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Yb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_7400_exp__le,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% exp_le
tff(fact_7401_exp__divide__power__eq,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb) = aa_real_real(exp_real,Xb) ) ) ).

% exp_divide_power_eq
tff(fact_7402_exp__divide__power__eq,axiom,
    ! [Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(Xb),aa_nat_complex(semiri8010041392384452111omplex,Nb)))),Nb) = aa_complex_complex(exp_complex,Xb) ) ) ).

% exp_divide_power_eq
tff(fact_7403_tanh__altdef,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))) ).

% tanh_altdef
tff(fact_7404_tanh__altdef,axiom,
    ! [Xb: complex] : aa_complex_complex(tanh_complex,Xb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)))) ).

% tanh_altdef
tff(fact_7405_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),Mb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_7406_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Mb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_7407_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Mb)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_7408_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Mb)),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,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_7409_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),Mb)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_7410_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Mb)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(Z,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_7411_real__le__lsqrt,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Yb) ) ) ) ).

% real_le_lsqrt
tff(fact_7412_real__sqrt__unique,axiom,
    ! [Yb: real,Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_real(sqrt,Xb) = Yb ) ) ) ).

% real_sqrt_unique
tff(fact_7413_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_7414_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Yb )
     => ( Xb = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_7415_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Xb )
     => ( Yb = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_7416_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% exp_half_le2
tff(fact_7417_real__sqrt__sum__squares__ge1,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_7418_real__sqrt__sum__squares__ge2,axiom,
    ! [Yb: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_7419_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C2: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_7420_sqrt__ge__absD,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,Yb))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Yb) ) ).

% sqrt_ge_absD
tff(fact_7421_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K2: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(R2),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R2)),K2)) = aa_rat_rat(times_times_rat(R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,R2)),one_one_rat)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7422_pochhammer__absorb__comp,axiom,
    ! [R2: int,K2: 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,K2))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K2)) = 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)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7423_pochhammer__absorb__comp,axiom,
    ! [R2: real,K2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,R2)),K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,R2)),one_one_real)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7424_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K2: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R2),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R2)),K2)) = aa_complex_complex(times_times_complex(R2),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R2)),one_one_complex)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7425_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R2,aa_nat_Code_integer(semiri4939895301339042750nteger,K2))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),K2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2),one_one_Code_integer)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7426_real__less__lsqrt,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),Yb) ) ) ) ).

% real_less_lsqrt
tff(fact_7427_sqrt__sum__squares__le__sum,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_7428_real__sqrt__ge__abs1,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_7429_real__sqrt__ge__abs2,axiom,
    ! [Yb: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_7430_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Yb))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_7431_ln__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_7432_arsinh__real__def,axiom,
    ! [Xb: real] : aa_real_real(arsinh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_7433_complex__norm,axiom,
    ! [Xb: real,Yb: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(Xb),Yb)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_7434_exp__bound__half,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_7435_exp__bound__half,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_7436_pochhammer__minus,axiom,
    ! [B: rat,K2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K2) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)) ).

% pochhammer_minus
tff(fact_7437_pochhammer__minus,axiom,
    ! [B: int,K2: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K2) = 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)),K2)),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,K2))),one_one_int)),K2)) ).

% pochhammer_minus
tff(fact_7438_pochhammer__minus,axiom,
    ! [B: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)) ).

% pochhammer_minus
tff(fact_7439_pochhammer__minus,axiom,
    ! [B: complex,K2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)) ).

% pochhammer_minus
tff(fact_7440_pochhammer__minus,axiom,
    ! [B: code_integer,K2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K2)),one_one_Code_integer)),K2)) ).

% pochhammer_minus
tff(fact_7441_pochhammer__minus_H,axiom,
    ! [B: rat,K2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K2)) ).

% pochhammer_minus'
tff(fact_7442_pochhammer__minus_H,axiom,
    ! [B: int,K2: 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,K2))),one_one_int)),K2) = 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)),K2)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K2)) ).

% pochhammer_minus'
tff(fact_7443_pochhammer__minus_H,axiom,
    ! [B: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K2)) ).

% pochhammer_minus'
tff(fact_7444_pochhammer__minus_H,axiom,
    ! [B: complex,K2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K2)) ).

% pochhammer_minus'
tff(fact_7445_pochhammer__minus_H,axiom,
    ! [B: code_integer,K2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K2)),one_one_Code_integer)),K2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K2)) ).

% pochhammer_minus'
tff(fact_7446_finite__psubset__def,axiom,
    finite_psubset_nat = collec6662362479098859352et_nat(produc6247414631856714078_nat_o(aTP_Lamm_qi)) ).

% finite_psubset_def
tff(fact_7447_finite__psubset__def,axiom,
    finite_psubset_int = collec957716948307931664et_int(produc4109468873575309990_int_o(aTP_Lamm_qj)) ).

% finite_psubset_def
tff(fact_7448_finite__psubset__def,axiom,
    finite8643634255014194347omplex = collec5108298041176329748omplex(produc3914248068834153634plex_o(aTP_Lamm_qk)) ).

% finite_psubset_def
tff(fact_7449_finite__psubset__def,axiom,
    finite469560695537375940at_nat = collec6321179662152712658at_nat(produc410239310623530412_nat_o(aTP_Lamm_ql)) ).

% finite_psubset_def
tff(fact_7450_arsinh__real__aux,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_7451_exp__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% exp_bound
tff(fact_7452_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xb: real,Yb: real,Xaa: real,Yaa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_7453_real__sqrt__power__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_7454_arith__geo__mean__sqrt,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Yb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Yb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_7455_real__exp__bound__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))) ) ) ).

% real_exp_bound_lemma
tff(fact_7456_cos__x__y__le__one,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_7457_real__sqrt__sum__squares__less,axiom,
    ! [Xb: real,U: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Yb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_7458_arcosh__real__def,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_real(arcosh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_7459_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_7460_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_7461_cos__arctan,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(arctan,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_7462_sin__arctan,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(arctan,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_7463_pochhammer__code,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_qm(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_rat)) ).

% pochhammer_code
tff(fact_7464_pochhammer__code,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = $ite(Nb = zero_zero_nat,one_one_int,aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_qn(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_int)) ).

% pochhammer_code
tff(fact_7465_pochhammer__code,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = $ite(Nb = zero_zero_nat,one_one_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_qo(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_real)) ).

% pochhammer_code
tff(fact_7466_pochhammer__code,axiom,
    ! [A: complex,Nb: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_qp(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_complex)) ).

% pochhammer_code
tff(fact_7467_pochhammer__code,axiom,
    ! [A: code_integer,Nb: nat] :
      aa_nat_Code_integer(comm_s8582702949713902594nteger(A),Nb) = $ite(Nb = zero_zero_nat,one_one_Code_integer,set_fo1084959871951514735nteger(aTP_Lamm_qq(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),one_one_Code_integer)) ).

% pochhammer_code
tff(fact_7468_pochhammer__code,axiom,
    ! [A: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_qr(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_nat)) ).

% pochhammer_code
tff(fact_7469_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(real_V7735802525324610683m_real,Z)))) ) ).

% exp_bound_lemma
tff(fact_7470_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(real_V1022390504157884413omplex,Z)))) ) ).

% exp_bound_lemma
tff(fact_7471_Maclaurin__exp__le,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qb,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_7472_sqrt__sum__squares__half__less,axiom,
    ! [Xb: real,U: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_7473_exp__lower__Taylor__quadratic,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,Xb)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_7474_sin__cos__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xb))
     => ( aa_real_real(sin_real,Xb) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_7475_arctan__half,axiom,
    ! [Xb: real] : aa_real_real(arctan,Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_7476_tanh__real__altdef,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)))) ).

% tanh_real_altdef
tff(fact_7477_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = groups73079841787564623at_rat(aTP_Lamm_qs(Z),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7478_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_qt,Z)),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7479_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_qu(Z)),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7480_cos__arcsin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xb)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_7481_sin__arccos__abs,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
     => ( aa_real_real(sin_real,aa_real_real(arccos,Yb)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_7482_sin__arccos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xb)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_7483_or__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L) = $ite(
        ( ( K2 = aa_int_int(uminus_uminus_int,one_one_int) )
        | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
        aa_int_int(uminus_uminus_int,one_one_int),
        $ite(
          K2 = zero_zero_int,
          L,
          $ite(L = zero_zero_int,K2,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,K2),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,K2),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_7484_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_qv(R2),set_or1269000886237332187st_nat(zero_zero_nat,Mb)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb))),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_7485_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_qw(R2)),set_or1269000886237332187st_nat(zero_zero_nat,Mb)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb))),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_7486_gchoose__row__sum__weighted,axiom,
    ! [R2: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qx,R2)),set_or1269000886237332187st_nat(zero_zero_nat,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_7487_or_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,zero_zero_int),A) = A ).

% or.left_neutral
tff(fact_7488_or_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,zero_zero_nat),A) = A ).

% or.left_neutral
tff(fact_7489_or_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),zero_zero_int) = A ).

% or.right_neutral
tff(fact_7490_or_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),zero_zero_nat) = A ).

% or.right_neutral
tff(fact_7491_arcsin__0,axiom,
    aa_real_real(arcsin,zero_zero_real) = zero_zero_real ).

% arcsin_0
tff(fact_7492_of__nat__prod,axiom,
    ! [F: fun_int_nat,A4: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A4)) = groups1705073143266064639nt_int(aTP_Lamm_ku(F),A4) ).

% of_nat_prod
tff(fact_7493_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_la(F)),A4) ).

% of_nat_prod
tff(fact_7494_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) = aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_kx(F)),A4) ).

% of_nat_prod
tff(fact_7495_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) = groups3455450783089532116nteger(aTP_Lamm_ky(F),A4) ).

% of_nat_prod
tff(fact_7496_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_kz(F)),A4) ).

% of_nat_prod
tff(fact_7497_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) = groups705719431365010083at_int(aTP_Lamm_kw(F),A4) ).

% of_nat_prod
tff(fact_7498_of__int__prod,axiom,
    ! [F: fun_nat_int,A4: set_nat] : aa_int_real(ring_1_of_int_real,groups705719431365010083at_int(F,A4)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_lc(F)),A4) ).

% of_int_prod
tff(fact_7499_of__int__prod,axiom,
    ! [F: fun_nat_int,A4: set_nat] : aa_int_rat(ring_1_of_int_rat,groups705719431365010083at_int(F,A4)) = groups73079841787564623at_rat(aTP_Lamm_qy(F),A4) ).

% of_int_prod
tff(fact_7500_of__int__prod,axiom,
    ! [F: fun_nat_int,A4: set_nat] : aa_int_complex(ring_17405671764205052669omplex,groups705719431365010083at_int(F,A4)) = aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_qz(F)),A4) ).

% of_int_prod
tff(fact_7501_of__int__prod,axiom,
    ! [F: fun_nat_int,A4: set_nat] : aa_int_int(ring_1_of_int_int,groups705719431365010083at_int(F,A4)) = groups705719431365010083at_int(aTP_Lamm_ra(F),A4) ).

% of_int_prod
tff(fact_7502_of__int__prod,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_real(ring_1_of_int_real,groups1705073143266064639nt_int(F,A4)) = groups2316167850115554303t_real(aTP_Lamm_ld(F),A4) ).

% of_int_prod
tff(fact_7503_of__int__prod,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(F,A4)) = groups1072433553688619179nt_rat(aTP_Lamm_le(F),A4) ).

% of_int_prod
tff(fact_7504_of__int__prod,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_complex(ring_17405671764205052669omplex,groups1705073143266064639nt_int(F,A4)) = groups7440179247065528705omplex(aTP_Lamm_lf(F),A4) ).

% of_int_prod
tff(fact_7505_of__int__prod,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A4)) = groups1705073143266064639nt_int(aTP_Lamm_lg(F),A4) ).

% of_int_prod
tff(fact_7506_prod__zero__iff,axiom,
    ! [A4: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_set_nat_complex(groups6464643781859351333omplex(F),A4) = zero_zero_complex )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & ( aa_nat_complex(F,X4) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_7507_prod__zero__iff,axiom,
    ! [A4: set_int,F: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( groups7440179247065528705omplex(F,A4) = zero_zero_complex )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A4)
            & ( aa_int_complex(F,X4) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_7508_prod__zero__iff,axiom,
    ! [A4: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( groups3708469109370488835omplex(F,A4) = zero_zero_complex )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A4)
            & ( aa_complex_complex(F,X4) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_7509_prod__zero__iff,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_set_nat_real(groups129246275422532515t_real(F),A4) = zero_zero_real )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & ( aa_nat_real(F,X4) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7510_prod__zero__iff,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( groups2316167850115554303t_real(F,A4) = zero_zero_real )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A4)
            & ( aa_int_real(F,X4) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7511_prod__zero__iff,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( groups766887009212190081x_real(F,A4) = zero_zero_real )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A4)
            & ( aa_complex_real(F,X4) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7512_prod__zero__iff,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( groups73079841787564623at_rat(F,A4) = zero_zero_rat )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & ( aa_nat_rat(F,X4) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7513_prod__zero__iff,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( groups1072433553688619179nt_rat(F,A4) = zero_zero_rat )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A4)
            & ( aa_int_rat(F,X4) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7514_prod__zero__iff,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( groups225925009352817453ex_rat(F,A4) = zero_zero_rat )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A4)
            & ( aa_complex_rat(F,X4) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7515_prod__zero__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( groups1707563613775114915nt_nat(F,A4) = zero_zero_nat )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A4)
            & ( aa_int_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_7516_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_complex] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_complex(groups6464643781859351333omplex(G),A4) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7517_prod_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_complex] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( groups7440179247065528705omplex(G,A4) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7518_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups3708469109370488835omplex(G,A4) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7519_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_real] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_real(groups129246275422532515t_real(G),A4) = one_one_real ) ) ).

% prod.infinite
tff(fact_7520_prod_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_real] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( groups2316167850115554303t_real(G,A4) = one_one_real ) ) ).

% prod.infinite
tff(fact_7521_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_real] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups766887009212190081x_real(G,A4) = one_one_real ) ) ).

% prod.infinite
tff(fact_7522_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( groups73079841787564623at_rat(G,A4) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7523_prod_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_rat] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( groups1072433553688619179nt_rat(G,A4) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7524_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_rat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups225925009352817453ex_rat(G,A4) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7525_prod_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_nat] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( groups1707563613775114915nt_nat(G,A4) = one_one_nat ) ) ).

% prod.infinite
tff(fact_7526_dvd__prod__eqI,axiom,
    ! [A4: set_real,A: real,B: nat,F: fun_real_nat] :
      ( finite_finite_real(A4)
     => ( member_real(A,A4)
       => ( ( B = aa_real_nat(F,A) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),groups4696554848551431203al_nat(F,A4)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7527_dvd__prod__eqI,axiom,
    ! [A4: set_int,A: int,B: nat,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(A),A4)
       => ( ( B = aa_int_nat(F,A) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),groups1707563613775114915nt_nat(F,A4)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7528_dvd__prod__eqI,axiom,
    ! [A4: set_complex,A: complex,B: nat,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(A),A4)
       => ( ( B = aa_complex_nat(F,A) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),groups861055069439313189ex_nat(F,A4)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7529_dvd__prod__eqI,axiom,
    ! [A4: set_real,A: real,B: int,F: fun_real_int] :
      ( finite_finite_real(A4)
     => ( member_real(A,A4)
       => ( ( B = aa_real_int(F,A) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),groups4694064378042380927al_int(F,A4)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7530_dvd__prod__eqI,axiom,
    ! [A4: set_complex,A: complex,B: int,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(A),A4)
       => ( ( B = aa_complex_int(F,A) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),groups858564598930262913ex_int(F,A4)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7531_dvd__prod__eqI,axiom,
    ! [A4: set_real,A: real,B: code_integer,F: fun_re6978948838062957830nteger] :
      ( finite_finite_real(A4)
     => ( member_real(A,A4)
       => ( ( B = aa_real_Code_integer(F,A) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),groups6225526099057966256nteger(F,A4)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7532_dvd__prod__eqI,axiom,
    ! [A4: set_nat,A: nat,B: code_integer,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => ( ( B = aa_nat_Code_integer(F,A) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),groups3455450783089532116nteger(F,A4)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7533_dvd__prod__eqI,axiom,
    ! [A4: set_int,A: int,B: code_integer,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(A),A4)
       => ( ( B = aa_int_Code_integer(F,A) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),groups3827104343326376752nteger(F,A4)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7534_dvd__prod__eqI,axiom,
    ! [A4: set_complex,A: complex,B: code_integer,F: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(A),A4)
       => ( ( B = aa_com4217951221296830281nteger(F,A) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),groups8682486955453173170nteger(F,A4)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7535_dvd__prod__eqI,axiom,
    ! [A4: set_nat,A: nat,B: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => ( ( B = aa_nat_nat(F,A) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7536_dvd__prodI,axiom,
    ! [A4: set_real,A: real,F: fun_real_nat] :
      ( finite_finite_real(A4)
     => ( member_real(A,A4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_real_nat(F,A)),groups4696554848551431203al_nat(F,A4)) ) ) ).

% dvd_prodI
tff(fact_7537_dvd__prodI,axiom,
    ! [A4: set_int,A: int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(A),A4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(F,A)),groups1707563613775114915nt_nat(F,A4)) ) ) ).

% dvd_prodI
tff(fact_7538_dvd__prodI,axiom,
    ! [A4: set_complex,A: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(A),A4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_complex_nat(F,A)),groups861055069439313189ex_nat(F,A4)) ) ) ).

% dvd_prodI
tff(fact_7539_dvd__prodI,axiom,
    ! [A4: set_real,A: real,F: fun_real_int] :
      ( finite_finite_real(A4)
     => ( member_real(A,A4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_real_int(F,A)),groups4694064378042380927al_int(F,A4)) ) ) ).

% dvd_prodI
tff(fact_7540_dvd__prodI,axiom,
    ! [A4: set_complex,A: complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(A),A4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_complex_int(F,A)),groups858564598930262913ex_int(F,A4)) ) ) ).

% dvd_prodI
tff(fact_7541_dvd__prodI,axiom,
    ! [A4: set_real,A: real,F: fun_re6978948838062957830nteger] :
      ( finite_finite_real(A4)
     => ( member_real(A,A4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_real_Code_integer(F,A)),groups6225526099057966256nteger(F,A4)) ) ) ).

% dvd_prodI
tff(fact_7542_dvd__prodI,axiom,
    ! [A4: set_nat,A: nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(F,A)),groups3455450783089532116nteger(F,A4)) ) ) ).

% dvd_prodI
tff(fact_7543_dvd__prodI,axiom,
    ! [A4: set_int,A: int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(A),A4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_int_Code_integer(F,A)),groups3827104343326376752nteger(F,A4)) ) ) ).

% dvd_prodI
tff(fact_7544_dvd__prodI,axiom,
    ! [A4: set_complex,A: complex,F: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(A),A4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_com4217951221296830281nteger(F,A)),groups8682486955453173170nteger(F,A4)) ) ) ).

% dvd_prodI
tff(fact_7545_dvd__prodI,axiom,
    ! [A4: set_nat,A: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(F,A)),aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) ) ) ).

% dvd_prodI
tff(fact_7546_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_complex(gbinomial_complex(zero_zero_complex),aa_nat_nat(suc,K2)) = zero_zero_complex ).

% gbinomial_0(2)
tff(fact_7547_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,zero_zero_real),aa_nat_nat(suc,K2)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_7548_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K2)) = zero_zero_rat ).

% gbinomial_0(2)
tff(fact_7549_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,zero_zero_nat),aa_nat_nat(suc,K2)) = zero_zero_nat ).

% gbinomial_0(2)
tff(fact_7550_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K2)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_7551_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_7552_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_7553_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_7554_gbinomial__0_I1_J,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_7555_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_7556_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_7557_or__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_7558_or__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_7559_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( finite_finite_real(S2)
     => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_rb(A),B),S2) = $ite(member_real(A,S2),aa_real_complex(B,A),one_one_complex) ) ) ).

% prod.delta'
tff(fact_7560_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_complex(groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_rc(A),B)),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_complex(B,A),one_one_complex) ) ) ).

% prod.delta'
tff(fact_7561_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_rd(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_complex(B,A),one_one_complex) ) ) ).

% prod.delta'
tff(fact_7562_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_re(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_complex(B,A),one_one_complex) ) ) ).

% prod.delta'
tff(fact_7563_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S2)
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_rf(A),B),S2) = $ite(member_real(A,S2),aa_real_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_7564_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_rg(A),B)),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_7565_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_rh(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_7566_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ri(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_7567_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( finite_finite_real(S2)
     => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_rj(A),B),S2) = $ite(member_real(A,S2),aa_real_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_7568_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_rk(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_7569_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( finite_finite_real(S2)
     => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_rl(A),B),S2) = $ite(member_real(A,S2),aa_real_complex(B,A),one_one_complex) ) ) ).

% prod.delta
tff(fact_7570_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_complex(groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_rm(A),B)),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_complex(B,A),one_one_complex) ) ) ).

% prod.delta
tff(fact_7571_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_rn(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_complex(B,A),one_one_complex) ) ) ).

% prod.delta
tff(fact_7572_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_ro(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_complex(B,A),one_one_complex) ) ) ).

% prod.delta
tff(fact_7573_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S2)
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_rp(A),B),S2) = $ite(member_real(A,S2),aa_real_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_7574_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_rq(A),B)),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_7575_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_rr(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_7576_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_rs(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_7577_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( finite_finite_real(S2)
     => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_rt(A),B),S2) = $ite(member_real(A,S2),aa_real_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_7578_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ru(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_7579_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_real(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_7580_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_rat(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_7581_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_nat(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_7582_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_int(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_7583_cos__arccos,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arccos,Yb)) = Yb ) ) ) ).

% cos_arccos
tff(fact_7584_sin__arcsin,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Yb)) = Yb ) ) ) ).

% sin_arcsin
tff(fact_7585_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups6464643781859351333omplex(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_complex,aa_complex_complex(times_times_complex(aa_set_nat_complex(groups6464643781859351333omplex(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_7586_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_7587_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] :
      groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_rat,aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_7588_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_7589_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      groups705719431365010083at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_7590_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_7591_or__numerals_I7_J,axiom,
    ! [Xb: num,Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Yb))) = 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(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Yb)))) ).

% or_numerals(7)
tff(fact_7592_or__numerals_I7_J,axiom,
    ! [Xb: num,Yb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Yb)))) ).

% or_numerals(7)
tff(fact_7593_or__numerals_I6_J,axiom,
    ! [Xb: num,Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Yb))) = 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(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Yb)))) ).

% or_numerals(6)
tff(fact_7594_or__numerals_I6_J,axiom,
    ! [Xb: num,Yb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Yb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Yb)))) ).

% or_numerals(6)
tff(fact_7595_or__numerals_I4_J,axiom,
    ! [Xb: num,Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Yb))) = 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(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Yb)))) ).

% or_numerals(4)
tff(fact_7596_or__numerals_I4_J,axiom,
    ! [Xb: num,Yb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Yb)))) ).

% or_numerals(4)
tff(fact_7597_or__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & ( B = zero_zero_int ) ) ) ).

% or_eq_0_iff
tff(fact_7598_or__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% or_eq_0_iff
tff(fact_7599_bit_Odisj__zero__right,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),zero_zero_int) = Xb ).

% bit.disj_zero_right
tff(fact_7600_prod_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_nat,G: fun_real_fun_nat_nat,R: fun_real_fun_nat_o] :
      ( finite_finite_real(A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups4696554848551431203al_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_rv(B4),G),R),A4) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_rw(A4),G),R)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7601_prod_Oswap__restrict,axiom,
    ! [A4: set_int,B4: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups1707563613775114915nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_rx(B4),G),R),A4) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_ry(A4),G),R)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7602_prod_Oswap__restrict,axiom,
    ! [A4: set_complex,B4: set_nat,G: fun_co636942507509095880at_nat,R: fun_co5676208753217392844_nat_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups861055069439313189ex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_rz(B4),G),R),A4) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_sa(A4),G),R)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7603_prod_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_nat,G: fun_real_fun_nat_int,R: fun_real_fun_nat_o] :
      ( finite_finite_real(A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups4694064378042380927al_int(aa_fun7318160320247150695al_int(aa_fun4230461673125498710al_int(aTP_Lamm_sb(B4),G),R),A4) = groups705719431365010083at_int(aa_fun3188195345082886411at_int(aa_fun8006801000864960506at_int(aTP_Lamm_sd(A4),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7604_prod_Oswap__restrict,axiom,
    ! [A4: set_complex,B4: set_nat,G: fun_co7370231399432144932at_int,R: fun_co5676208753217392844_nat_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups858564598930262913ex_int(aa_fun7310784931243472999ex_int(aa_fun7210183792663747796ex_int(aTP_Lamm_se(B4),G),R),A4) = groups705719431365010083at_int(aa_fun1349975166126329225at_int(aa_fun7503395680861046006at_int(aTP_Lamm_sg(A4),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7605_prod_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_int,G: fun_real_fun_int_int,R: fun_real_fun_int_o] :
      ( finite_finite_real(A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups4694064378042380927al_int(aa_fun7555737461897725707al_int(aa_fun1679075639499870878al_int(aTP_Lamm_sh(B4),G),R),A4) = groups1705073143266064639nt_int(aa_fun3996843141751676939nt_int(aa_fun129049606478405534nt_int(aTP_Lamm_si(A4),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7606_prod_Oswap__restrict,axiom,
    ! [A4: set_complex,B4: set_int,G: fun_co333780937270390912nt_int,R: fun_co3431063785546946928_int_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups858564598930262913ex_int(aa_fun5055871773053899019ex_int(aa_fun7361901416012365340ex_int(aTP_Lamm_sj(B4),G),R),A4) = groups1705073143266064639nt_int(aa_fun2158622962795119753nt_int(aa_fun607866698265288346nt_int(aTP_Lamm_sk(A4),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7607_prod_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_real,G: fun_nat_fun_real_nat,R: fun_nat_fun_real_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( finite_finite_real(B4)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun6477086565944264623at_nat(aa_fun888394026836741882at_nat(aTP_Lamm_sm(B4),G),R)),A4) = groups4696554848551431203al_nat(aa_fun1455215566758156043al_nat(aa_fun1180509458258223190al_nat(aTP_Lamm_sp(A4),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7608_prod_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_int,G: fun_nat_fun_int_nat,R: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun8666973529875159599at_nat(aa_fun3399039426249668090at_nat(aTP_Lamm_sq(B4),G),R)),A4) = groups1707563613775114915nt_nat(aa_fun442955204921438347nt_nat(aa_fun5585961000942689878nt_nat(aTP_Lamm_ss(A4),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7609_prod_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_complex,G: fun_na8457291176974909128ex_nat,R: fun_na1554462459066888140plex_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun1031781741659083821at_nat(aa_fun7518049653624490486at_nat(aTP_Lamm_st(B4),G),R)),A4) = groups861055069439313189ex_nat(aa_fun1649587401935388427ex_nat(aa_fun3336366540202158548ex_nat(aTP_Lamm_sv(A4),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7610_prod__nonneg,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) ) ).

% prod_nonneg
tff(fact_7611_prod__nonneg,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A4)) ) ).

% prod_nonneg
tff(fact_7612_prod__nonneg,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A4)) ) ).

% prod_nonneg
tff(fact_7613_prod__mono,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I2)),aa_complex_real(G,I2)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A4)),groups766887009212190081x_real(G,A4)) ) ).

% prod_mono
tff(fact_7614_prod__mono,axiom,
    ! [A4: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I2: real] :
          ( member_real(I2,A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I2)),aa_real_real(G,I2)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A4)),groups1681761925125756287l_real(G,A4)) ) ).

% prod_mono
tff(fact_7615_prod__mono,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I2)),aa_nat_real(G,I2)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A4)),aa_set_nat_real(groups129246275422532515t_real(G),A4)) ) ).

% prod_mono
tff(fact_7616_prod__mono,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I2)),aa_int_real(G,I2)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A4)),groups2316167850115554303t_real(G,A4)) ) ).

% prod_mono
tff(fact_7617_prod__mono,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A4)),groups225925009352817453ex_rat(G,A4)) ) ).

% prod_mono
tff(fact_7618_prod__mono,axiom,
    ! [A4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( member_real(I2,A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A4)),groups4061424788464935467al_rat(G,A4)) ) ).

% prod_mono
tff(fact_7619_prod__mono,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A4)),groups73079841787564623at_rat(G,A4)) ) ).

% prod_mono
tff(fact_7620_prod__mono,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A4)),groups1072433553688619179nt_rat(G,A4)) ) ).

% prod_mono
tff(fact_7621_prod__mono,axiom,
    ! [A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),A4)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups861055069439313189ex_nat(F,A4)),groups861055069439313189ex_nat(G,A4)) ) ).

% prod_mono
tff(fact_7622_prod__mono,axiom,
    ! [A4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( member_real(I2,A4)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4696554848551431203al_nat(F,A4)),groups4696554848551431203al_nat(G,A4)) ) ).

% prod_mono
tff(fact_7623_prod__pos,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) ) ).

% prod_pos
tff(fact_7624_prod__pos,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups705719431365010083at_int(F,A4)) ) ).

% prod_pos
tff(fact_7625_prod__pos,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A4)) ) ).

% prod_pos
tff(fact_7626_prod__ge__1,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups766887009212190081x_real(F,A4)) ) ).

% prod_ge_1
tff(fact_7627_prod__ge__1,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A4)) ) ).

% prod_ge_1
tff(fact_7628_prod__ge__1,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),A4)) ) ).

% prod_ge_1
tff(fact_7629_prod__ge__1,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A4)) ) ).

% prod_ge_1
tff(fact_7630_prod__ge__1,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups225925009352817453ex_rat(F,A4)) ) ).

% prod_ge_1
tff(fact_7631_prod__ge__1,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( member_real(X3,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A4)) ) ).

% prod_ge_1
tff(fact_7632_prod__ge__1,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A4)) ) ).

% prod_ge_1
tff(fact_7633_prod__ge__1,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A4)) ) ).

% prod_ge_1
tff(fact_7634_prod__ge__1,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_complex_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups861055069439313189ex_nat(F,A4)) ) ).

% prod_ge_1
tff(fact_7635_prod__ge__1,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_real_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups4696554848551431203al_nat(F,A4)) ) ).

% prod_ge_1
tff(fact_7636_prod__zero,axiom,
    ! [A4: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
            & ( aa_nat_complex(F,X) = zero_zero_complex ) )
       => ( aa_set_nat_complex(groups6464643781859351333omplex(F),A4) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_7637_prod__zero,axiom,
    ! [A4: set_int,F: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ? [X: int] :
            ( aa_set_int_o(member_int(X),A4)
            & ( aa_int_complex(F,X) = zero_zero_complex ) )
       => ( groups7440179247065528705omplex(F,A4) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_7638_prod__zero,axiom,
    ! [A4: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ? [X: complex] :
            ( aa_set_complex_o(member_complex(X),A4)
            & ( aa_complex_complex(F,X) = zero_zero_complex ) )
       => ( groups3708469109370488835omplex(F,A4) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_7639_prod__zero,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
            & ( aa_nat_real(F,X) = zero_zero_real ) )
       => ( aa_set_nat_real(groups129246275422532515t_real(F),A4) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7640_prod__zero,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ? [X: int] :
            ( aa_set_int_o(member_int(X),A4)
            & ( aa_int_real(F,X) = zero_zero_real ) )
       => ( groups2316167850115554303t_real(F,A4) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7641_prod__zero,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ? [X: complex] :
            ( aa_set_complex_o(member_complex(X),A4)
            & ( aa_complex_real(F,X) = zero_zero_real ) )
       => ( groups766887009212190081x_real(F,A4) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7642_prod__zero,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
            & ( aa_nat_rat(F,X) = zero_zero_rat ) )
       => ( groups73079841787564623at_rat(F,A4) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7643_prod__zero,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ? [X: int] :
            ( aa_set_int_o(member_int(X),A4)
            & ( aa_int_rat(F,X) = zero_zero_rat ) )
       => ( groups1072433553688619179nt_rat(F,A4) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7644_prod__zero,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ? [X: complex] :
            ( aa_set_complex_o(member_complex(X),A4)
            & ( aa_complex_rat(F,X) = zero_zero_rat ) )
       => ( groups225925009352817453ex_rat(F,A4) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7645_prod__zero,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ? [X: int] :
            ( aa_set_int_o(member_int(X),A4)
            & ( aa_int_nat(F,X) = zero_zero_nat ) )
       => ( groups1707563613775114915nt_nat(F,A4) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_7646_or__greater__eq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L)) ) ).

% or_greater_eq
tff(fact_7647_OR__lower,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Yb)) ) ) ).

% OR_lower
tff(fact_7648_prod_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_complex,P2: fun_real_o] :
      ( finite_finite_real(A4)
     => ( groups713298508707869441omplex(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bf(A4),P2))) = groups713298508707869441omplex(aa_fun1098856492951973088omplex(aTP_Lamm_sw(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7649_prod_Ointer__filter,axiom,
    ! [A4: set_nat,G: fun_nat_complex,P2: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_complex(groups6464643781859351333omplex(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bi(A4),P2))) = aa_set_nat_complex(groups6464643781859351333omplex(aa_fun1400689748890728928omplex(aTP_Lamm_sx(G),P2)),A4) ) ) ).

% prod.inter_filter
tff(fact_7650_prod_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_complex,P2: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups7440179247065528705omplex(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bj(A4),P2))) = groups7440179247065528705omplex(aa_fun6439487526662521568omplex(aTP_Lamm_sy(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7651_prod_Ointer__filter,axiom,
    ! [A4: set_complex,G: fun_complex_complex,P2: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups3708469109370488835omplex(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(A4),P2))) = groups3708469109370488835omplex(aa_fun3681620399087435360omplex(aTP_Lamm_sz(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7652_prod_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_real,P2: fun_real_o] :
      ( finite_finite_real(A4)
     => ( groups1681761925125756287l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bf(A4),P2))) = groups1681761925125756287l_real(aa_fun4448911596306054750l_real(aTP_Lamm_ta(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7653_prod_Ointer__filter,axiom,
    ! [A4: set_nat,G: fun_nat_real,P2: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_real(groups129246275422532515t_real(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bi(A4),P2))) = aa_set_nat_real(groups129246275422532515t_real(aa_fun5510454223943551838t_real(aTP_Lamm_tb(G),P2)),A4) ) ) ).

% prod.inter_filter
tff(fact_7654_prod_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_real,P2: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups2316167850115554303t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bj(A4),P2))) = groups2316167850115554303t_real(aa_fun7506669883404309598t_real(aTP_Lamm_tc(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7655_prod_Ointer__filter,axiom,
    ! [A4: set_complex,G: fun_complex_real,P2: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups766887009212190081x_real(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(A4),P2))) = groups766887009212190081x_real(aa_fun8261057609414086622x_real(aTP_Lamm_td(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7656_prod_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_rat,P2: fun_real_o] :
      ( finite_finite_real(A4)
     => ( groups4061424788464935467al_rat(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bf(A4),P2))) = groups4061424788464935467al_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_te(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7657_prod_Ointer__filter,axiom,
    ! [A4: set_nat,G: fun_nat_rat,P2: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups73079841787564623at_rat(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bi(A4),P2))) = groups73079841787564623at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_tf(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7658_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ja(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_7659_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = groups705719431365010083at_int(aTP_Lamm_jm(G),set_or1269000886237332187st_nat(Mb,Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_7660_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K2: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jc(G),K2)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_7661_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,Mb: nat,K2: nat,Nb: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_tg(G),K2),set_or1269000886237332187st_nat(Mb,Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_7662_prod__le__1,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X3)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A4)),one_one_real) ) ).

% prod_le_1
tff(fact_7663_prod__le__1,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X3)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A4)),one_one_real) ) ).

% prod_le_1
tff(fact_7664_prod__le__1,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X3)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A4)),one_one_real) ) ).

% prod_le_1
tff(fact_7665_prod__le__1,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A4)),one_one_real) ) ).

% prod_le_1
tff(fact_7666_prod__le__1,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_7667_prod__le__1,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( member_real(X3,A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_7668_prod__le__1,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_7669_prod__le__1,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_7670_prod__le__1,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A4)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X3)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups861055069439313189ex_nat(F,A4)),one_one_nat) ) ).

% prod_le_1
tff(fact_7671_prod__le__1,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A4)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X3)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4696554848551431203al_nat(F,A4)),one_one_nat) ) ).

% prod_le_1
tff(fact_7672_prod_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,one_one_complex),one_one_complex)
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_nat_complex(H,X3)),aa_nat_complex(G,X3)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_set_nat_complex(groups6464643781859351333omplex(H),S2)),aa_set_nat_complex(groups6464643781859351333omplex(G),S2)) ) ) ) ) ).

% prod.related
tff(fact_7673_prod_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S2: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,one_one_complex),one_one_complex)
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_int_complex(H,X3)),aa_int_complex(G,X3)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,groups7440179247065528705omplex(H,S2)),groups7440179247065528705omplex(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_7674_prod_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S2: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,one_one_complex),one_one_complex)
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(H,X3)),aa_complex_complex(G,X3)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,groups3708469109370488835omplex(H,S2)),groups3708469109370488835omplex(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_7675_prod_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_nat,H: fun_nat_real,G: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_nat_real(H,X3)),aa_nat_real(G,X3)) )
           => aa_real_o(aa_real_fun_real_o(R,aa_set_nat_real(groups129246275422532515t_real(H),S2)),aa_set_nat_real(groups129246275422532515t_real(G),S2)) ) ) ) ) ).

% prod.related
tff(fact_7676_prod_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_int_real(H,X3)),aa_int_real(G,X3)) )
           => aa_real_o(aa_real_fun_real_o(R,groups2316167850115554303t_real(H,S2)),groups2316167850115554303t_real(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_7677_prod_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_complex_real(H,X3)),aa_complex_real(G,X3)) )
           => aa_real_o(aa_real_fun_real_o(R,groups766887009212190081x_real(H,S2)),groups766887009212190081x_real(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_7678_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X15),Y15)),aa_rat_rat(times_times_rat(X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H,X3)),aa_nat_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups73079841787564623at_rat(H,S2)),groups73079841787564623at_rat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_7679_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X15),Y15)),aa_rat_rat(times_times_rat(X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_int_rat(H,X3)),aa_int_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups1072433553688619179nt_rat(H,S2)),groups1072433553688619179nt_rat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_7680_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X15),Y15)),aa_rat_rat(times_times_rat(X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_complex_rat(H,X3)),aa_complex_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups225925009352817453ex_rat(H,S2)),groups225925009352817453ex_rat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_7681_prod_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,one_one_nat),one_one_nat)
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_int_nat(H,X3)),aa_int_nat(G,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups1707563613775114915nt_nat(H,S2)),groups1707563613775114915nt_nat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_7682_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [A3: real] :
              ( member_real(A3,A4)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_real_nat(F,A3)),aa_real_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups4696554848551431203al_nat(F,A4)),groups4696554848551431203al_nat(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7683_prod__dvd__prod__subset2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [A3: complex] :
              ( aa_set_complex_o(member_complex(A3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_complex_nat(F,A3)),aa_complex_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups861055069439313189ex_nat(F,A4)),groups861055069439313189ex_nat(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7684_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_int,G: fun_real_int] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [A3: real] :
              ( member_real(A3,A4)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_real_int(F,A3)),aa_real_int(G,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups4694064378042380927al_int(F,A4)),groups4694064378042380927al_int(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7685_prod__dvd__prod__subset2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [A3: complex] :
              ( aa_set_complex_o(member_complex(A3),A4)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_complex_int(F,A3)),aa_complex_int(G,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups858564598930262913ex_int(F,A4)),groups858564598930262913ex_int(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7686_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_re6978948838062957830nteger,G: fun_re6978948838062957830nteger] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [A3: real] :
              ( member_real(A3,A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_real_Code_integer(F,A3)),aa_real_Code_integer(G,A3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups6225526099057966256nteger(F,A4)),groups6225526099057966256nteger(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7687_prod__dvd__prod__subset2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat(A3),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(F,A3)),aa_nat_Code_integer(G,A3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3455450783089532116nteger(F,A4)),groups3455450783089532116nteger(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7688_prod__dvd__prod__subset2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_co8153928105989009288nteger,G: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [A3: complex] :
              ( aa_set_complex_o(member_complex(A3),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_com4217951221296830281nteger(F,A3)),aa_com4217951221296830281nteger(G,A3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups8682486955453173170nteger(F,A4)),groups8682486955453173170nteger(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7689_prod__dvd__prod__subset2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(F,A3)),aa_int_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A4)),groups1707563613775114915nt_nat(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7690_prod__dvd__prod__subset2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_int_Code_integer(F,A3)),aa_int_Code_integer(G,A3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3827104343326376752nteger(F,A4)),groups3827104343326376752nteger(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7691_prod__dvd__prod__subset2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat(A3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(F,A3)),aa_nat_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)),aa_set_nat_nat(groups708209901874060359at_nat(G),B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7692_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups861055069439313189ex_nat(F,A4)),groups861055069439313189ex_nat(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_7693_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups858564598930262913ex_int(F,A4)),groups858564598930262913ex_int(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_7694_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3455450783089532116nteger(F,A4)),groups3455450783089532116nteger(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_7695_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups8682486955453173170nteger(F,A4)),groups8682486955453173170nteger(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_7696_prod__dvd__prod__subset,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A4)),groups1707563613775114915nt_nat(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_7697_prod__dvd__prod__subset,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3827104343326376752nteger(F,A4)),groups3827104343326376752nteger(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_7698_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)),aa_set_nat_nat(groups708209901874060359at_nat(F),B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_7699_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups705719431365010083at_int(F,A4)),groups705719431365010083at_int(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_7700_prod__dvd__prod__subset,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups1705073143266064639nt_int(F,A4)),groups1705073143266064639nt_int(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_7701_prod__dvd__prod__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups4077766827762148844at_nat(F,A4)),groups4077766827762148844at_nat(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_7702_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(S5)
     => ( finite_finite_real(T3)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S2,S5))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S2,S5))
               => member_real(aa_real_real(J,A3),minus_minus_set_real(T,T3)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T3))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T3))
                   => member_real(aa_real_real(I,B3),minus_minus_set_real(S2,S5)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S5)
                     => ( aa_real_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T3)
                       => ( aa_real_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S2)
                         => ( aa_real_complex(H,aa_real_real(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups713298508707869441omplex(G,S2) = groups713298508707869441omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7703_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( finite_finite_real(S5)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S2,S5))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S2,S5))
               => aa_set_int_o(member_int(aa_real_int(J,A3)),minus_minus_set_int(T,T3)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(T,T3))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int(B3),minus_minus_set_int(T,T3))
                   => member_real(aa_int_real(I,B3),minus_minus_set_real(S2,S5)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S5)
                     => ( aa_real_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int(B3),T3)
                       => ( aa_int_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S2)
                         => ( aa_int_complex(H,aa_real_int(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups713298508707869441omplex(G,S2) = groups7440179247065528705omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7704_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_complex,S2: set_real,I: fun_complex_real,J: fun_real_complex,T: set_complex,G: fun_real_complex,H: fun_complex_complex] :
      ( finite_finite_real(S5)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S2,S5))
             => ( aa_complex_real(I,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S2,S5))
               => aa_set_complex_o(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T,T3)) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T,T3))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T,T3))
                   => member_real(aa_complex_real(I,B3),minus_minus_set_real(S2,S5)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S5)
                     => ( aa_real_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( aa_set_complex_o(member_complex(B3),T3)
                       => ( aa_complex_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S2)
                         => ( aa_complex_complex(H,aa_real_complex(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups713298508707869441omplex(G,S2) = groups3708469109370488835omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7705_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( aa_set_int_o(finite_finite_int,S5)
     => ( finite_finite_real(T3)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
               => member_real(aa_int_real(J,A3),minus_minus_set_real(T,T3)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T3))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T3))
                   => aa_set_int_o(member_int(aa_real_int(I,B3)),minus_minus_set_int(S2,S5)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S5)
                     => ( aa_int_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T3)
                       => ( aa_real_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S2)
                         => ( aa_real_complex(H,aa_int_real(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups7440179247065528705omplex(G,S2) = groups713298508707869441omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7706_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S5)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
               => aa_set_int_o(member_int(aa_int_int(J,A3)),minus_minus_set_int(T,T3)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(T,T3))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int(B3),minus_minus_set_int(T,T3))
                   => aa_set_int_o(member_int(aa_int_int(I,B3)),minus_minus_set_int(S2,S5)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S5)
                     => ( aa_int_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int(B3),T3)
                       => ( aa_int_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S2)
                         => ( aa_int_complex(H,aa_int_int(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups7440179247065528705omplex(G,S2) = groups7440179247065528705omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7707_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_complex,S2: set_int,I: fun_complex_int,J: fun_int_complex,T: set_complex,G: fun_int_complex,H: fun_complex_complex] :
      ( aa_set_int_o(finite_finite_int,S5)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
             => ( aa_complex_int(I,aa_int_complex(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S2,S5))
               => aa_set_complex_o(member_complex(aa_int_complex(J,A3)),minus_811609699411566653omplex(T,T3)) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T,T3))
                 => ( aa_int_complex(J,aa_complex_int(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T,T3))
                   => aa_set_int_o(member_int(aa_complex_int(I,B3)),minus_minus_set_int(S2,S5)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S5)
                     => ( aa_int_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( aa_set_complex_o(member_complex(B3),T3)
                       => ( aa_complex_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S2)
                         => ( aa_complex_complex(H,aa_int_complex(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups7440179247065528705omplex(G,S2) = groups3708469109370488835omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7708_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T3: set_real,S2: set_complex,I: fun_real_complex,J: fun_complex_real,T: set_real,G: fun_complex_complex,H: fun_real_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S5)
     => ( finite_finite_real(T3)
       => ( ! [A3: complex] :
              ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(S2,S5))
             => ( aa_real_complex(I,aa_complex_real(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(S2,S5))
               => member_real(aa_complex_real(J,A3),minus_minus_set_real(T,T3)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T3))
                 => ( aa_complex_real(J,aa_real_complex(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T3))
                   => aa_set_complex_o(member_complex(aa_real_complex(I,B3)),minus_811609699411566653omplex(S2,S5)) )
               => ( ! [A3: complex] :
                      ( aa_set_complex_o(member_complex(A3),S5)
                     => ( aa_complex_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T3)
                       => ( aa_real_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: complex] :
                          ( aa_set_complex_o(member_complex(A3),S2)
                         => ( aa_real_complex(H,aa_complex_real(J,A3)) = aa_complex_complex(G,A3) ) )
                     => ( groups3708469109370488835omplex(G,S2) = groups713298508707869441omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7709_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T3: set_int,S2: set_complex,I: fun_int_complex,J: fun_complex_int,T: set_int,G: fun_complex_complex,H: fun_int_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S5)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A3: complex] :
              ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(S2,S5))
             => ( aa_int_complex(I,aa_complex_int(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(S2,S5))
               => aa_set_int_o(member_int(aa_complex_int(J,A3)),minus_minus_set_int(T,T3)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(T,T3))
                 => ( aa_complex_int(J,aa_int_complex(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int(B3),minus_minus_set_int(T,T3))
                   => aa_set_complex_o(member_complex(aa_int_complex(I,B3)),minus_811609699411566653omplex(S2,S5)) )
               => ( ! [A3: complex] :
                      ( aa_set_complex_o(member_complex(A3),S5)
                     => ( aa_complex_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int(B3),T3)
                       => ( aa_int_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: complex] :
                          ( aa_set_complex_o(member_complex(A3),S2)
                         => ( aa_int_complex(H,aa_complex_int(J,A3)) = aa_complex_complex(G,A3) ) )
                     => ( groups3708469109370488835omplex(G,S2) = groups7440179247065528705omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7710_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T3: set_complex,S2: set_complex,I: fun_complex_complex,J: fun_complex_complex,T: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S5)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A3: complex] :
              ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(S2,S5))
             => ( aa_complex_complex(I,aa_complex_complex(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(S2,S5))
               => aa_set_complex_o(member_complex(aa_complex_complex(J,A3)),minus_811609699411566653omplex(T,T3)) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T,T3))
                 => ( aa_complex_complex(J,aa_complex_complex(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T,T3))
                   => aa_set_complex_o(member_complex(aa_complex_complex(I,B3)),minus_811609699411566653omplex(S2,S5)) )
               => ( ! [A3: complex] :
                      ( aa_set_complex_o(member_complex(A3),S5)
                     => ( aa_complex_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( aa_set_complex_o(member_complex(B3),T3)
                       => ( aa_complex_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: complex] :
                          ( aa_set_complex_o(member_complex(A3),S2)
                         => ( aa_complex_complex(H,aa_complex_complex(J,A3)) = aa_complex_complex(G,A3) ) )
                     => ( groups3708469109370488835omplex(G,S2) = groups3708469109370488835omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7711_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(S5)
     => ( finite_finite_real(T3)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S2,S5))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S2,S5))
               => member_real(aa_real_real(J,A3),minus_minus_set_real(T,T3)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T3))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T3))
                   => member_real(aa_real_real(I,B3),minus_minus_set_real(S2,S5)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S5)
                     => ( aa_real_real(G,A3) = one_one_real ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T3)
                       => ( aa_real_real(H,B3) = one_one_real ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S2)
                         => ( aa_real_real(H,aa_real_real(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups1681761925125756287l_real(G,S2) = groups1681761925125756287l_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7712_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(A),K2)),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_7713_gbinomial__Suc__Suc,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_7714_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(A),K2)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_7715_gbinomial__of__nat__symmetric,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),K2) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_7716_gbinomial__of__nat__symmetric,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),K2) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_7717_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_complex] :
      ( finite_finite_real(A4)
     => ( groups713298508707869441omplex(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_th(G)))) = groups713298508707869441omplex(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7718_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups7440179247065528705omplex(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_ti(G)))) = groups7440179247065528705omplex(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7719_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups3708469109370488835omplex(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_tj(G)))) = groups3708469109370488835omplex(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7720_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_real] :
      ( finite_finite_real(A4)
     => ( groups1681761925125756287l_real(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_tk(G)))) = groups1681761925125756287l_real(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7721_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups2316167850115554303t_real(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_tl(G)))) = groups2316167850115554303t_real(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7722_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups766887009212190081x_real(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_tm(G)))) = groups766887009212190081x_real(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7723_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_rat] :
      ( finite_finite_real(A4)
     => ( groups4061424788464935467al_rat(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_tn(G)))) = groups4061424788464935467al_rat(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7724_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups1072433553688619179nt_rat(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_to(G)))) = groups1072433553688619179nt_rat(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7725_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_tp(G)))) = groups225925009352817453ex_rat(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7726_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_nat] :
      ( finite_finite_real(A4)
     => ( groups4696554848551431203al_nat(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_tq(G)))) = groups4696554848551431203al_nat(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7727_exp__sum,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( aa_real_real(exp_real,groups8778361861064173332t_real(F,I6)) = groups2316167850115554303t_real(aTP_Lamm_tr(F),I6) ) ) ).

% exp_sum
tff(fact_7728_exp__sum,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( aa_real_real(exp_real,groups5808333547571424918x_real(F,I6)) = groups766887009212190081x_real(aTP_Lamm_ts(F),I6) ) ) ).

% exp_sum
tff(fact_7729_exp__sum,axiom,
    ! [I6: set_Pr1261947904930325089at_nat,F: fun_Pr672294278902479040t_real] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,I6)
     => ( aa_real_real(exp_real,groups4567486121110086003t_real(F,I6)) = groups6036352826371341000t_real(aTP_Lamm_tt(F),I6) ) ) ).

% exp_sum
tff(fact_7730_exp__sum,axiom,
    ! [I6: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( aa_complex_complex(exp_complex,groups7754918857620584856omplex(F,I6)) = groups3708469109370488835omplex(aTP_Lamm_tu(F),I6) ) ) ).

% exp_sum
tff(fact_7731_exp__sum,axiom,
    ! [I6: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( aa_real_real(exp_real,aa_set_nat_real(groups6591440286371151544t_real(F),I6)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_tv(F)),I6) ) ) ).

% exp_sum
tff(fact_7732_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mh(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nat_diff_reindex
tff(fact_7733_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_tw(G),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nat_diff_reindex
tff(fact_7734_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(Nb,Mb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ji(G),Nb),Mb)),set_or1269000886237332187st_nat(Nb,Mb)) ).

% prod.atLeastAtMost_rev
tff(fact_7735_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,Nb: nat,Mb: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(Nb,Mb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_tx(G),Nb),Mb),set_or1269000886237332187st_nat(Nb,Mb)) ).

% prod.atLeastAtMost_rev
tff(fact_7736_gbinomial__Suc,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)) = aa_rat_rat(divide_divide_rat(groups73079841787564623at_rat(aTP_Lamm_ty(A),set_or1269000886237332187st_nat(zero_zero_nat,K2))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_7737_gbinomial__Suc,axiom,
    ! [A: code_integer,K2: nat] : aa_nat_Code_integer(gbinom8545251970709558553nteger(A),aa_nat_nat(suc,K2)) = divide6298287555418463151nteger(groups3455450783089532116nteger(aTP_Lamm_tz(A),set_or1269000886237332187st_nat(zero_zero_nat,K2)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_7738_gbinomial__Suc,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_ua,A)),set_or1269000886237332187st_nat(zero_zero_nat,K2))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_7739_gbinomial__Suc,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2)) = aa_complex_complex(divide1717551699836669952omplex(aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_ub(A)),set_or1269000886237332187st_nat(zero_zero_nat,K2))),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_7740_gbinomial__Suc,axiom,
    ! [A: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uc,A)),set_or1269000886237332187st_nat(zero_zero_nat,K2))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_7741_gbinomial__Suc,axiom,
    ! [A: int,K2: nat] : aa_nat_int(gbinomial_int(A),aa_nat_nat(suc,K2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,groups705719431365010083at_int(aTP_Lamm_ud(A),set_or1269000886237332187st_nat(zero_zero_nat,K2))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_7742_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: fun_real_real] :
      ( finite_finite_real(I6)
     => ( member_real(I,I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7743_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( aa_set_nat_o(member_nat(I),I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7744_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( aa_set_int_o(member_int(I),I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7745_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( aa_set_complex_o(member_complex(I),I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_complex_real(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups766887009212190081x_real(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7746_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: fun_real_rat] :
      ( finite_finite_real(I6)
     => ( member_real(I,I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_real_rat(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7747_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( aa_set_nat_o(member_nat(I),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7748_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( aa_set_int_o(member_int(I),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7749_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( aa_set_complex_o(member_complex(I),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_complex_rat(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7750_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: fun_real_int] :
      ( finite_finite_real(I6)
     => ( member_real(I,I6)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I6)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups4694064378042380927al_int(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7751_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( aa_set_complex_o(member_complex(I),I6)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_complex_int(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I6)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_complex_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups858564598930262913ex_int(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7752_arccos__le__arccos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Yb)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_le_arccos
tff(fact_7753_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_complex_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups766887009212190081x_real(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_7754_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_7755_less__1__prod,axiom,
    ! [I6: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),I6)) ) ) ) ).

% less_1_prod
tff(fact_7756_less__1__prod,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_7757_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_complex_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_7758_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_rat] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_real_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_7759_less__1__prod,axiom,
    ! [I6: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_7760_less__1__prod,axiom,
    ! [I6: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_7761_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_complex_int(F,I2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups858564598930262913ex_int(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_7762_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_int] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(F,I2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups4694064378042380927al_int(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_7763_arccos__le__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb) ) ) ) ).

% arccos_le_mono
tff(fact_7764_arccos__eq__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real) )
     => ( ( aa_real_real(arccos,Xb) = aa_real_real(arccos,Yb) )
      <=> ( Xb = Yb ) ) ) ).

% arccos_eq_iff
tff(fact_7765_arcsin__le__arcsin,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Yb)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_7766_arcsin__minus,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,Xb)) ) ) ) ).

% arcsin_minus
tff(fact_7767_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups766887009212190081x_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G,minus_811609699411566653omplex(A4,B4))),groups766887009212190081x_real(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7768_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set_nat_real(groups129246275422532515t_real(G),A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),minus_minus_set_nat(A4,B4))),aa_set_nat_real(groups129246275422532515t_real(G),B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7769_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups225925009352817453ex_rat(G,A4) = aa_rat_rat(times_times_rat(groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A4,B4))),groups225925009352817453ex_rat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7770_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups73079841787564623at_rat(G,A4) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,minus_minus_set_nat(A4,B4))),groups73079841787564623at_rat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7771_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups861055069439313189ex_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A4,B4))),groups861055069439313189ex_nat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7772_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups858564598930262913ex_int(G,A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G,minus_811609699411566653omplex(A4,B4))),groups858564598930262913ex_int(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7773_prod_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_real] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups2316167850115554303t_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(G,minus_minus_set_int(A4,B4))),groups2316167850115554303t_real(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7774_prod_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_rat] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups1072433553688619179nt_rat(G,A4) = aa_rat_rat(times_times_rat(groups1072433553688619179nt_rat(G,minus_minus_set_int(A4,B4))),groups1072433553688619179nt_rat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7775_prod_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups1707563613775114915nt_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,minus_minus_set_int(A4,B4))),groups1707563613775114915nt_nat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7776_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),minus_minus_set_nat(A4,B4))),aa_set_nat_nat(groups708209901874060359at_nat(G),B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7777_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_complex(G,X3) = one_one_complex ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_complex(G,X3) = aa_real_complex(H,X3) ) )
           => ( groups713298508707869441omplex(G,T) = groups713298508707869441omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7778_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_complex(G,X3) = one_one_complex ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_complex(G,X3) = aa_complex_complex(H,X3) ) )
           => ( groups3708469109370488835omplex(G,T) = groups3708469109370488835omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7779_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_real(G,X3) = one_one_real ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_real(G,X3) = aa_real_real(H,X3) ) )
           => ( groups1681761925125756287l_real(G,T) = groups1681761925125756287l_real(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7780_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(G,X3) = one_one_real ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_real(G,X3) = aa_complex_real(H,X3) ) )
           => ( groups766887009212190081x_real(G,T) = groups766887009212190081x_real(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7781_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_rat(G,X3) = one_one_rat ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_rat(G,X3) = aa_real_rat(H,X3) ) )
           => ( groups4061424788464935467al_rat(G,T) = groups4061424788464935467al_rat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7782_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(G,X3) = one_one_rat ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_rat(G,X3) = aa_complex_rat(H,X3) ) )
           => ( groups225925009352817453ex_rat(G,T) = groups225925009352817453ex_rat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7783_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_nat(G,X3) = one_one_nat ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_nat(G,X3) = aa_real_nat(H,X3) ) )
           => ( groups4696554848551431203al_nat(G,T) = groups4696554848551431203al_nat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7784_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(G,X3) = one_one_nat ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_nat(G,X3) = aa_complex_nat(H,X3) ) )
           => ( groups861055069439313189ex_nat(G,T) = groups861055069439313189ex_nat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7785_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_int(G,X3) = one_one_int ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_int(G,X3) = aa_real_int(H,X3) ) )
           => ( groups4694064378042380927al_int(G,T) = groups4694064378042380927al_int(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7786_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(G,X3) = one_one_int ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_int(G,X3) = aa_complex_int(H,X3) ) )
           => ( groups858564598930262913ex_int(G,T) = groups858564598930262913ex_int(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7787_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_complex(H,X3) = one_one_complex ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_complex(G,X3) = aa_real_complex(H,X3) ) )
           => ( groups713298508707869441omplex(G,S2) = groups713298508707869441omplex(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7788_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_complex(H,X3) = one_one_complex ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_complex(G,X3) = aa_complex_complex(H,X3) ) )
           => ( groups3708469109370488835omplex(G,S2) = groups3708469109370488835omplex(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7789_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_real(H,X3) = one_one_real ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_real(G,X3) = aa_real_real(H,X3) ) )
           => ( groups1681761925125756287l_real(G,S2) = groups1681761925125756287l_real(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7790_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(H,X3) = one_one_real ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_real(G,X3) = aa_complex_real(H,X3) ) )
           => ( groups766887009212190081x_real(G,S2) = groups766887009212190081x_real(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7791_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_rat(H,X3) = one_one_rat ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_rat(G,X3) = aa_real_rat(H,X3) ) )
           => ( groups4061424788464935467al_rat(G,S2) = groups4061424788464935467al_rat(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7792_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(H,X3) = one_one_rat ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_rat(G,X3) = aa_complex_rat(H,X3) ) )
           => ( groups225925009352817453ex_rat(G,S2) = groups225925009352817453ex_rat(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7793_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_nat(H,X3) = one_one_nat ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_nat(G,X3) = aa_real_nat(H,X3) ) )
           => ( groups4696554848551431203al_nat(G,S2) = groups4696554848551431203al_nat(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7794_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(H,X3) = one_one_nat ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_nat(G,X3) = aa_complex_nat(H,X3) ) )
           => ( groups861055069439313189ex_nat(G,S2) = groups861055069439313189ex_nat(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7795_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_int,G: fun_real_int] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X3: real] :
              ( member_real(X3,minus_minus_set_real(T,S2))
             => ( aa_real_int(H,X3) = one_one_int ) )
         => ( ! [X3: real] :
                ( member_real(X3,S2)
               => ( aa_real_int(G,X3) = aa_real_int(H,X3) ) )
           => ( groups4694064378042380927al_int(G,S2) = groups4694064378042380927al_int(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7796_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(H,X3) = one_one_int ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_int(G,X3) = aa_complex_int(H,X3) ) )
           => ( groups858564598930262913ex_int(G,S2) = groups858564598930262913ex_int(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7797_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_complex(G,X3) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,T) = groups3708469109370488835omplex(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7798_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(G,X3) = one_one_real ) )
         => ( groups766887009212190081x_real(G,T) = groups766887009212190081x_real(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7799_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(G,X3) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,T) = groups225925009352817453ex_rat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7800_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(G,X3) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,T) = groups861055069439313189ex_nat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7801_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(G,X3) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,T) = groups858564598930262913ex_int(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7802_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T,S2))
             => ( aa_nat_complex(G,X3) = one_one_complex ) )
         => ( aa_set_nat_complex(groups6464643781859351333omplex(G),T) = aa_set_nat_complex(groups6464643781859351333omplex(G),S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7803_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T,S2))
             => ( aa_nat_real(G,X3) = one_one_real ) )
         => ( aa_set_nat_real(groups129246275422532515t_real(G),T) = aa_set_nat_real(groups129246275422532515t_real(G),S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7804_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T,S2))
             => ( aa_nat_rat(G,X3) = one_one_rat ) )
         => ( groups73079841787564623at_rat(G,T) = groups73079841787564623at_rat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7805_prod_Omono__neutral__right,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T,S2))
             => ( aa_int_complex(G,X3) = one_one_complex ) )
         => ( groups7440179247065528705omplex(G,T) = groups7440179247065528705omplex(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7806_prod_Omono__neutral__right,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T,S2))
             => ( aa_int_real(G,X3) = one_one_real ) )
         => ( groups2316167850115554303t_real(G,T) = groups2316167850115554303t_real(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7807_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_complex(G,X3) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,S2) = groups3708469109370488835omplex(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7808_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(G,X3) = one_one_real ) )
         => ( groups766887009212190081x_real(G,S2) = groups766887009212190081x_real(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7809_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(G,X3) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,S2) = groups225925009352817453ex_rat(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7810_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(G,X3) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,S2) = groups861055069439313189ex_nat(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7811_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(G,X3) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,S2) = groups858564598930262913ex_int(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7812_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T,S2))
             => ( aa_nat_complex(G,X3) = one_one_complex ) )
         => ( aa_set_nat_complex(groups6464643781859351333omplex(G),S2) = aa_set_nat_complex(groups6464643781859351333omplex(G),T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7813_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T,S2))
             => ( aa_nat_real(G,X3) = one_one_real ) )
         => ( aa_set_nat_real(groups129246275422532515t_real(G),S2) = aa_set_nat_real(groups129246275422532515t_real(G),T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7814_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T,S2))
             => ( aa_nat_rat(G,X3) = one_one_rat ) )
         => ( groups73079841787564623at_rat(G,S2) = groups73079841787564623at_rat(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7815_prod_Omono__neutral__left,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T,S2))
             => ( aa_int_complex(G,X3) = one_one_complex ) )
         => ( groups7440179247065528705omplex(G,S2) = groups7440179247065528705omplex(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7816_prod_Omono__neutral__left,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T,S2))
             => ( aa_int_real(G,X3) = one_one_real ) )
         => ( groups2316167850115554303t_real(G,S2) = groups2316167850115554303t_real(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7817_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_complex(H,B3) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,C4) = groups713298508707869441omplex(H,C4) )
               => ( groups713298508707869441omplex(G,A4) = groups713298508707869441omplex(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7818_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_complex(H,B3) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,C4) = groups3708469109370488835omplex(H,C4) )
               => ( groups3708469109370488835omplex(G,A4) = groups3708469109370488835omplex(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7819_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) )
               => ( groups1681761925125756287l_real(G,A4) = groups1681761925125756287l_real(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7820_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_real(G,A3) = one_one_real ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_real(H,B3) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,C4) = groups766887009212190081x_real(H,C4) )
               => ( groups766887009212190081x_real(G,A4) = groups766887009212190081x_real(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7821_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_rat(H,B3) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,C4) = groups4061424788464935467al_rat(H,C4) )
               => ( groups4061424788464935467al_rat(G,A4) = groups4061424788464935467al_rat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7822_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_rat(H,B3) = one_one_rat ) )
             => ( ( groups225925009352817453ex_rat(G,C4) = groups225925009352817453ex_rat(H,C4) )
               => ( groups225925009352817453ex_rat(G,A4) = groups225925009352817453ex_rat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7823_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_nat(H,B3) = one_one_nat ) )
             => ( ( groups4696554848551431203al_nat(G,C4) = groups4696554848551431203al_nat(H,C4) )
               => ( groups4696554848551431203al_nat(G,A4) = groups4696554848551431203al_nat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7824_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_nat(H,B3) = one_one_nat ) )
             => ( ( groups861055069439313189ex_nat(G,C4) = groups861055069439313189ex_nat(H,C4) )
               => ( groups861055069439313189ex_nat(G,A4) = groups861055069439313189ex_nat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7825_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_int(G,A3) = one_one_int ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_int(H,B3) = one_one_int ) )
             => ( ( groups4694064378042380927al_int(G,C4) = groups4694064378042380927al_int(H,C4) )
               => ( groups4694064378042380927al_int(G,A4) = groups4694064378042380927al_int(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7826_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_int(G,A3) = one_one_int ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_int(H,B3) = one_one_int ) )
             => ( ( groups858564598930262913ex_int(G,C4) = groups858564598930262913ex_int(H,C4) )
               => ( groups858564598930262913ex_int(G,A4) = groups858564598930262913ex_int(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7827_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_complex(H,B3) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,A4) = groups713298508707869441omplex(H,B4) )
              <=> ( groups713298508707869441omplex(G,C4) = groups713298508707869441omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7828_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_complex(H,B3) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,A4) = groups3708469109370488835omplex(H,B4) )
              <=> ( groups3708469109370488835omplex(G,C4) = groups3708469109370488835omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7829_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,A4) = groups1681761925125756287l_real(H,B4) )
              <=> ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7830_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_real(G,A3) = one_one_real ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_real(H,B3) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,A4) = groups766887009212190081x_real(H,B4) )
              <=> ( groups766887009212190081x_real(G,C4) = groups766887009212190081x_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7831_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_rat(H,B3) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,A4) = groups4061424788464935467al_rat(H,B4) )
              <=> ( groups4061424788464935467al_rat(G,C4) = groups4061424788464935467al_rat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7832_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_rat(H,B3) = one_one_rat ) )
             => ( ( groups225925009352817453ex_rat(G,A4) = groups225925009352817453ex_rat(H,B4) )
              <=> ( groups225925009352817453ex_rat(G,C4) = groups225925009352817453ex_rat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7833_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_nat(H,B3) = one_one_nat ) )
             => ( ( groups4696554848551431203al_nat(G,A4) = groups4696554848551431203al_nat(H,B4) )
              <=> ( groups4696554848551431203al_nat(G,C4) = groups4696554848551431203al_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7834_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_nat(H,B3) = one_one_nat ) )
             => ( ( groups861055069439313189ex_nat(G,A4) = groups861055069439313189ex_nat(H,B4) )
              <=> ( groups861055069439313189ex_nat(G,C4) = groups861055069439313189ex_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7835_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_int(G,A3) = one_one_int ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_int(H,B3) = one_one_int ) )
             => ( ( groups4694064378042380927al_int(G,A4) = groups4694064378042380927al_int(H,B4) )
              <=> ( groups4694064378042380927al_int(G,C4) = groups4694064378042380927al_int(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7836_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_int(G,A3) = one_one_int ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_int(H,B3) = one_one_int ) )
             => ( ( groups858564598930262913ex_int(G,A4) = groups858564598930262913ex_int(H,B4) )
              <=> ( groups858564598930262913ex_int(G,C4) = groups858564598930262913ex_int(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7837_arcsin__le__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ) ).

% arcsin_le_mono
tff(fact_7838_arcsin__eq__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
       => ( ( aa_real_real(arcsin,Xb) = aa_real_real(arcsin,Yb) )
        <=> ( Xb = Yb ) ) ) ) ).

% arcsin_eq_iff
tff(fact_7839_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7840_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7841_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7842_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7843_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,aa_nat_nat(suc,Nb))),aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7844_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7845_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7846_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),groups705719431365010083at_int(G,set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7847_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(Mb,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Mb)),aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7848_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(Mb,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,Mb)),groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7849_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7850_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(Mb,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7851_gbinomial__addition__formula,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(suc,K2))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ).

% gbinomial_addition_formula
tff(fact_7852_gbinomial__addition__formula,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K2)) ).

% gbinomial_addition_formula
tff(fact_7853_gbinomial__addition__formula,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K2)) ).

% gbinomial_addition_formula
tff(fact_7854_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K2)),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K2))),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_7855_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K2))),K2)),aa_nat_rat(gbinomial_rat(A),K2)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_7856_gbinomial__mult__1,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(A),K2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K2)),aa_nat_rat(gbinomial_rat(A),K2))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1
tff(fact_7857_gbinomial__mult__1,axiom,
    ! [A: real,K2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1
tff(fact_7858_gbinomial__mult__1,axiom,
    ! [A: complex,K2: nat] : aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(A),K2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K2)),aa_nat_complex(gbinomial_complex(A),K2))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1
tff(fact_7859_gbinomial__mult__1_H,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K2)),A) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K2)),aa_nat_rat(gbinomial_rat(A),K2))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1'
tff(fact_7860_gbinomial__mult__1_H,axiom,
    ! [A: real,K2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1'
tff(fact_7861_gbinomial__mult__1_H,axiom,
    ! [A: complex,K2: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K2)),A) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K2)),aa_nat_complex(gbinomial_complex(A),K2))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1'
tff(fact_7862_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_jb(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_7863_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_jl(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_7864_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ja(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_7865_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_jm(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_7866_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Mb)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_jb(G)),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7867_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,Mb)),groups73079841787564623at_rat(aTP_Lamm_jl(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7868_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ja(G)),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7869_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),groups705719431365010083at_int(aTP_Lamm_jm(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7870_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ja(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.atLeast1_atMost_eq
tff(fact_7871_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = groups705719431365010083at_int(aTP_Lamm_jm(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.atLeast1_atMost_eq
tff(fact_7872_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : aa_set_nat_complex(groups6464643781859351333omplex(F),set_or1269000886237332187st_nat(A,B)) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_ue(F),A),B),one_one_complex) ).

% prod_atLeastAtMost_code
tff(fact_7873_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : aa_set_nat_real(groups129246275422532515t_real(F),set_or1269000886237332187st_nat(A,B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_uf(F),A),B),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_7874_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups73079841787564623at_rat(F,set_or1269000886237332187st_nat(A,B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_ug(F),A,B),one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_7875_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups708209901874060359at_nat(F),set_or1269000886237332187st_nat(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_uh(F),A),B),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_7876_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups705719431365010083at_int(F,set_or1269000886237332187st_nat(A,B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_ui(F),A),B),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_7877_prod__mono__strict,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),A4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,I2)),aa_complex_real(G,I2)) ) )
       => ( ( A4 != bot_bot_set_complex )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups766887009212190081x_real(F,A4)),groups766887009212190081x_real(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7878_prod__mono__strict,axiom,
    ! [A4: set_real,F: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(A4)
     => ( ! [I2: real] :
            ( member_real(I2,A4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,I2)),aa_real_real(G,I2)) ) )
       => ( ( A4 != bot_bot_set_real )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1681761925125756287l_real(F,A4)),groups1681761925125756287l_real(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7879_prod__mono__strict,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),A4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,I2)),aa_nat_real(G,I2)) ) )
       => ( ( A4 != bot_bot_set_nat )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups129246275422532515t_real(F),A4)),aa_set_nat_real(groups129246275422532515t_real(G),A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7880_prod__mono__strict,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),A4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,I2)),aa_int_real(G,I2)) ) )
       => ( ( A4 != bot_bot_set_int )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups2316167850115554303t_real(F,A4)),groups2316167850115554303t_real(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7881_prod__mono__strict,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),A4)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2)) ) )
       => ( ( A4 != bot_bot_set_complex )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups225925009352817453ex_rat(F,A4)),groups225925009352817453ex_rat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7882_prod__mono__strict,axiom,
    ! [A4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(A4)
     => ( ! [I2: real] :
            ( member_real(I2,A4)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2)) ) )
       => ( ( A4 != bot_bot_set_real )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups4061424788464935467al_rat(F,A4)),groups4061424788464935467al_rat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7883_prod__mono__strict,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),A4)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) ) )
       => ( ( A4 != bot_bot_set_nat )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups73079841787564623at_rat(F,A4)),groups73079841787564623at_rat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7884_prod__mono__strict,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),A4)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) ) )
       => ( ( A4 != bot_bot_set_int )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1072433553688619179nt_rat(F,A4)),groups1072433553688619179nt_rat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7885_prod__mono__strict,axiom,
    ! [A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),A4)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2))
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) ) )
       => ( ( A4 != bot_bot_set_complex )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups861055069439313189ex_nat(F,A4)),groups861055069439313189ex_nat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7886_prod__mono__strict,axiom,
    ! [A4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(A4)
     => ( ! [I2: real] :
            ( member_real(I2,A4)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2))
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) ) )
       => ( ( A4 != bot_bot_set_real )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4696554848551431203al_nat(F,A4)),groups4696554848551431203al_nat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7887_even__prod__iff,axiom,
    ! [A4: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups3455450783089532116nteger(F,A4))
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_7888_even__prod__iff,axiom,
    ! [A4: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups3827104343326376752nteger(F,A4))
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A4)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_7889_even__prod__iff,axiom,
    ! [A4: set_complex,F: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups8682486955453173170nteger(F,A4))
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A4)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_com4217951221296830281nteger(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_7890_even__prod__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups1707563613775114915nt_nat(F,A4))
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A4)
            & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_7891_even__prod__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups861055069439313189ex_nat(F,A4))
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A4)
            & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_complex_nat(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_7892_even__prod__iff,axiom,
    ! [A4: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups858564598930262913ex_int(F,A4))
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A4)
            & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_complex_int(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_7893_even__prod__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups708209901874060359at_nat(F),A4))
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_7894_even__prod__iff,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups705719431365010083at_int(F,A4))
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_7895_even__prod__iff,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups1705073143266064639nt_int(F,A4))
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A4)
            & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_7896_even__prod__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr6026251430685297nteger] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups1230400874837758585nteger(F,A4))
      <=> ? [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Pro4764841431850000930nteger(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_7897_arccos__lbound,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Yb)) ) ) ).

% arccos_lbound
tff(fact_7898_arccos__less__arccos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Yb)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_less_arccos
tff(fact_7899_arccos__less__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ) ) ).

% arccos_less_mono
tff(fact_7900_arccos__ubound,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Yb)),pi) ) ) ).

% arccos_ubound
tff(fact_7901_arccos__cos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xb)) = Xb ) ) ) ).

% arccos_cos
tff(fact_7902_arcsin__less__arcsin,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Yb)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_7903_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_7904_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))),groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_7905_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_7906_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(Mb,Nb))),groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_7907_arcsin__less__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ) ).

% arcsin_less_mono
tff(fact_7908_cos__arccos__abs,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
     => ( aa_real_real(cos_real,aa_real_real(arccos,Yb)) = Yb ) ) ).

% cos_arccos_abs
tff(fact_7909_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi)
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_7910_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K2))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K2)) ).

% Suc_times_gbinomial
tff(fact_7911_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) ).

% Suc_times_gbinomial
tff(fact_7912_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2))),aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(gbinomial_complex(A),K2)) ).

% Suc_times_gbinomial
tff(fact_7913_gbinomial__absorption,axiom,
    ! [K2: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K2)) ).

% gbinomial_absorption
tff(fact_7914_gbinomial__absorption,axiom,
    ! [K2: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K2)) ).

% gbinomial_absorption
tff(fact_7915_gbinomial__absorption,axiom,
    ! [K2: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ).

% gbinomial_absorption
tff(fact_7916_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,Mb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),Mb)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),K2)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7917_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,Mb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7918_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,Mb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),Mb)),aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),K2)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K2)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7919_norm__prod__diff,axiom,
    ! [I6: set_complex,Z: fun_complex_real,W2: fun_complex_real] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_complex_real(Z,I2))),one_one_real) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_complex_real(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups766887009212190081x_real(Z,I6)),groups766887009212190081x_real(W2,I6)))),groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_uj(Z),W2),I6)) ) ) ).

% norm_prod_diff
tff(fact_7920_norm__prod__diff,axiom,
    ! [I6: set_real,Z: fun_real_real,W2: fun_real_real] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(Z,I2))),one_one_real) )
     => ( ! [I2: real] :
            ( member_real(I2,I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups1681761925125756287l_real(Z,I6)),groups1681761925125756287l_real(W2,I6)))),groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_uk(Z),W2),I6)) ) ) ).

% norm_prod_diff
tff(fact_7921_norm__prod__diff,axiom,
    ! [I6: set_set_nat,Z: fun_set_nat_real,W2: fun_set_nat_real] :
      ( ! [I2: set_nat] :
          ( member_set_nat(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(Z,I2))),one_one_real) )
     => ( ! [I2: set_nat] :
            ( member_set_nat(I2,I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups3619160379726066777t_real(Z,I6)),groups3619160379726066777t_real(W2,I6)))),groups5107569545109728110t_real(aa_fun3569254276315353500t_real(aTP_Lamm_ul(Z),W2),I6)) ) ) ).

% norm_prod_diff
tff(fact_7922_norm__prod__diff,axiom,
    ! [I6: set_int,Z: fun_int_real,W2: fun_int_real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_int_real(Z,I2))),one_one_real) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_int_real(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups2316167850115554303t_real(Z,I6)),groups2316167850115554303t_real(W2,I6)))),groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_um(Z),W2),I6)) ) ) ).

% norm_prod_diff
tff(fact_7923_norm__prod__diff,axiom,
    ! [I6: set_complex,Z: fun_complex_complex,W2: fun_complex_complex] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Z,I2))),one_one_real) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(groups3708469109370488835omplex(Z,I6)),groups3708469109370488835omplex(W2,I6)))),groups5808333547571424918x_real(aa_fun1953249243402654830x_real(aTP_Lamm_un(Z),W2),I6)) ) ) ).

% norm_prod_diff
tff(fact_7924_norm__prod__diff,axiom,
    ! [I6: set_real,Z: fun_real_complex,W2: fun_real_complex] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(Z,I2))),one_one_real) )
     => ( ! [I2: real] :
            ( member_real(I2,I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(groups713298508707869441omplex(Z,I6)),groups713298508707869441omplex(W2,I6)))),groups8097168146408367636l_real(aa_fun2786211375485122666l_real(aTP_Lamm_uo(Z),W2),I6)) ) ) ).

% norm_prod_diff
tff(fact_7925_norm__prod__diff,axiom,
    ! [I6: set_set_nat,Z: fun_set_nat_complex,W2: fun_set_nat_complex] :
      ( ! [I2: set_nat] :
          ( member_set_nat(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(Z,I2))),one_one_real) )
     => ( ! [I2: set_nat] :
            ( member_set_nat(I2,I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(groups1092910753850256091omplex(Z,I6)),groups1092910753850256091omplex(W2,I6)))),groups5107569545109728110t_real(aa_fun5767801665285160478t_real(aTP_Lamm_up(Z),W2),I6)) ) ) ).

% norm_prod_diff
tff(fact_7926_norm__prod__diff,axiom,
    ! [I6: set_int,Z: fun_int_complex,W2: fun_int_complex] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(Z,I2))),one_one_real) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(groups7440179247065528705omplex(Z,I6)),groups7440179247065528705omplex(W2,I6)))),groups8778361861064173332t_real(aa_fun1985759105522019690t_real(aTP_Lamm_uq(Z),W2),I6)) ) ) ).

% norm_prod_diff
tff(fact_7927_norm__prod__diff,axiom,
    ! [I6: set_nat,Z: fun_nat_real,W2: fun_nat_real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Z,I2))),one_one_real) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups129246275422532515t_real(Z),I6)),aa_set_nat_real(groups129246275422532515t_real(W2),I6)))),aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ur(Z),W2)),I6)) ) ) ).

% norm_prod_diff
tff(fact_7928_norm__prod__diff,axiom,
    ! [I6: set_nat,Z: fun_nat_complex,W2: fun_nat_complex] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Z,I2))),one_one_real) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_set_nat_complex(groups6464643781859351333omplex(Z),I6)),aa_set_nat_complex(groups6464643781859351333omplex(W2),I6)))),aa_set_nat_real(groups6591440286371151544t_real(aa_fun7054779078796324018t_real(aTP_Lamm_us(Z),W2)),I6)) ) ) ).

% norm_prod_diff
tff(fact_7929_fact__eq__fact__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_kr),set_or1269000886237332187st_nat(aa_nat_nat(suc,Nb),Mb))) ) ) ).

% fact_eq_fact_times
tff(fact_7930_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_real] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,B3)) )
         => ( ! [A3: real] :
                ( member_real(A3,A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A4)),groups1681761925125756287l_real(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7931_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,B3)) )
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A4)),groups766887009212190081x_real(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7932_prod__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,B3)) )
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A4)),aa_set_nat_real(groups129246275422532515t_real(F),B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7933_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_rat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,B3)) )
         => ( ! [A3: real] :
                ( member_real(A3,A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A4)),groups4061424788464935467al_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7934_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,B3)) )
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A4)),groups225925009352817453ex_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7935_prod__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,B3)) )
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A4)),groups73079841787564623at_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7936_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_int] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B4,A4))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(F,B3)) )
         => ( ! [A3: real] :
                ( member_real(A3,A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,A3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4694064378042380927al_int(F,A4)),groups4694064378042380927al_int(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7937_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B4,A4))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_complex_int(F,B3)) )
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,A3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups858564598930262913ex_int(F,A4)),groups858564598930262913ex_int(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7938_prod__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( ! [B3: int] :
              ( aa_set_int_o(member_int(B3),minus_minus_set_int(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,B3)) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A4)),groups2316167850115554303t_real(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7939_prod__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( ! [B3: int] :
              ( aa_set_int_o(member_int(B3),minus_minus_set_int(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,B3)) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A4)),groups1072433553688619179nt_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7940_arccos__lt__bounded,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arccos,Yb))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Yb)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_7941_arccos__bounded,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Yb))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Yb)),pi) ) ) ) ).

% arccos_bounded
tff(fact_7942_gbinomial__rec,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K2)),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_7943_gbinomial__rec,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_7944_gbinomial__rec,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K2)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_7945_gbinomial__factors,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2)))),aa_nat_rat(gbinomial_rat(A),K2)) ).

% gbinomial_factors
tff(fact_7946_gbinomial__factors,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) ).

% gbinomial_factors
tff(fact_7947_gbinomial__factors,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2)))),aa_nat_complex(gbinomial_complex(A),K2)) ).

% gbinomial_factors
tff(fact_7948_sin__arccos__nonzero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xb)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_7949_arccos__cos2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Xb)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xb)) = aa_real_real(uminus_uminus_real,Xb) ) ) ) ).

% arccos_cos2
tff(fact_7950_arccos__minus,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_minus
tff(fact_7951_cos__arcsin__nonzero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xb)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_7952_pochhammer__Suc__prod,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = groups73079841787564623at_rat(aTP_Lamm_ut(A),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% pochhammer_Suc_prod
tff(fact_7953_pochhammer__Suc__prod,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_uu,A)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% pochhammer_Suc_prod
tff(fact_7954_pochhammer__Suc__prod,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_uv(A)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% pochhammer_Suc_prod
tff(fact_7955_pochhammer__Suc__prod,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,Nb)) = groups3455450783089532116nteger(aTP_Lamm_uw(A),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% pochhammer_Suc_prod
tff(fact_7956_pochhammer__Suc__prod,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ux,A)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% pochhammer_Suc_prod
tff(fact_7957_pochhammer__Suc__prod,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = groups705719431365010083at_int(aTP_Lamm_uy(A),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% pochhammer_Suc_prod
tff(fact_7958_pochhammer__prod__rev,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_uz(A),Nb),set_or1269000886237332187st_nat(one_one_nat,Nb)) ).

% pochhammer_prod_rev
tff(fact_7959_pochhammer__prod__rev,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_va(A),Nb)),set_or1269000886237332187st_nat(one_one_nat,Nb)) ).

% pochhammer_prod_rev
tff(fact_7960_pochhammer__prod__rev,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = aa_set_nat_complex(groups6464643781859351333omplex(aa_nat6454746962644518025omplex(aTP_Lamm_vb(A),Nb)),set_or1269000886237332187st_nat(one_one_nat,Nb)) ).

% pochhammer_prod_rev
tff(fact_7961_pochhammer__prod__rev,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),Nb) = groups3455450783089532116nteger(aa_nat2010205248571845816nteger(aTP_Lamm_vc(A),Nb),set_or1269000886237332187st_nat(one_one_nat,Nb)) ).

% pochhammer_prod_rev
tff(fact_7962_pochhammer__prod__rev,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vd(A),Nb)),set_or1269000886237332187st_nat(one_one_nat,Nb)) ).

% pochhammer_prod_rev
tff(fact_7963_pochhammer__prod__rev,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ve(A),Nb),set_or1269000886237332187st_nat(one_one_nat,Nb)) ).

% pochhammer_prod_rev
tff(fact_7964_fact__div__fact,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_kr),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),Mb)) ) ) ).

% fact_div_fact
tff(fact_7965_one__or__eq,axiom,
    ! [A: code_integer] : bit_se1080825931792720795nteger(one_one_Code_integer,A) = plus_p5714425477246183910nteger(A,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).

% one_or_eq
tff(fact_7966_one__or__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) ).

% one_or_eq
tff(fact_7967_one__or__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) ).

% one_or_eq
tff(fact_7968_or__one__eq,axiom,
    ! [A: code_integer] : bit_se1080825931792720795nteger(A,one_one_Code_integer) = plus_p5714425477246183910nteger(A,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).

% or_one_eq
tff(fact_7969_or__one__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) ).

% or_one_eq
tff(fact_7970_or__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) ).

% or_one_eq
tff(fact_7971_OR__upper,axiom,
    ! [Xb: int,Nb: nat,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Yb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% OR_upper
tff(fact_7972_gbinomial__minus,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K2) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)) ).

% gbinomial_minus
tff(fact_7973_gbinomial__minus,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,A)),K2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)) ).

% gbinomial_minus
tff(fact_7974_gbinomial__minus,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)) ).

% gbinomial_minus
tff(fact_7975_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_vf(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% prod.in_pairs
tff(fact_7976_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] : groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups73079841787564623at_rat(aTP_Lamm_vg(G),set_or1269000886237332187st_nat(Mb,Nb)) ).

% prod.in_pairs
tff(fact_7977_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_vh(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% prod.in_pairs
tff(fact_7978_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups705719431365010083at_int(aTP_Lamm_vi(G),set_or1269000886237332187st_nat(Mb,Nb)) ).

% prod.in_pairs
tff(fact_7979_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7980_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7981_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7982_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_uz(A),Nb),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_7983_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_va(A),Nb)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_7984_pochhammer__Suc__prod__rev,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = aa_set_nat_complex(groups6464643781859351333omplex(aa_nat6454746962644518025omplex(aTP_Lamm_vb(A),Nb)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_7985_pochhammer__Suc__prod__rev,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,Nb)) = groups3455450783089532116nteger(aa_nat2010205248571845816nteger(aTP_Lamm_vc(A),Nb),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_7986_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vd(A),Nb)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_7987_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ve(A),Nb),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_7988_arccos,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Yb))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Yb)),pi)
          & ( aa_real_real(cos_real,aa_real_real(arccos,Yb)) = Yb ) ) ) ) ).

% arccos
tff(fact_7989_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)),aa_nat_rat(semiri773545260158071498ct_rat,K2)) ).

% gbinomial_pochhammer'
tff(fact_7990_gbinomial__pochhammer_H,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)),aa_nat_real(semiri2265585572941072030t_real,K2)) ).

% gbinomial_pochhammer'
tff(fact_7991_gbinomial__pochhammer_H,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)),aa_nat_complex(semiri5044797733671781792omplex,K2)) ).

% gbinomial_pochhammer'
tff(fact_7992_arccos__minus__abs,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xb)) ) ) ).

% arccos_minus_abs
tff(fact_7993_or__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
            | ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,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,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_7994_gbinomial__sum__up__index,axiom,
    ! [K2: nat,Nb: nat] : groups2906978787729119204at_rat(aTP_Lamm_vj(K2),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_7995_gbinomial__sum__up__index,axiom,
    ! [K2: nat,Nb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_vk(K2)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_7996_gbinomial__sum__up__index,axiom,
    ! [K2: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_vl,K2)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_7997_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7998_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7999_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8000_arccos__le__pi2,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Yb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_8001_arcsin__lbound,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Yb)) ) ) ).

% arcsin_lbound
tff(fact_8002_arcsin__ubound,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Yb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_8003_arcsin__bounded,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Yb))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Yb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_bounded
tff(fact_8004_arcsin__sin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,Xb)) = Xb ) ) ) ).

% arcsin_sin
tff(fact_8005_arcsin,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Yb))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Yb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Yb)) = Yb ) ) ) ) ).

% arcsin
tff(fact_8006_arcsin__pi,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Yb))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Yb)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Yb)) = Yb ) ) ) ) ).

% arcsin_pi
tff(fact_8007_arcsin__le__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),Yb)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sin_real,Yb)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_8008_le__arcsin__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(arcsin,Xb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Yb)),Xb) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_8009_gbinomial__code,axiom,
    ! [A: rat,K2: nat] :
      aa_nat_rat(gbinomial_rat(A),K2) = $ite(K2 = zero_zero_nat,one_one_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_vm(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat)),one_one_rat)),aa_nat_rat(semiri773545260158071498ct_rat,K2))) ).

% gbinomial_code
tff(fact_8010_gbinomial__code,axiom,
    ! [A: real,K2: nat] :
      aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = $ite(K2 = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_vn(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat)),one_one_real)),aa_nat_real(semiri2265585572941072030t_real,K2))) ).

% gbinomial_code
tff(fact_8011_gbinomial__code,axiom,
    ! [A: complex,K2: nat] :
      aa_nat_complex(gbinomial_complex(A),K2) = $ite(K2 = zero_zero_nat,one_one_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_vo(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat)),one_one_complex)),aa_nat_complex(semiri5044797733671781792omplex,K2))) ).

% gbinomial_code
tff(fact_8012_gbinomial__partial__row__sum,axiom,
    ! [A: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_qv(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),one_one_rat)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8013_gbinomial__partial__row__sum,axiom,
    ! [A: complex,Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_qw(A)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),one_one_complex)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8014_gbinomial__partial__row__sum,axiom,
    ! [A: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qx,A)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8015_gbinomial__r__part__sum,axiom,
    ! [Mb: nat] : groups2906978787729119204at_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Mb))),one_one_rat)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% gbinomial_r_part_sum
tff(fact_8016_gbinomial__r__part__sum,axiom,
    ! [Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Mb))),one_one_complex))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% gbinomial_r_part_sum
tff(fact_8017_gbinomial__r__part__sum,axiom,
    ! [Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),one_one_real))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% gbinomial_r_part_sum
tff(fact_8018_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_8019_Maclaurin__sin__bound,axiom,
    ! [Xb: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_py,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Xb)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_8020_divmod__BitM__2__eq,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(aa_num_num(bitM,Mb),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_8021_atMost__eq__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( aa_nat_set_nat(set_ord_atMost_nat,Xb) = aa_nat_set_nat(set_ord_atMost_nat,Yb) )
    <=> ( Xb = Yb ) ) ).

% atMost_eq_iff
tff(fact_8022_atMost__eq__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( ( set_ord_atMost_int(Xb) = set_ord_atMost_int(Yb) )
    <=> ( Xb = Yb ) ) ).

% atMost_eq_iff
tff(fact_8023_inverse__nonzero__iff__nonzero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_8024_inverse__nonzero__iff__nonzero,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_8025_inverse__nonzero__iff__nonzero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_8026_inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% inverse_zero
tff(fact_8027_inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% inverse_zero
tff(fact_8028_inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% inverse_zero
tff(fact_8029_atMost__iff,axiom,
    ! [I: real,K2: real] :
      ( member_real(I,set_ord_atMost_real(K2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),K2) ) ).

% atMost_iff
tff(fact_8030_atMost__iff,axiom,
    ! [I: set_nat,K2: set_nat] :
      ( member_set_nat(I,set_or4236626031148496127et_nat(K2))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,I),K2) ) ).

% atMost_iff
tff(fact_8031_atMost__iff,axiom,
    ! [I: set_int,K2: set_int] :
      ( member_set_int(I,set_or58775011639299419et_int(K2))
    <=> aa_set_int_o(ord_less_eq_set_int(I),K2) ) ).

% atMost_iff
tff(fact_8032_atMost__iff,axiom,
    ! [I: rat,K2: rat] :
      ( member_rat(I,set_ord_atMost_rat(K2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),K2) ) ).

% atMost_iff
tff(fact_8033_atMost__iff,axiom,
    ! [I: num,K2: num] :
      ( member_num(I,set_ord_atMost_num(K2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),K2) ) ).

% atMost_iff
tff(fact_8034_atMost__iff,axiom,
    ! [I: nat,K2: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_ord_atMost_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2) ) ).

% atMost_iff
tff(fact_8035_atMost__iff,axiom,
    ! [I: int,K2: int] :
      ( aa_set_int_o(member_int(I),set_ord_atMost_int(K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K2) ) ).

% atMost_iff
tff(fact_8036_binomial__Suc__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Nb) = aa_nat_nat(suc,Nb) ).

% binomial_Suc_n
tff(fact_8037_finite__atMost,axiom,
    ! [K2: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_ord_atMost_nat,K2)) ).

% finite_atMost
tff(fact_8038_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_8039_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_8040_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_8041_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_8042_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% inverse_positive_iff_positive
tff(fact_8043_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% inverse_positive_iff_positive
tff(fact_8044_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% inverse_negative_iff_negative
tff(fact_8045_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% inverse_negative_iff_negative
tff(fact_8046_inverse__less__iff__less__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_8047_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_8048_inverse__less__iff__less,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% inverse_less_iff_less
tff(fact_8049_inverse__less__iff__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% inverse_less_iff_less
tff(fact_8050_atMost__subset__iff,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( ord_le4403425263959731960et_int(set_or58775011639299419et_int(Xb),set_or58775011639299419et_int(Yb))
    <=> aa_set_int_o(ord_less_eq_set_int(Xb),Yb) ) ).

% atMost_subset_iff
tff(fact_8051_atMost__subset__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(Xb),set_ord_atMost_rat(Yb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ).

% atMost_subset_iff
tff(fact_8052_atMost__subset__iff,axiom,
    ! [Xb: num,Yb: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(Xb),set_ord_atMost_num(Yb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb) ) ).

% atMost_subset_iff
tff(fact_8053_atMost__subset__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,Xb)),aa_nat_set_nat(set_ord_atMost_nat,Yb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ).

% atMost_subset_iff
tff(fact_8054_atMost__subset__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_atMost_int(Xb)),set_ord_atMost_int(Yb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ).

% atMost_subset_iff
tff(fact_8055_binomial__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,zero_zero_nat)) = Nb ).

% binomial_1
tff(fact_8056_binomial__0__Suc,axiom,
    ! [K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K2)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_8057_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% binomial_eq_0_iff
tff(fact_8058_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2))) ).

% binomial_Suc_Suc
tff(fact_8059_binomial__n__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_8060_prod__eq__1__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( groups1707563613775114915nt_nat(F,A4) = one_one_nat )
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),A4)
           => ( aa_int_nat(F,X4) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8061_prod__eq__1__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( groups861055069439313189ex_nat(F,A4) = one_one_nat )
      <=> ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A4)
           => ( aa_complex_nat(F,X4) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8062_prod__eq__1__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( ( groups4077766827762148844at_nat(F,A4) = one_one_nat )
      <=> ! [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
           => ( aa_Pro4243173770044397973at_nat(F,X4) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8063_prod__eq__1__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_set_nat_nat(groups708209901874060359at_nat(F),A4) = one_one_nat )
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
           => ( aa_nat_nat(F,X4) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8064_inverse__le__iff__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% inverse_le_iff_le
tff(fact_8065_inverse__le__iff__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% inverse_le_iff_le
tff(fact_8066_inverse__le__iff__le__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_8067_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_8068_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,A)) = one_one_real ) ) ).

% right_inverse
tff(fact_8069_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,A)) = one_one_complex ) ) ).

% right_inverse
tff(fact_8070_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,A)) = one_one_rat ) ) ).

% right_inverse
tff(fact_8071_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% left_inverse
tff(fact_8072_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% left_inverse
tff(fact_8073_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% left_inverse
tff(fact_8074_or__nat__numerals_I2_J,axiom,
    ! [Yb: 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,Yb))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb)) ).

% or_nat_numerals(2)
tff(fact_8075_or__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_nat_numerals(4)
tff(fact_8076_Icc__subset__Iic__iff,axiom,
    ! [L: set_int,H: set_int,H2: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(L,H),set_or58775011639299419et_int(H2))
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(L),H)
        | aa_set_int_o(ord_less_eq_set_int(H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8077_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H: rat,H2: rat] :
      ( ord_less_eq_set_rat(set_or633870826150836451st_rat(L,H),set_ord_atMost_rat(H2))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),H)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8078_Icc__subset__Iic__iff,axiom,
    ! [L: num,H: num,H2: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(L,H),set_ord_atMost_num(H2))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),H)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8079_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_or1269000886237332187st_nat(L,H)),aa_nat_set_nat(set_ord_atMost_nat,H2))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8080_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),H)),set_ord_atMost_int(H2))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8081_Icc__subset__Iic__iff,axiom,
    ! [L: real,H: real,H2: real] :
      ( ord_less_eq_set_real(set_or1222579329274155063t_real(L,H),set_ord_atMost_real(H2))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),H)
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8082_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8083_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8084_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8085_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8086_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ).

% zero_less_binomial_iff
tff(fact_8087_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_8088_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_8089_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_8090_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_8091_prod__pos__nat__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(F,A4))
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8092_prod__pos__nat__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups861055069439313189ex_nat(F,A4))
      <=> ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8093_prod__pos__nat__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4077766827762148844at_nat(F,A4))
      <=> ! [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Pro4243173770044397973at_nat(F,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8094_prod__pos__nat__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A4))
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8095_or__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_nat_numerals(3)
tff(fact_8096_or__nat__numerals_I1_J,axiom,
    ! [Yb: 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,Yb))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb)) ).

% or_nat_numerals(1)
tff(fact_8097_int__prod,axiom,
    ! [F: fun_int_nat,A4: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A4)) = groups1705073143266064639nt_int(aTP_Lamm_ku(F),A4) ).

% int_prod
tff(fact_8098_int__prod,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) = groups705719431365010083at_int(aTP_Lamm_kw(F),A4) ).

% int_prod
tff(fact_8099_field__class_Ofield__inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% field_class.field_inverse_zero
tff(fact_8100_field__class_Ofield__inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% field_class.field_inverse_zero
tff(fact_8101_field__class_Ofield__inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% field_class.field_inverse_zero
tff(fact_8102_inverse__zero__imp__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
     => ( A = zero_zero_real ) ) ).

% inverse_zero_imp_zero
tff(fact_8103_inverse__zero__imp__zero,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = zero_zero_complex )
     => ( A = zero_zero_complex ) ) ).

% inverse_zero_imp_zero
tff(fact_8104_inverse__zero__imp__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
     => ( A = zero_zero_rat ) ) ).

% inverse_zero_imp_zero
tff(fact_8105_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
     => ( ( A != zero_zero_real )
       => ( ( B != zero_zero_real )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_8106_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B) )
     => ( ( A != zero_zero_complex )
       => ( ( B != zero_zero_complex )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_8107_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
     => ( ( A != zero_zero_rat )
       => ( ( B != zero_zero_rat )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_8108_nonzero__inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_8109_nonzero__inverse__inverse__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_8110_nonzero__inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(inverse_inverse_rat,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_8111_nonzero__imp__inverse__nonzero,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) != zero_zero_real ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_8112_nonzero__imp__inverse__nonzero,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) != zero_zero_complex ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_8113_nonzero__imp__inverse__nonzero,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) != zero_zero_rat ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_8114_nonzero__norm__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(real_V7735802525324610683m_real,A)) ) ) ).

% nonzero_norm_inverse
tff(fact_8115_nonzero__norm__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_real_real(inverse_inverse_real,aa_complex_real(real_V1022390504157884413omplex,A)) ) ) ).

% nonzero_norm_inverse
tff(fact_8116_sum__choose__upper,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vp,Mb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb)) ).

% sum_choose_upper
tff(fact_8117_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: real] : bot_bot_set_real != set_ord_atMost_real(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_8118_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: nat] : bot_bot_set_nat != aa_nat_set_nat(set_ord_atMost_nat,H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_8119_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: int] : bot_bot_set_int != set_ord_atMost_int(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_8120_infinite__Iic,axiom,
    ! [A: int] : ~ aa_set_int_o(finite_finite_int,set_ord_atMost_int(A)) ).

% infinite_Iic
tff(fact_8121_not__Iic__eq__Icc,axiom,
    ! [H2: int,L: int,H: int] : set_ord_atMost_int(H2) != aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),H) ).

% not_Iic_eq_Icc
tff(fact_8122_not__Iic__eq__Icc,axiom,
    ! [H2: real,L: real,H: real] : set_ord_atMost_real(H2) != set_or1222579329274155063t_real(L,H) ).

% not_Iic_eq_Icc
tff(fact_8123_atMost__def,axiom,
    ! [U: real] : set_ord_atMost_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_vq,U)) ).

% atMost_def
tff(fact_8124_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_av,U)) ).

% atMost_def
tff(fact_8125_atMost__def,axiom,
    ! [U: set_int] : set_or58775011639299419et_int(U) = collect_set_int(aTP_Lamm_ay(U)) ).

% atMost_def
tff(fact_8126_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_vr,U)) ).

% atMost_def
tff(fact_8127_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_vs,U)) ).

% atMost_def
tff(fact_8128_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bc,U)) ).

% atMost_def
tff(fact_8129_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_vt,U)) ).

% atMost_def
tff(fact_8130_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vu,R2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),Nb))),Nb) ).

% sum_choose_lower
tff(fact_8131_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vv,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),one_one_nat)),Mb) ).

% choose_rising_sum(2)
tff(fact_8132_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vv,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_8133_norm__inverse__le__norm,axiom,
    ! [R2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_real_real(real_V7735802525324610683m_real,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,Xb))),aa_real_real(inverse_inverse_real,R2)) ) ) ).

% norm_inverse_le_norm
tff(fact_8134_norm__inverse__le__norm,axiom,
    ! [R2: real,Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_complex_real(real_V1022390504157884413omplex,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,Xb))),aa_real_real(inverse_inverse_real,R2)) ) ) ).

% norm_inverse_le_norm
tff(fact_8135_binomial__eq__0,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_8136_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)) ) ).

% positive_imp_inverse_positive
tff(fact_8137_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ).

% positive_imp_inverse_positive
tff(fact_8138_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real) ) ).

% negative_imp_inverse_negative
tff(fact_8139_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat) ) ).

% negative_imp_inverse_negative
tff(fact_8140_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
     => ( ( A != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ).

% inverse_positive_imp_positive
tff(fact_8141_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
     => ( ( A != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ).

% inverse_positive_imp_positive
tff(fact_8142_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
     => ( ( A != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% inverse_negative_imp_negative
tff(fact_8143_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
     => ( ( A != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% inverse_negative_imp_negative
tff(fact_8144_less__imp__inverse__less__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% less_imp_inverse_less_neg
tff(fact_8145_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% less_imp_inverse_less_neg
tff(fact_8146_inverse__less__imp__less__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% inverse_less_imp_less_neg
tff(fact_8147_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% inverse_less_imp_less_neg
tff(fact_8148_less__imp__inverse__less,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% less_imp_inverse_less
tff(fact_8149_less__imp__inverse__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% less_imp_inverse_less
tff(fact_8150_inverse__less__imp__less,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% inverse_less_imp_less
tff(fact_8151_inverse__less__imp__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% inverse_less_imp_less
tff(fact_8152_nonzero__inverse__mult__distrib,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_8153_nonzero__inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_8154_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_8155_nonzero__inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_8156_nonzero__inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_8157_nonzero__inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_8158_Suc__times__binomial,axiom,
    ! [K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ).

% Suc_times_binomial
tff(fact_8159_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2))),aa_nat_nat(suc,K2)) ).

% Suc_times_binomial_eq
tff(fact_8160_binomial__symmetric,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% binomial_symmetric
tff(fact_8161_power__mult__power__inverse__commute,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xb)),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xb)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)) ).

% power_mult_power_inverse_commute
tff(fact_8162_power__mult__power__inverse__commute,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Nb)),aa_nat_complex(power_power_complex(Xb),Mb)) ).

% power_mult_power_inverse_commute
tff(fact_8163_power__mult__power__inverse__commute,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xb)),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xb)),Nb)),aa_nat_rat(power_power_rat(Xb),Mb)) ).

% power_mult_power_inverse_commute
tff(fact_8164_power__mult__inverse__distrib,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)) ).

% power_mult_inverse_distrib
tff(fact_8165_power__mult__inverse__distrib,axiom,
    ! [Xb: complex,Mb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),aa_nat_complex(power_power_complex(Xb),Mb)) ).

% power_mult_inverse_distrib
tff(fact_8166_power__mult__inverse__distrib,axiom,
    ! [Xb: rat,Mb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_rat_rat(inverse_inverse_rat,Xb)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,Xb)),aa_nat_rat(power_power_rat(Xb),Mb)) ).

% power_mult_inverse_distrib
tff(fact_8167_choose__mult__lemma,axiom,
    ! [Mb: nat,R2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),K2)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),Mb)) ).

% choose_mult_lemma
tff(fact_8168_nonzero__abs__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(abs_abs_real,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_8169_nonzero__abs__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(abs_abs_rat,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_8170_binomial__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% binomial_le_pow
tff(fact_8171_mult__inverse__of__int__commute,axiom,
    ! [Xaa: int,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xaa))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xaa))) ).

% mult_inverse_of_int_commute
tff(fact_8172_mult__inverse__of__int__commute,axiom,
    ! [Xaa: int,Xb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xaa))),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xaa))) ).

% mult_inverse_of_int_commute
tff(fact_8173_mult__inverse__of__int__commute,axiom,
    ! [Xaa: int,Xb: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xaa))),Xb) = aa_rat_rat(times_times_rat(Xb),aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xaa))) ).

% mult_inverse_of_int_commute
tff(fact_8174_atMost__atLeast0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,Nb) = set_or1269000886237332187st_nat(zero_zero_nat,Nb) ).

% atMost_atLeast0
tff(fact_8175_lessThan__Suc__atMost,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2)) = aa_nat_set_nat(set_ord_atMost_nat,K2) ).

% lessThan_Suc_atMost
tff(fact_8176_sum__choose__diagonal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vw(Mb),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Mb) ) ) ).

% sum_choose_diagonal
tff(fact_8177_vandermonde,axiom,
    ! [Mb: nat,Nb: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vx(Mb),Nb),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),R2) ).

% vandermonde
tff(fact_8178_not__Iic__le__Icc,axiom,
    ! [H: int,L2: int,H2: int] : ~ aa_set_int_o(ord_less_eq_set_int(set_ord_atMost_int(H)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L2),H2)) ).

% not_Iic_le_Icc
tff(fact_8179_not__Iic__le__Icc,axiom,
    ! [H: real,L2: real,H2: real] : ~ ord_less_eq_set_real(set_ord_atMost_real(H),set_or1222579329274155063t_real(L2,H2)) ).

% not_Iic_le_Icc
tff(fact_8180_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_vy(Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_wa(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.triangle_reindex_eq
tff(fact_8181_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_int,Nb: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_vy(Nb)))) = groups705719431365010083at_int(aTP_Lamm_wc(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.triangle_reindex_eq
tff(fact_8182_finite__nat__iff__bounded__le,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
    <=> ? [K3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_atMost_nat,K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_8183_binomial,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wd(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial
tff(fact_8184_inverse__le__imp__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% inverse_le_imp_le
tff(fact_8185_inverse__le__imp__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% inverse_le_imp_le
tff(fact_8186_le__imp__inverse__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% le_imp_inverse_le
tff(fact_8187_le__imp__inverse__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% le_imp_inverse_le
tff(fact_8188_inverse__le__imp__le__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% inverse_le_imp_le_neg
tff(fact_8189_inverse__le__imp__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% inverse_le_imp_le_neg
tff(fact_8190_le__imp__inverse__le__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% le_imp_inverse_le_neg
tff(fact_8191_le__imp__inverse__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% le_imp_inverse_le_neg
tff(fact_8192_inverse__le__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,Xb)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% inverse_le_1_iff
tff(fact_8193_inverse__le__1__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,Xb)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ) ).

% inverse_le_1_iff
tff(fact_8194_one__less__inverse__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,Xb))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% one_less_inverse_iff
tff(fact_8195_one__less__inverse__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,Xb))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),one_one_rat) ) ) ).

% one_less_inverse_iff
tff(fact_8196_one__less__inverse,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,A)) ) ) ).

% one_less_inverse
tff(fact_8197_one__less__inverse,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% one_less_inverse
tff(fact_8198_zero__less__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ) ).

% zero_less_binomial
tff(fact_8199_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% field_class.field_inverse
tff(fact_8200_field__class_Ofield__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% field_class.field_inverse
tff(fact_8201_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% field_class.field_inverse
tff(fact_8202_division__ring__inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_8203_division__ring__inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(plus_plus_complex(A),B))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_8204_division__ring__inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(plus_plus_rat(A),B))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_8205_inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(inverse_inverse_real,A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% inverse_add
tff(fact_8206_inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(invers8013647133539491842omplex,A))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% inverse_add
tff(fact_8207_inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(inverse_inverse_rat,A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% inverse_add
tff(fact_8208_division__ring__inverse__diff,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_8209_division__ring__inverse__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(B),A))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_8210_division__ring__inverse__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(minus_minus_rat(B),A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_8211_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_8212_nonzero__inverse__eq__divide,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_8213_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_8214_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_8215_binomial__ring,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(A),B)),Nb) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_we(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8216_binomial__ring,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_wf(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8217_binomial__ring,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(A),B)),Nb) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_wg(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8218_binomial__ring,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(plus_p5714425477246183910nteger(A,B)),Nb) = groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_wh(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8219_binomial__ring,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wd(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8220_binomial__ring,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),Nb) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_wi(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8221_choose__mult,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Mb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2))) ) ) ) ).

% choose_mult
tff(fact_8222_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))),aa_nat_nat(suc,K2)) ).

% binomial_Suc_Suc_eq_times
tff(fact_8223_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),B)),Nb) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_wj(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_8224_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_wk(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_8225_pochhammer__binomial__sum,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),B)),Nb) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_wl(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_8226_pochhammer__binomial__sum,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(A,B)),Nb) = groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_wm(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_8227_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),Nb) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_wn(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_8228_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_wo(Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_wa(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.triangle_reindex
tff(fact_8229_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_int,Nb: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_wo(Nb)))) = groups705719431365010083at_int(aTP_Lamm_wc(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.triangle_reindex
tff(fact_8230_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(A),set_ord_lessThan_rat(B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8231_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(A),set_ord_lessThan_num(B))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8232_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,A)),aa_nat_set_nat(set_ord_lessThan_nat,B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8233_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_atMost_int(A)),set_ord_lessThan_int(B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8234_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( ord_less_eq_set_real(set_ord_atMost_real(A),set_or5984915006950818249n_real(B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8235_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,J)) = groups1705073143266064639nt_int(aTP_Lamm_eg,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_8236_eval__nat__numeral_I2_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,Nb))) ).

% eval_nat_numeral(2)
tff(fact_8237_inverse__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ).

% inverse_le_iff
tff(fact_8238_inverse__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ).

% inverse_le_iff
tff(fact_8239_inverse__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ) ).

% inverse_less_iff
tff(fact_8240_inverse__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ) ).

% inverse_less_iff
tff(fact_8241_one__le__inverse,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,A)) ) ) ).

% one_le_inverse
tff(fact_8242_one__le__inverse,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% one_le_inverse
tff(fact_8243_inverse__less__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,Xb)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% inverse_less_1_iff
tff(fact_8244_inverse__less__1__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,Xb)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ) ).

% inverse_less_1_iff
tff(fact_8245_one__le__inverse__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,Xb))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% one_le_inverse_iff
tff(fact_8246_one__le__inverse__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,Xb))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat) ) ) ).

% one_le_inverse_iff
tff(fact_8247_BitM__plus__one,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bitM,Nb)),one) = aa_num_num(bit0,Nb) ).

% BitM_plus_one
tff(fact_8248_one__plus__BitM,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% one_plus_BitM
tff(fact_8249_inverse__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(inverse_inverse_real,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_8250_inverse__diff__inverse,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(A),B))),aa_complex_complex(invers8013647133539491842omplex,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_8251_inverse__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(inverse_inverse_rat,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_8252_reals__Archimedean,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))),Xb) ) ).

% reals_Archimedean
tff(fact_8253_reals__Archimedean,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),Xb) ) ).

% reals_Archimedean
tff(fact_8254_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups2906978787729119204at_rat(aTP_Lamm_wp(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_8255_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_wq,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_8256_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_wr(Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_8257_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups7501900531339628137nteger(aTP_Lamm_ws(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_8258_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_wt,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_8259_binomial__absorption,axiom,
    ! [K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K2)) ).

% binomial_absorption
tff(fact_8260_binomial__r__part__sum,axiom,
    ! [Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% binomial_r_part_sum
tff(fact_8261_forall__pos__mono__1,axiom,
    ! [P2: fun_real_o,E2: real] :
      ( ! [D2: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E)
         => ( aa_real_o(P2,D2)
           => aa_real_o(P2,E) ) )
     => ( ! [N2: nat] : aa_real_o(P2,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P2,E2) ) ) ) ).

% forall_pos_mono_1
tff(fact_8262_binomial__fact__lemma,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_8263_real__arch__inverse,axiom,
    ! [E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
    <=> ? [N3: nat] :
          ( ( N3 != zero_zero_nat )
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3)))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3))),E2) ) ) ).

% real_arch_inverse
tff(fact_8264_forall__pos__mono,axiom,
    ! [P2: fun_real_o,E2: real] :
      ( ! [D2: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E)
         => ( aa_real_o(P2,D2)
           => aa_real_o(P2,E) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => aa_real_o(P2,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2))) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P2,E2) ) ) ) ).

% forall_pos_mono
tff(fact_8265_sqrt__divide__self__eq,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,Xb)),Xb) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb)) ) ) ).

% sqrt_divide_self_eq
tff(fact_8266_ln__inverse,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_inverse
tff(fact_8267_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups2906978787729119204at_rat(aTP_Lamm_wu(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_8268_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_wv,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_8269_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_ww(Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ).

% choose_alternating_sum
tff(fact_8270_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups7501900531339628137nteger(aTP_Lamm_wx(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_8271_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_wy,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_8272_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_jl(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_8273_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_jm(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_8274_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ja(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_8275_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jb(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_8276_sum__telescope,axiom,
    ! [F: fun_nat_rat,I: nat] : groups2906978787729119204at_rat(aTP_Lamm_mj(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_8277_sum__telescope,axiom,
    ! [F: fun_nat_int,I: nat] : groups3539618377306564664at_int(aTP_Lamm_mk(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_8278_sum__telescope,axiom,
    ! [F: fun_nat_real,I: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ml(F)),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_8279_polyfun__eq__coeffs,axiom,
    ! [C2: fun_nat_complex,Nb: nat,D: fun_nat_complex] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oq(C2),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oq(D),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb))
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
         => ( aa_nat_complex(C2,I3) = aa_nat_complex(D,I3) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_8280_polyfun__eq__coeffs,axiom,
    ! [C2: fun_nat_real,Nb: nat,D: fun_nat_real] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_os(C2),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_os(D),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb))
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
         => ( aa_nat_real(C2,I3) = aa_nat_real(D,I3) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_8281_bounded__imp__summable,axiom,
    ! [A: fun_nat_int,B4: int] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(A,N2))
     => ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(A,aa_nat_set_nat(set_ord_atMost_nat,N2))),B4)
       => summable_int(A) ) ) ).

% bounded_imp_summable
tff(fact_8282_bounded__imp__summable,axiom,
    ! [A: fun_nat_nat,B4: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(A,N2))
     => ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_ord_atMost_nat,N2))),B4)
       => summable_nat(A) ) ) ).

% bounded_imp_summable
tff(fact_8283_bounded__imp__summable,axiom,
    ! [A: fun_nat_real,B4: real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(A),aa_nat_set_nat(set_ord_atMost_nat,N2))),B4)
       => summable_real(A) ) ) ).

% bounded_imp_summable
tff(fact_8284_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_jb(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_8285_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_jl(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_8286_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ja(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_8287_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_jm(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_8288_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_eg,aa_int_set_int(aa_int8639817059373041269et_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_8289_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_wz(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xb(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nested_swap'
tff(fact_8290_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xc(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_xe(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nested_swap'
tff(fact_8291_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_xf(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xg(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nested_swap'
tff(fact_8292_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,Nb: nat] : groups705719431365010083at_int(aTP_Lamm_xh(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_xj(A),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nested_swap'
tff(fact_8293_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,K2))),K2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8294_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(semiri681578069525770553at_rat,K2))),K2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8295_binomial__maximum_H,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_8296_binomial__mono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_mono
tff(fact_8297_binomial__antimono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ) ) ) ).

% binomial_antimono
tff(fact_8298_binomial__maximum,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_8299_binomial__le__pow2,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% binomial_le_pow2
tff(fact_8300_choose__reduce__nat,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K2)) ) ) ) ).

% choose_reduce_nat
tff(fact_8301_ex__inverse__of__nat__less,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2))),Xb) ) ) ).

% ex_inverse_of_nat_less
tff(fact_8302_ex__inverse__of__nat__less,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),Xb) ) ) ).

% ex_inverse_of_nat_less
tff(fact_8303_times__binomial__minus1__eq,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_8304_power__diff__conv__inverse,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xb)),Mb)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_8305_power__diff__conv__inverse,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_complex(power_power_complex(Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Mb)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_8306_power__diff__conv__inverse,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_rat(power_power_rat(Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xb)),Mb)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_8307_binomial__altdef__nat,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_altdef_nat
tff(fact_8308_zero__polynom__imp__zero__coeffs,axiom,
    ! [C2: fun_nat_complex,Nb: nat,K2: nat] :
      ( ! [W: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oq(C2),W)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_complex(C2,K2) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_8309_zero__polynom__imp__zero__coeffs,axiom,
    ! [C2: fun_nat_real,Nb: nat,K2: nat] :
      ( ! [W: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_os(C2),W)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_real(C2,K2) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_8310_polyfun__eq__0,axiom,
    ! [C2: fun_nat_complex,Nb: nat] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oq(C2),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
         => ( aa_nat_complex(C2,I3) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_8311_polyfun__eq__0,axiom,
    ! [C2: fun_nat_real,Nb: nat] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_os(C2),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
         => ( aa_nat_real(C2,I3) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_8312_ln__prod,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( finite_finite_real(I6)
     => ( ! [I2: real] :
            ( member_real(I2,I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( aa_real_real(ln_ln_real,groups1681761925125756287l_real(F,I6)) = groups8097168146408367636l_real(aTP_Lamm_xk(F),I6) ) ) ) ).

% ln_prod
tff(fact_8313_ln__prod,axiom,
    ! [I6: set_set_nat,F: fun_set_nat_real] :
      ( finite1152437895449049373et_nat(I6)
     => ( ! [I2: set_nat] :
            ( member_set_nat(I2,I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_set_nat_real(F,I2)) )
       => ( aa_real_real(ln_ln_real,groups3619160379726066777t_real(F,I6)) = groups5107569545109728110t_real(aTP_Lamm_xl(F),I6) ) ) ) ).

% ln_prod
tff(fact_8314_ln__prod,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( aa_real_real(ln_ln_real,groups2316167850115554303t_real(F,I6)) = groups8778361861064173332t_real(aTP_Lamm_xm(F),I6) ) ) ) ).

% ln_prod
tff(fact_8315_ln__prod,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( aa_real_real(ln_ln_real,groups766887009212190081x_real(F,I6)) = groups5808333547571424918x_real(aTP_Lamm_xn(F),I6) ) ) ) ).

% ln_prod
tff(fact_8316_ln__prod,axiom,
    ! [I6: set_Pr1261947904930325089at_nat,F: fun_Pr672294278902479040t_real] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,I6)
     => ( ! [I2: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I2),I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_Pro6168199777849092593t_real(F,I2)) )
       => ( aa_real_real(ln_ln_real,groups6036352826371341000t_real(F,I6)) = groups4567486121110086003t_real(aTP_Lamm_xo(F),I6) ) ) ) ).

% ln_prod
tff(fact_8317_ln__prod,axiom,
    ! [I6: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I2)) )
       => ( aa_real_real(ln_ln_real,aa_set_nat_real(groups129246275422532515t_real(F),I6)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xp(F)),I6) ) ) ) ).

% ln_prod
tff(fact_8318_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_jl(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_8319_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_jm(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_8320_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ja(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_8321_sum_OatMost__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jb(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_8322_sum__up__index__split,axiom,
    ! [F: fun_nat_rat,Mb: nat,Nb: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_8323_sum__up__index__split,axiom,
    ! [F: fun_nat_int,Mb: nat,Nb: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_8324_sum__up__index__split,axiom,
    ! [F: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_8325_sum__up__index__split,axiom,
    ! [F: fun_nat_real,Mb: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_8326_prod_OatMost__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_jb(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_8327_prod_OatMost__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_jl(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_8328_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ja(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_8329_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_jm(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_8330_gbinomial__parallel__sum,axiom,
    ! [A: rat,Nb: nat] : groups2906978787729119204at_rat(aTP_Lamm_xq(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,Nb))),one_one_rat)),Nb) ).

% gbinomial_parallel_sum
tff(fact_8331_gbinomial__parallel__sum,axiom,
    ! [A: complex,Nb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_xr(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,Nb))),one_one_complex)),Nb) ).

% gbinomial_parallel_sum
tff(fact_8332_gbinomial__parallel__sum,axiom,
    ! [A: real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xs,A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,Nb))),one_one_real)),Nb) ).

% gbinomial_parallel_sum
tff(fact_8333_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_vy(Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_xt(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.triangle_reindex_eq
tff(fact_8334_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_real,Nb: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_vy(Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xv(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.triangle_reindex_eq
tff(fact_8335_binomial__less__binomial__Suc,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2))) ) ).

% binomial_less_binomial_Suc
tff(fact_8336_binomial__strict__mono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_strict_mono
tff(fact_8337_binomial__strict__antimono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ) ) ) ).

% binomial_strict_antimono
tff(fact_8338_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_8339_binomial__addition__formula,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(suc,K2))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K2)) ) ) ).

% binomial_addition_formula
tff(fact_8340_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_xw(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_8341_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_xx,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_8342_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_xy(Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_8343_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups7501900531339628137nteger(aTP_Lamm_xz(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_8344_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ya,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_8345_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_yb(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8346_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_yc,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8347_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_yd(Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8348_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups7501900531339628137nteger(aTP_Lamm_ye(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8349_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_yf,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8350_binomial__fact,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_fact
tff(fact_8351_binomial__fact,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_fact
tff(fact_8352_binomial__fact,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Nb)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K2)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_fact
tff(fact_8353_fact__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))) ) ) ).

% fact_binomial
tff(fact_8354_fact__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))) ) ) ).

% fact_binomial
tff(fact_8355_fact__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K2)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Nb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))) ) ) ).

% fact_binomial
tff(fact_8356_sum__gp__basic,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_8357_sum__gp__basic,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_8358_sum__gp__basic,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_8359_sum__gp__basic,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_8360_exp__plus__inverse__exp,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))) ).

% exp_plus_inverse_exp
tff(fact_8361_polyfun__roots__finite,axiom,
    ! [C2: fun_nat_complex,K2: nat,Nb: nat] :
      ( ( aa_nat_complex(C2,K2) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_yg(C2),Nb))) ) ) ).

% polyfun_roots_finite
tff(fact_8362_polyfun__roots__finite,axiom,
    ! [C2: fun_nat_real,K2: nat,Nb: nat] :
      ( ( aa_nat_real(C2,K2) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => finite_finite_real(collect_real(aa_nat_fun_real_o(aTP_Lamm_yh(C2),Nb))) ) ) ).

% polyfun_roots_finite
tff(fact_8363_polyfun__finite__roots,axiom,
    ! [C2: fun_nat_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_yg(C2),Nb)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
          & ( aa_nat_complex(C2,I3) != zero_zero_complex ) ) ) ).

% polyfun_finite_roots
tff(fact_8364_polyfun__finite__roots,axiom,
    ! [C2: fun_nat_real,Nb: nat] :
      ( finite_finite_real(collect_real(aa_nat_fun_real_o(aTP_Lamm_yh(C2),Nb)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
          & ( aa_nat_real(C2,I3) != zero_zero_real ) ) ) ).

% polyfun_finite_roots
tff(fact_8365_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_complex,A: complex,Nb: nat] :
      ( ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oq(C2),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex )
     => ~ ! [B3: fun_nat_complex] :
            ~ ! [Z3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oq(C2),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z3),A)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oq(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_8366_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_rat,A: rat,Nb: nat] :
      ( ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_yi(C2),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat )
     => ~ ! [B3: fun_nat_rat] :
            ~ ! [Z3: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_yi(C2),Z3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z3),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_yi(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_8367_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_int,A: int,Nb: nat] :
      ( ( groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_yj(C2),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int )
     => ~ ! [B3: fun_nat_int] :
            ~ ! [Z3: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_yj(C2),Z3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z3),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_yj(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_8368_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_real,A: real,Nb: nat] :
      ( ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_os(C2),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real )
     => ~ ! [B3: fun_nat_real] :
            ~ ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_os(C2),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z3),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_os(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_8369_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_complex,Nb: nat,A: complex] :
    ? [B3: fun_nat_complex] :
    ! [Z3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oq(C2),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z3),A)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oq(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oq(C2),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_8370_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_rat,Nb: nat,A: rat] :
    ? [B3: fun_nat_rat] :
    ! [Z3: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_yi(C2),Z3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z3),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_yi(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_yi(C2),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_8371_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_int,Nb: nat,A: int] :
    ? [B3: fun_nat_int] :
    ! [Z3: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_yj(C2),Z3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = 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,Z3),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_yj(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_yj(C2),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_8372_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_real,Nb: nat,A: real] :
    ? [B3: fun_nat_real] :
    ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_os(C2),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z3),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_os(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_os(C2),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_8373_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),set_or1269000886237332187st_nat(Mb,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_8374_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(Mb,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_8375_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(power_power_int(Xb),set_or1269000886237332187st_nat(Mb,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Mb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_8376_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(Mb,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_8377_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_wo(Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_xt(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.triangle_reindex
tff(fact_8378_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_real,Nb: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_wo(Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xv(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.triangle_reindex
tff(fact_8379_plus__inverse__ge__2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(inverse_inverse_real,Xb))) ) ).

% plus_inverse_ge_2
tff(fact_8380_real__inv__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,Xb) ) ) ).

% real_inv_sqrt_pow2
tff(fact_8381_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_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))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_kn(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_8382_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_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))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_ko(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_8383_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_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))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kp(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_8384_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_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))),Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kq(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_8385_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_complex,Nb: nat,B: fun_nat_complex,Xb: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_complex(A,I2) = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_complex(B,J2) = zero_zero_complex ) )
       => ( aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oq(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oq(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_yl(A),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_8386_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_rat,Nb: nat,B: fun_nat_rat,Xb: rat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_rat(A,I2) = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_rat(B,J2) = zero_zero_rat ) )
       => ( aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_yi(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_yi(B),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_yn(A),B),Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_8387_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_int,Nb: nat,B: fun_nat_int,Xb: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_int(A,I2) = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_int(B,J2) = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_yj(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_yj(B),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_yp(A),B),Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_8388_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_real,Nb: nat,B: fun_nat_real,Xb: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_real(A,I2) = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_real(B,J2) = zero_zero_real ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_os(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_os(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_yr(A),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_8389_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_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))),Nb)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_vf(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_8390_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_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))),Nb)))) = groups73079841787564623at_rat(aTP_Lamm_vg(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_8391_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_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))),Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_vh(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_8392_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_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))),Nb)))) = groups705719431365010083at_int(aTP_Lamm_vi(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_8393_polyfun__eq__const,axiom,
    ! [C2: fun_nat_complex,Nb: nat,K2: complex] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oq(C2),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = K2
    <=> ( ( aa_nat_complex(C2,zero_zero_nat) = K2 )
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),set_or1269000886237332187st_nat(one_one_nat,Nb))
           => ( aa_nat_complex(C2,X4) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_8394_polyfun__eq__const,axiom,
    ! [C2: fun_nat_real,Nb: nat,K2: real] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_os(C2),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = K2
    <=> ( ( aa_nat_real(C2,zero_zero_nat) = K2 )
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),set_or1269000886237332187st_nat(one_one_nat,Nb))
           => ( aa_nat_real(C2,X4) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_8395_polynomial__product__nat,axiom,
    ! [Mb: nat,A: fun_nat_nat,Nb: nat,B: fun_nat_nat,Xb: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_nat(A,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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_ys(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ys(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_yu(A),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_8396_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_or_eq
tff(fact_8397_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% or_Suc_0_eq
tff(fact_8398_or__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_8399_real__le__x__sinh,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_8400_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P)
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_yv(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_yw(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8401_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P)
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_yx(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P)) = groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_yy(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8402_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P)
       => ( groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_yz(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P)) = groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_za(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8403_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_zb(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_zc(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8404_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P)
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_zd(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_ze(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8405_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P)
       => ( aa_set_nat_complex(groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_zf(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_complex(groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_yw(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8406_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P)
       => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_zg(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_ze(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8407_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P)
       => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_zh(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P)) = groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_yy(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8408_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_zi(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_zc(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8409_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P)
       => ( groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_zj(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P)) = groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_za(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8410_real__le__abs__sinh,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_8411_gbinomial__partial__sum__poly,axiom,
    ! [Mb: nat,A: rat,Xb: rat,Yb: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_zk(Mb),A),Xb),Yb),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_zl(Mb),A),Xb),Yb),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly
tff(fact_8412_gbinomial__partial__sum__poly,axiom,
    ! [Mb: nat,A: complex,Xb: complex,Yb: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_zm(Mb),A),Xb),Yb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_zn(Mb),A),Xb),Yb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly
tff(fact_8413_gbinomial__partial__sum__poly,axiom,
    ! [Mb: nat,A: real,Xb: real,Yb: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_zo(Mb),A),Xb),Yb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_zp(Mb),A),Xb),Yb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly
tff(fact_8414_or__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_8415_binomial__code,axiom,
    ! [Nb: nat,K2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),one_one_nat)),Nb),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K2))) ) ).

% binomial_code
tff(fact_8416_cot__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,Xb)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_8417_of__nat__id,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = Nb ).

% of_nat_id
tff(fact_8418_cot__pi,axiom,
    aa_real_real(cot_real,pi) = zero_zero_real ).

% cot_pi
tff(fact_8419_cot__npi,axiom,
    ! [Nb: nat] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_8420_cot__periodic,axiom,
    ! [Xb: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,Xb) ).

% cot_periodic
tff(fact_8421_complex__inverse,axiom,
    ! [A: real,B: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_8422_cot__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,Xb)) ) ) ).

% cot_gt_zero
tff(fact_8423_log__base__10__eq1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,Xb)) ) ) ).

% log_base_10_eq1
tff(fact_8424_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_zq(D))) ).

% int_ge_less_than_def
tff(fact_8425_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_zr(D))) ).

% int_ge_less_than2_def
tff(fact_8426_sinh__real__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sinh_real,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% sinh_real_zero_iff
tff(fact_8427_sinh__real__le__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ).

% sinh_real_le_iff
tff(fact_8428_sinh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% sinh_real_neg_iff
tff(fact_8429_sinh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sinh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% sinh_real_pos_iff
tff(fact_8430_sinh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% sinh_real_nonpos_iff
tff(fact_8431_sinh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% sinh_real_nonneg_iff
tff(fact_8432_log__one,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_8433_log__eq__one,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_8434_log__less__cancel__iff,axiom,
    ! [A: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Yb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_8435_log__less__one__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_8436_one__less__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_8437_log__less__zero__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_8438_zero__less__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_8439_log__le__cancel__iff,axiom,
    ! [A: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Yb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_8440_log__le__one__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_8441_one__le__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_8442_log__le__zero__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_8443_zero__le__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_8444_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_8445_cosh__real__nonzero,axiom,
    ! [Xb: real] : aa_real_real(cosh_real,Xb) != zero_zero_real ).

% cosh_real_nonzero
tff(fact_8446_sinh__le__cosh__real,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) ).

% sinh_le_cosh_real
tff(fact_8447_cosh__real__pos,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_pos
tff(fact_8448_cosh__real__nonpos__le__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_8449_cosh__real__nonneg__le__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_8450_cosh__real__nonneg,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_nonneg
tff(fact_8451_cosh__real__ge__1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_ge_1
tff(fact_8452_cosh__real__strict__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Yb)) ) ) ).

% cosh_real_strict_mono
tff(fact_8453_cosh__real__nonneg__less__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_8454_cosh__real__nonpos__less__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_8455_arcosh__cosh__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,Xb)) = Xb ) ) ).

% arcosh_cosh_real
tff(fact_8456_log__base__change,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B)) ) ) ) ).

% log_base_change
tff(fact_8457_log__mult,axiom,
    ! [A: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Yb)) ) ) ) ) ) ).

% log_mult
tff(fact_8458_log__divide,axiom,
    ! [A: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Yb)) ) ) ) ) ) ).

% log_divide
tff(fact_8459_le__log__of__power,axiom,
    ! [B: real,Nb: nat,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Mb)) ) ) ).

% le_log_of_power
tff(fact_8460_log__base__pow,axiom,
    ! [A: real,Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log_base_pow
tff(fact_8461_log__nat__power,axiom,
    ! [Xb: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ).

% log_nat_power
tff(fact_8462_log__inverse,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)) ) ) ) ) ).

% log_inverse
tff(fact_8463_log__of__power__less,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_less
tff(fact_8464_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( ( B != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_8465_log__of__power__le,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_le
tff(fact_8466_le__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% le_log2_of_power
tff(fact_8467_log2__of__power__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_less
tff(fact_8468_cosh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(inverse_inverse_real,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_8469_log2__of__power__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_le
tff(fact_8470_log__base__10__eq2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,Xb)) ) ) ).

% log_base_10_eq2
tff(fact_8471_sinh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(inverse_inverse_real,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_8472_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_8473_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_8474_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_8475_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_8476_floor__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_nat_int(semiri1314217659103216013at_int,Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_8477_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_8478_floor__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Nb ) ) ) ).

% floor_eq
tff(fact_8479_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).

% real_of_int_floor_add_one_ge
tff(fact_8480_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).

% real_of_int_floor_ge_diff_one
tff(fact_8481_floor__eq2,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Nb ) ) ) ).

% floor_eq2
tff(fact_8482_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_8483_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_8484_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B: real,K2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K2)),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,K2))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_8485_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P2: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [I2: int,J2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I2),J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
               => aa_int_o(aa_int_fun_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P2,I2),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P2,A0),A1) ) ) ).

% upto.pinduct
tff(fact_8486_arcsin__def,axiom,
    ! [Yb: real] : aa_real_real(arcsin,Yb) = the_real(aa_real_fun_real_o(aTP_Lamm_zs,Yb)) ).

% arcsin_def
tff(fact_8487_modulo__int__unfold,axiom,
    ! [K2: int,Mb: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,Mb)),
        $ite(aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))))) ) ).

% modulo_int_unfold
tff(fact_8488_powr__gt__zero,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A))
    <=> ( Xb != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_8489_powr__nonneg__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_8490_powr__eq__one__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb) = one_one_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_8491_powr__one__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),one_one_real) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% powr_one_gt_zero_iff
tff(fact_8492_powr__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),one_one_real) = Xb ) ) ).

% powr_one
tff(fact_8493_powr__le__cancel__iff,axiom,
    ! [Xb: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% powr_le_cancel_iff
tff(fact_8494_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),L)),K2)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        & ( ( R2 = zero_zero_int )
         => ( K2 = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_8495_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R2))),K2)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        & ( ( R2 = zero_zero_int )
         => ( K2 = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_8496_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),K2))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_8497_dvd__mult__sgn__iff,axiom,
    ! [L: int,K2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_int_int(sgn_sgn_int,R2)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_8498_powr__log__cancel,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)) = Xb ) ) ) ) ).

% powr_log_cancel
tff(fact_8499_log__powr__cancel,axiom,
    ! [A: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Yb)) = Yb ) ) ) ).

% log_powr_cancel
tff(fact_8500_powr__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,Nb)) ) ) ).

% powr_numeral
tff(fact_8501_powr__non__neg,axiom,
    ! [A: real,Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb)),zero_zero_real) ).

% powr_non_neg
tff(fact_8502_powr__less__mono2__neg,axiom,
    ! [A: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Yb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_8503_powr__mono2,axiom,
    ! [A: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Yb),A)) ) ) ) ).

% powr_mono2
tff(fact_8504_powr__ge__pzero,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Yb)) ).

% powr_ge_pzero
tff(fact_8505_powr__mono,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B)) ) ) ).

% powr_mono
tff(fact_8506_int__sgnE,axiom,
    ! [K2: int] :
      ~ ! [N2: nat,L3: int] : K2 != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_sgnE
tff(fact_8507_powr__less__mono2,axiom,
    ! [A: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Yb),A)) ) ) ) ).

% powr_less_mono2
tff(fact_8508_powr__mono2_H,axiom,
    ! [A: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Yb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ) ) ) ).

% powr_mono2'
tff(fact_8509_powr__inj,axiom,
    ! [A: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Yb) )
        <=> ( Xb = Yb ) ) ) ) ).

% powr_inj
tff(fact_8510_gr__one__powr,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Yb)) ) ) ).

% gr_one_powr
tff(fact_8511_ge__one__powr__ge__zero,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_8512_powr__mono__both,axiom,
    ! [A: real,B: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Yb),B)) ) ) ) ) ).

% powr_mono_both
tff(fact_8513_powr__le1,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_8514_powr__divide,axiom,
    ! [Xb: real,Yb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Yb)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Yb),A)) ) ) ) ).

% powr_divide
tff(fact_8515_powr__mult,axiom,
    ! [Xb: real,Yb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Yb)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Yb),A)) ) ) ) ).

% powr_mult
tff(fact_8516_inverse__powr,axiom,
    ! [Yb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(inverse_inverse_real,Yb)),A) = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Yb),A)) ) ) ).

% inverse_powr
tff(fact_8517_log__base__powr,axiom,
    ! [A: real,B: real,Xb: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),B)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),B) ) ) ).

% log_base_powr
tff(fact_8518_ln__powr,axiom,
    ! [Xb: real,Yb: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_powr
tff(fact_8519_log__powr,axiom,
    ! [Xb: real,B: real,Yb: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ).

% log_powr
tff(fact_8520_sgn__mod,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
       => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_8521_ln__neg__is__const,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_real(ln_ln_real,Xb) = the_real(aTP_Lamm_zt) ) ) ).

% ln_neg_is_const
tff(fact_8522_powr__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb) ) ) ).

% powr_realpow
tff(fact_8523_powr__less__iff,axiom,
    ! [B: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Yb)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ).

% powr_less_iff
tff(fact_8524_less__powr__iff,axiom,
    ! [B: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Yb) ) ) ) ).

% less_powr_iff
tff(fact_8525_log__less__iff,axiom,
    ! [B: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Yb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Yb)) ) ) ) ).

% log_less_iff
tff(fact_8526_less__log__iff,axiom,
    ! [B: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Yb)),Xb) ) ) ) ).

% less_log_iff
tff(fact_8527_zsgn__def,axiom,
    ! [I: int] :
      aa_int_int(sgn_sgn_int,I) = $ite(
        I = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_8528_div__sgn__abs__cancel,axiom,
    ! [V: int,K2: 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,K2))),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,K2)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_8529_powr__neg__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb) ) ) ).

% powr_neg_one
tff(fact_8530_powr__mult__base,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Yb)) ) ) ).

% powr_mult_base
tff(fact_8531_le__log__iff,axiom,
    ! [B: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Yb)),Xb) ) ) ) ).

% le_log_iff
tff(fact_8532_log__le__iff,axiom,
    ! [B: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Yb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Yb)) ) ) ) ).

% log_le_iff
tff(fact_8533_le__powr__iff,axiom,
    ! [B: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Yb) ) ) ) ).

% le_powr_iff
tff(fact_8534_powr__le__iff,axiom,
    ! [B: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Yb)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ).

% powr_le_iff
tff(fact_8535_ln__powr__bound,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),A)) ) ) ).

% ln_powr_bound
tff(fact_8536_ln__powr__bound2,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,Xb)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_8537_add__log__eq__powr,axiom,
    ! [B: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Yb)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_8538_log__add__eq__powr,axiom,
    ! [B: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Yb) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Yb))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_8539_minus__log__eq__powr,axiom,
    ! [B: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Yb),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Yb)),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_8540_arccos__def,axiom,
    ! [Yb: real] : aa_real_real(arccos,Yb) = the_real(aa_real_fun_real_o(aTP_Lamm_zu,Yb)) ).

% arccos_def
tff(fact_8541_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K2: int,Q3: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L))
       => ( ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L)),R2) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_8542_log__minus__eq__powr,axiom,
    ! [B: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Yb) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,Yb)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_8543_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K3) ) )
        | ? [L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_8544_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A1) ) )
       => ( ! [Q2: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q2),A22) ) ) )
         => ~ ! [R3: int,Q2: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q2),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_8545_div__noneq__sgn__abs,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K2) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),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,K2)),aa_int_int(abs_abs_int,L)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_8546_powr__half__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,Xb) ) ) ).

% powr_half_sqrt
tff(fact_8547_powr__neg__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,Nb))) ) ) ).

% powr_neg_numeral
tff(fact_8548_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_zv) ).

% pi_half
tff(fact_8549_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_zv)) ).

% pi_def
tff(fact_8550_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B: real,K2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = K2 )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,K2))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_8551_divide__int__unfold,axiom,
    ! [K2: int,Mb: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)))))) ) ).

% divide_int_unfold
tff(fact_8552_modulo__int__def,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = $ite(
        L = zero_zero_int,
        K2,
        $ite(aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,L)),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)))),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,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))))) ) ).

% modulo_int_def
tff(fact_8553_divide__int__def,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),zero_n2687167440665602831ol_nat(~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)))))) ) ).

% divide_int_def
tff(fact_8554_sgn__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sgn_sgn_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% sgn_le_0_iff
tff(fact_8555_zero__le__sgn__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_sgn_iff
tff(fact_8556_nat__int,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,Nb)) = Nb ).

% nat_int
tff(fact_8557_nat__numeral,axiom,
    ! [K2: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K2)) = aa_num_nat(numeral_numeral_nat,K2) ).

% nat_numeral
tff(fact_8558_nat__of__bool,axiom,
    ! [P2: $o] : aa_int_nat(nat2,aa_o_int(zero_n2684676970156552555ol_int,(P2))) = zero_n2687167440665602831ol_nat((P2)) ).

% nat_of_bool
tff(fact_8559_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_8560_nat__le__0,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_8561_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int) ) ).

% nat_0_iff
tff(fact_8562_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% zless_nat_conj
tff(fact_8563_nat__neg__numeral,axiom,
    ! [K2: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_8564_nat__zminus__int,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_8565_int__nat__eq,axiom,
    ! [Z: int] :
      aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),Z,zero_zero_int) ).

% int_nat_eq
tff(fact_8566_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% zero_less_nat_eq
tff(fact_8567_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_8568_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_int_nat(nat2,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_8569_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_nat(nat2,Yb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_8570_dvd__nat__abs__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_int_nat(nat2,aa_int_int(abs_abs_int,K2)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K2) ) ).

% dvd_nat_abs_iff
tff(fact_8571_nat__abs__dvd__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),Nb)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% nat_abs_dvd_iff
tff(fact_8572_nat__ceiling__le__eq,axiom,
    ! [Xb: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_8573_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% one_less_nat_eq
tff(fact_8574_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_8575_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_8576_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_8577_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_8578_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_8579_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_8580_nat__mono,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Yb)) ) ).

% nat_mono
tff(fact_8581_eq__nat__nat__iff,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z5) )
        <=> ( Z = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_8582_all__nat,axiom,
    ! [P2: fun_nat_o] :
      ( ! [X_13: nat] : aa_nat_o(P2,X_13)
    <=> ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => aa_nat_o(P2,aa_int_nat(nat2,X4)) ) ) ).

% all_nat
tff(fact_8583_ex__nat,axiom,
    ! [P2: fun_nat_o] :
      ( ? [X_13: nat] : aa_nat_o(P2,X_13)
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
          & aa_nat_o(P2,aa_int_nat(nat2,X4)) ) ) ).

% ex_nat
tff(fact_8584_complex__of__real__def,axiom,
    ! [R2: real] : aa_real_complex(real_V4546457046886955230omplex,R2) = aa_real_complex(complex2(R2),zero_zero_real) ).

% complex_of_real_def
tff(fact_8585_complex__of__real__code,axiom,
    ! [X: real] : aa_real_complex(real_V4546457046886955230omplex,X) = aa_real_complex(complex2(X),zero_zero_real) ).

% complex_of_real_code
tff(fact_8586_complex__eq__cancel__iff2,axiom,
    ! [Xb: real,Yb: real,Xaa: real] :
      ( ( aa_real_complex(complex2(Xb),Yb) = aa_real_complex(real_V4546457046886955230omplex,Xaa) )
    <=> ( ( Xb = Xaa )
        & ( Yb = zero_zero_real ) ) ) ).

% complex_eq_cancel_iff2
tff(fact_8587_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% nat_mono_iff
tff(fact_8588_zless__nat__eq__int__zless,axiom,
    ! [Mb: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_8589_nat__le__iff,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),Nb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% nat_le_iff
tff(fact_8590_int__eq__iff,axiom,
    ! [Mb: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = Z )
    <=> ( ( Mb = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_8591_nat__0__le,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_8592_nat__int__add,axiom,
    ! [A: nat,B: nat] : aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) ).

% nat_int_add
tff(fact_8593_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_8594_Complex__add__complex__of__real,axiom,
    ! [Xb: real,Yb: real,R2: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(complex2(Xb),Yb)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),R2)),Yb) ).

% Complex_add_complex_of_real
tff(fact_8595_complex__of__real__add__Complex,axiom,
    ! [R2: real,Xb: real,Yb: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(Xb),Yb)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),Xb)),Yb) ).

% complex_of_real_add_Complex
tff(fact_8596_nat__plus__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_8597_real__nat__ceiling__ge,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb)))) ).

% real_nat_ceiling_ge
tff(fact_8598_sgn__real__def,axiom,
    ! [A: real] :
      aa_real_real(sgn_sgn_real,A) = $ite(
        A = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_real_def
tff(fact_8599_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% nat_less_eq_zless
tff(fact_8600_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W2)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ) ).

% nat_le_eq_zle
tff(fact_8601_nat__eq__iff2,axiom,
    ! [Mb: nat,W2: int] :
      ( ( Mb = aa_int_nat(nat2,W2) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2),W2 = aa_nat_int(semiri1314217659103216013at_int,Mb),Mb = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_8602_nat__eq__iff,axiom,
    ! [W2: int,Mb: nat] :
      ( ( aa_int_nat(nat2,W2) = Mb )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2),W2 = aa_nat_int(semiri1314217659103216013at_int,Mb),Mb = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_8603_split__nat,axiom,
    ! [P2: fun_nat_o,I: int] :
      ( aa_nat_o(P2,aa_int_nat(nat2,I))
    <=> ( ! [N3: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => aa_nat_o(P2,N3) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int)
         => aa_nat_o(P2,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_8604_le__nat__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_int_nat(nat2,K2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K2) ) ) ).

% le_nat_iff
tff(fact_8605_nat__add__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_8606_nat__mult__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_8607_Suc__as__int,axiom,
    ! [X: nat] : aa_nat_nat(suc,X) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X)),one_one_int)) ).

% Suc_as_int
tff(fact_8608_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z5),Z)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_8609_nat__diff__distrib_H,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Yb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Yb)) ) ) ) ).

% nat_diff_distrib'
tff(fact_8610_nat__abs__triangle__ineq,axiom,
    ! [K2: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_8611_nat__div__distrib_H,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Yb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Yb)) ) ) ).

% nat_div_distrib'
tff(fact_8612_nat__div__distrib,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Yb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Yb)) ) ) ).

% nat_div_distrib
tff(fact_8613_nat__floor__neg,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_8614_nat__power__eq,axiom,
    ! [Z: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_8615_nat__mod__distrib,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Yb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Yb)) ) ) ) ).

% nat_mod_distrib
tff(fact_8616_floor__eq3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).

% floor_eq3
tff(fact_8617_le__nat__floor,axiom,
    ! [Xb: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_8618_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_8619_sgn__power__injE,axiom,
    ! [A: real,Nb: nat,Xb: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) = Xb )
     => ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),Nb)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_8620_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_8621_nat__less__iff,axiom,
    ! [W2: int,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),Mb)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_nat_int(semiri1314217659103216013at_int,Mb)) ) ) ).

% nat_less_iff
tff(fact_8622_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z5)) = 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,Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_8623_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) ).

% nat_abs_int_diff
tff(fact_8624_floor__eq4,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).

% floor_eq4
tff(fact_8625_diff__nat__eq__if,axiom,
    ! [Z: int,Z5: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5),
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_8626_nat__dvd__iff,axiom,
    ! [Z: int,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),Mb)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,Mb)),Mb = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_8627_floor__real__def,axiom,
    ! [Xb: real] : aa_real_int(archim6058952711729229775r_real,Xb) = the_int(aa_real_fun_int_o(aTP_Lamm_zw,Xb)) ).

% floor_real_def
tff(fact_8628_even__nat__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) ) ) ).

% even_nat_iff
tff(fact_8629_powr__real__of__int,axiom,
    ! [Xb: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,Nb)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Nb))))) ) ) ).

% powr_real_of_int
tff(fact_8630_arctan__inverse,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Xb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,Xb)) ) ) ).

% arctan_inverse
tff(fact_8631_powr__int,axiom,
    ! [Xb: real,I: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,I)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,I)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I))))) ) ) ).

% powr_int
tff(fact_8632_and__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = $ite(
        ( ( K2 = zero_zero_int )
        | ( L = zero_zero_int ) ),
        zero_zero_int,
        $ite(
          K2 = aa_int_int(uminus_uminus_int,one_one_int),
          L,
          $ite(L = aa_int_int(uminus_uminus_int,one_one_int),K2,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,K2),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,K2),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_8633_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Yb: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
      <=> (Yb) )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( (Yb)
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Yb)
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Yb)
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
                   => ( (Yb)
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                     => ( ~ (Yb)
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_8634_set__encode__insert,axiom,
    ! [A4: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ~ aa_set_nat_o(member_nat(Nb),A4)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Nb,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_set_nat_nat(nat_set_encode,A4)) ) ) ) ).

% set_encode_insert
tff(fact_8635_mask__nat__positive__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% mask_nat_positive_iff
tff(fact_8636_concat__bit__of__zero__2,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),zero_zero_int) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2) ).

% concat_bit_of_zero_2
tff(fact_8637_and__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_8638_and__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_8639_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_Suc_0
tff(fact_8640_atMost__0,axiom,
    aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = insert_nat(zero_zero_nat,bot_bot_set_nat) ).

% atMost_0
tff(fact_8641_and__minus__numerals_I5_J,axiom,
    ! [Nb: 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,Nb)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_8642_and__minus__numerals_I1_J,axiom,
    ! [Nb: 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,Nb)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_8643_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb) ).

% take_bit_nat_less_eq_self
tff(fact_8644_take__bit__tightened__less__eq__nat,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Q3)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_8645_less__eq__mask,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% less_eq_mask
tff(fact_8646_take__bit__nat__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_int_nat(nat2,K2)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2)) ) ) ).

% take_bit_nat_eq
tff(fact_8647_nat__take__bit__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_int_nat(nat2,K2)) ) ) ).

% nat_take_bit_eq
tff(fact_8648_take__bit__eq__mask__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_8649_AND__lower,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Yb)) ) ).

% AND_lower
tff(fact_8650_AND__upper1,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Yb)),Xb) ) ).

% AND_upper1
tff(fact_8651_AND__upper2,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Yb)),Yb) ) ).

% AND_upper2
tff(fact_8652_AND__upper1_H,axiom,
    ! [Yb: int,Z: int,Yaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Yb),Yaa)),Z) ) ) ).

% AND_upper1'
tff(fact_8653_AND__upper2_H,axiom,
    ! [Yb: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Yb)),Z) ) ) ).

% AND_upper2'
tff(fact_8654_take__bit__tightened__less__eq__int,axiom,
    ! [Mb: nat,Nb: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Mb),K2)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_8655_take__bit__nonnegative,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2)) ).

% take_bit_nonnegative
tff(fact_8656_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_8657_take__bit__int__greater__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_8658_not__take__bit__negative,axiom,
    ! [Nb: nat,K2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_8659_plus__and__or,axiom,
    ! [Xb: int,Yb: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Yb)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Yb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Yb) ).

% plus_and_or
tff(fact_8660_lessThan__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2)) = insert_nat(K2,aa_nat_set_nat(set_ord_lessThan_nat,K2)) ).

% lessThan_Suc
tff(fact_8661_atMost__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K2)) = insert_nat(aa_nat_nat(suc,K2),aa_nat_set_nat(set_ord_atMost_nat,K2)) ).

% atMost_Suc
tff(fact_8662_mask__nonnegative__int,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% mask_nonnegative_int
tff(fact_8663_not__mask__negative__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_8664_AND__upper2_H_H,axiom,
    ! [Yb: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Yb)),Z) ) ) ).

% AND_upper2''
tff(fact_8665_AND__upper1_H_H,axiom,
    ! [Yb: int,Z: int,Yaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Yb),Yaa)),Z) ) ) ).

% AND_upper1''
tff(fact_8666_and__less__eq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L)),K2) ) ).

% and_less_eq
tff(fact_8667_take__bit__decr__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2) != zero_zero_int )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_8668_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% atLeast0_atMost_Suc
tff(fact_8669_Icc__eq__insert__lb__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( set_or1269000886237332187st_nat(Mb,Nb) = insert_nat(Mb,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_8670_atLeastAtMostSuc__conv,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),set_or1269000886237332187st_nat(Mb,Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_8671_atLeastAtMost__insertL,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( insert_nat(Mb,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) = set_or1269000886237332187st_nat(Mb,Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_8672_lessThan__nat__numeral,axiom,
    ! [K2: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K2)) = insert_nat(aa_num_nat(pred_numeral,K2),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K2))) ).

% lessThan_nat_numeral
tff(fact_8673_atMost__nat__numeral,axiom,
    ! [K2: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K2)) = insert_nat(aa_num_nat(numeral_numeral_nat,K2),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K2))) ).

% atMost_nat_numeral
tff(fact_8674_less__mask,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ) ).

% less_mask
tff(fact_8675_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_8676_atLeast1__atMost__eq__remove0,axiom,
    ! [Nb: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_8677_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb) ) ).

% take_bit_nat_less_self_iff
tff(fact_8678_Suc__mask__eq__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_mask_eq_exp
tff(fact_8679_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_8680_take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_8681_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2) ) ).

% take_bit_int_less_self_iff
tff(fact_8682_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(Nb),nat_set_decode(Z))
     => ( nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Z)) = insert_nat(Nb,nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_8683_take__bit__int__eq__self,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2) = K2 ) ) ) ).

% take_bit_int_eq_self
tff(fact_8684_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2) = K2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_8685_take__bit__incr__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2) != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2)) ) ) ).

% take_bit_incr_eq
tff(fact_8686_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ! [Uv2: $o] :
              ( ( Xb = vEBT_Leaf($true,(Uv2)) )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) )
         => ( ! [Uu2: $o] :
                ( ( Xb = vEBT_Leaf((Uu2),$true) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_8687_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_8688_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_8689_take__bit__int__greater__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2)) ) ).

% take_bit_int_greater_eq
tff(fact_8690_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_8691_and__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,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,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_8692_take__bit__minus__small__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(uminus_uminus_int,K2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_8693_floor__rat__def,axiom,
    ! [Xb: rat] : aa_rat_int(archim3151403230148437115or_rat,Xb) = the_int(aTP_Lamm_zx(Xb)) ).

% floor_rat_def
tff(fact_8694_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_8695_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Yb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Xaa) = Yb )
     => ( Yb = $ite(
            ( aa_set_int_o(member_int(Xb),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(Xaa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_8696_and__int_Osimps,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = $ite(
        ( aa_set_int_o(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
        aa_int_int(uminus_uminus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,
            aa_o_int(zero_n2684676970156552555ol_int,
              ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
              & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,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,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ).

% and_int.simps
tff(fact_8697_and__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_8698_and__nat__numerals_I1_J,axiom,
    ! [Yb: 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,Yb))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_8699_and__nat__numerals_I2_J,axiom,
    ! [Yb: 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,Yb))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_8700_and__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_8701_and__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_8702_Suc__0__and__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_8703_sgn__rat__def,axiom,
    ! [A: rat] :
      aa_rat_rat(sgn_sgn_rat,A) = $ite(
        A = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_rat_def
tff(fact_8704_abs__rat__def,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_rat_def
tff(fact_8705_less__eq__rat__def,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% less_eq_rat_def
tff(fact_8706_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
     => ~ ! [S: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S)
           => ! [T4: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T4)
               => ( R2 != aa_rat_rat(plus_plus_rat(S),T4) ) ) ) ) ).

% obtain_pos_sum
tff(fact_8707_add__inc,axiom,
    ! [Xb: num,Yb: num] : aa_num_num(plus_plus_num(Xb),aa_num_num(inc,Yb)) = aa_num_num(inc,aa_num_num(plus_plus_num(Xb),Yb)) ).

% add_inc
tff(fact_8708_add__One,axiom,
    ! [Xb: num] : aa_num_num(plus_plus_num(Xb),one) = aa_num_num(inc,Xb) ).

% add_One
tff(fact_8709_mult__inc,axiom,
    ! [Xb: num,Yb: num] : aa_num_num(times_times_num(Xb),aa_num_num(inc,Yb)) = aa_num_num(plus_plus_num(aa_num_num(times_times_num(Xb),Yb)),Xb) ).

% mult_inc
tff(fact_8710_atLeastAtMostPlus1__int__conv,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb))
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,Mb),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,Mb),Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_8711_simp__from__to,axiom,
    ! [I: int,J: int] :
      aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,I),J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),bot_bot_set_int,insert_int(I,aa_int_set_int(aa_int8639817059373041269et_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_8712_and__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = $ite(
        ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_8713_and__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_8714_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_8715_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Yb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Xaa) = Yb )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Yb = $ite(
                  ( aa_set_int_o(member_int(Xb),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & aa_set_int_o(member_int(Xaa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
                  aa_int_int(aa_int_fun_int_int(plus_plus_int,
                      aa_o_int(zero_n2684676970156552555ol_int,
                        ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_8716_and__int_Opsimps,axiom,
    ! [K2: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K2),L))
     => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = $ite(
            ( aa_set_int_o(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,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,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.psimps
tff(fact_8717_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P2: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [K: int,L3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L3))
           => ( ( ~ ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & aa_set_int_o(member_int(L3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P2,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,L3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P2,K),L3) ) )
       => aa_int_o(aa_int_fun_int_o(P2,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_8718_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),K2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_8719_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K2))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_8720_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K2)),zero_zero_int)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_8721_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2)))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W2))),Nb) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_8722_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W2)))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W2)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_8723_diff__rat__def,axiom,
    ! [Q3: rat,R2: rat] : aa_rat_rat(minus_minus_rat(Q3),R2) = aa_rat_rat(plus_plus_rat(Q3),aa_rat_rat(uminus_uminus_rat,R2)) ).

% diff_rat_def
tff(fact_8724_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Mb: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Mb),K2)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_8725_bit__concat__bit__iff,axiom,
    ! [Mb: nat,K2: int,L: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(Mb),K2),L)),Nb)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),Nb) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_8726_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N2: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M2)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),M2)
              <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),N2) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))
              <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),N2) ) ) ) ).

% int_bit_bound
tff(fact_8727_set__bit__eq,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% set_bit_eq
tff(fact_8728_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),Nb)))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2)) ).

% take_bit_Suc_from_most
tff(fact_8729_rat__inverse__code,axiom,
    ! [P: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_zy),quotient_of(P)) ).

% rat_inverse_code
tff(fact_8730_cmod__complex__polar,axiom,
    ! [R2: real,A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A)))))) = aa_real_real(abs_abs_real,R2) ).

% cmod_complex_polar
tff(fact_8731_cmod__unit__one,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A))))) = one_one_real ).

% cmod_unit_one
tff(fact_8732_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_xor_eq
tff(fact_8733_quotient__of__number_I3_J,axiom,
    ! [K2: num] : quotient_of(aa_num_rat(numeral_numeral_rat,K2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K2)),one_one_int) ).

% quotient_of_number(3)
tff(fact_8734_rat__one__code,axiom,
    quotient_of(one_one_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int) ).

% rat_one_code
tff(fact_8735_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% rat_zero_code
tff(fact_8736_xor__nat__numerals_I1_J,axiom,
    ! [Yb: 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,Yb))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb)) ).

% xor_nat_numerals(1)
tff(fact_8737_xor__nat__numerals_I2_J,axiom,
    ! [Yb: 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,Yb))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Yb)) ).

% xor_nat_numerals(2)
tff(fact_8738_xor__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% xor_nat_numerals(3)
tff(fact_8739_xor__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb)) ).

% xor_nat_numerals(4)
tff(fact_8740_quotient__of__number_I5_J,axiom,
    ! [K2: num] : quotient_of(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int) ).

% quotient_of_number(5)
tff(fact_8741_quotient__of__number_I4_J,axiom,
    quotient_of(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% quotient_of_number(4)
tff(fact_8742_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero_complex ).

% complex_i_not_zero
tff(fact_8743_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Nb)) ).

% not_bit_Suc_0_Suc
tff(fact_8744_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_8745_not__bit__Suc__0__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_8746_quotient__of__div,axiom,
    ! [R2: rat,Nb: int,D: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Nb),D) )
     => ( R2 = aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,Nb)),aa_int_rat(ring_1_of_int_rat,D)) ) ) ).

% quotient_of_div
tff(fact_8747_Complex__eq__i,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_complex(complex2(Xb),Yb) = imaginary_unit )
    <=> ( ( Xb = zero_zero_real )
        & ( Yb = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_8748_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_8749_quotient__of__denom__pos,axiom,
    ! [R2: rat,P: int,Q3: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q3) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3) ) ).

% quotient_of_denom_pos
tff(fact_8750_bit__nat__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K2)),Nb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),Nb) ) ) ).

% bit_nat_iff
tff(fact_8751_i__complex__of__real,axiom,
    ! [R2: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(zero_zero_real),R2) ).

% i_complex_of_real
tff(fact_8752_complex__of__real__i,axiom,
    ! [R2: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),imaginary_unit) = aa_real_complex(complex2(zero_zero_real),R2) ).

% complex_of_real_i
tff(fact_8753_Complex__eq,axiom,
    ! [A: real,B: real] : aa_real_complex(complex2(A),B) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,A)),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% Complex_eq
tff(fact_8754_rat__uminus__code,axiom,
    ! [P: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_zz),quotient_of(P)) ).

% rat_uminus_code
tff(fact_8755_rat__abs__code,axiom,
    ! [P: rat] : quotient_of(aa_rat_rat(abs_abs_rat,P)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aaa),quotient_of(P)) ).

% rat_abs_code
tff(fact_8756_rat__less__eq__code,axiom,
    ! [P: rat,Q3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P),Q3)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_aac(Q3)),quotient_of(P)) ) ).

% rat_less_eq_code
tff(fact_8757_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R3: real,A3: real] : Z = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A3))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A3))))) ).

% complex_split_polar
tff(fact_8758_xor__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_8759_xor__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_8760_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% xor_Suc_0_eq
tff(fact_8761_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_8762_quotient__of__int,axiom,
    ! [A: int] : quotient_of(aa_int_rat(of_int,A)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),one_one_int) ).

% quotient_of_int
tff(fact_8763_rat__minus__code,axiom,
    ! [P: rat,Q3: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P),Q3)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aae(Q3)),quotient_of(P)) ).

% rat_minus_code
tff(fact_8764_rat__plus__code,axiom,
    ! [P: rat,Q3: rat] : quotient_of(aa_rat_rat(plus_plus_rat(P),Q3)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aag(Q3)),quotient_of(P)) ).

% rat_plus_code
tff(fact_8765_xor__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_8766_xor__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_8767_csqrt__0,axiom,
    aa_complex_complex(csqrt,zero_zero_complex) = zero_zero_complex ).

% csqrt_0
tff(fact_8768_csqrt__eq__0,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(csqrt,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_complex ) ) ).

% csqrt_eq_0
tff(fact_8769_normalize__denom__zero,axiom,
    ! [P: int] : aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),zero_zero_int)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% normalize_denom_zero
tff(fact_8770_normalize__negative,axiom,
    ! [Q3: int,P: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q3),zero_zero_int)
     => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q3)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,P)),aa_int_int(uminus_uminus_int,Q3))) ) ) ).

% normalize_negative
tff(fact_8771_XOR__lower,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xb),Yb)) ) ) ).

% XOR_lower
tff(fact_8772_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P: int,Q3: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q3) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3) ) ).

% normalize_denom_pos
tff(fact_8773_normalize__crossproduct,axiom,
    ! [Q3: int,S3: int,P: int,R2: int] :
      ( ( Q3 != zero_zero_int )
     => ( ( S3 != zero_zero_int )
       => ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q3)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S3)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P),S3) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q3) ) ) ) ) ).

% normalize_crossproduct
tff(fact_8774_of__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,Xb)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,Xb)) ) ) ).

% of_real_sqrt
tff(fact_8775_XOR__upper,axiom,
    ! [Xb: int,Nb: nat,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xb),Yb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_8776_rat__divide__code,axiom,
    ! [P: rat,Q3: rat] : quotient_of(aa_rat_rat(divide_divide_rat(P),Q3)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aai(Q3)),quotient_of(P)) ).

% rat_divide_code
tff(fact_8777_rat__times__code,axiom,
    ! [P: rat,Q3: rat] : quotient_of(aa_rat_rat(times_times_rat(P),Q3)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aak(Q3)),quotient_of(P)) ).

% rat_times_code
tff(fact_8778_xor__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) != ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,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,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_8779_Frct__code__post_I5_J,axiom,
    ! [K2: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),aa_num_int(numeral_numeral_int,K2))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,K2)) ).

% Frct_code_post(5)
tff(fact_8780_xor__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L) = $ite(
        K2 = aa_int_int(uminus_uminus_int,one_one_int),
        aa_int_int(bit_ri7919022796975470100ot_int,L),
        $ite(
          L = aa_int_int(uminus_uminus_int,one_one_int),
          aa_int_int(bit_ri7919022796975470100ot_int,K2),
          $ite(
            K2 = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K2,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,K2),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,K2),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_8781_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list_int] :
      upto_aux(I,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),Js,upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_8782_not__negative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% not_negative_int_iff
tff(fact_8783_not__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% not_nonnegative_int_iff
tff(fact_8784_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_8785_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_8786_and__not__numerals_I3_J,axiom,
    ! [Nb: 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,Nb)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_8787_or__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_8788_Frct__code__post_I2_J,axiom,
    ! [A: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),zero_zero_int)) = zero_zero_rat ).

% Frct_code_post(2)
tff(fact_8789_Frct__code__post_I1_J,axiom,
    ! [A: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A)) = zero_zero_rat ).

% Frct_code_post(1)
tff(fact_8790_Frct__code__post_I7_J,axiom,
    ! [A: int,B: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,A)),B)) = aa_rat_rat(uminus_uminus_rat,frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ).

% Frct_code_post(7)
tff(fact_8791_Frct__code__post_I8_J,axiom,
    ! [A: int,B: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),aa_int_int(uminus_uminus_int,B))) = aa_rat_rat(uminus_uminus_rat,frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ).

% Frct_code_post(8)
tff(fact_8792_Frct__code__post_I3_J,axiom,
    frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_8793_or__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(5)
tff(fact_8794_Frct__code__post_I4_J,axiom,
    ! [K2: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K2)),one_one_int)) = aa_num_rat(numeral_numeral_rat,K2) ).

% Frct_code_post(4)
tff(fact_8795_and__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% and_not_numerals(8)
tff(fact_8796_or__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(8)
tff(fact_8797_or__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(9)
tff(fact_8798_not__int__rec,axiom,
    ! [K2: int] : aa_int_int(bit_ri7919022796975470100ot_int,K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_8799_Frct__code__post_I6_J,axiom,
    ! [K2: num,L: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K2)),aa_num_int(numeral_numeral_int,L))) = aa_rat_rat(divide_divide_rat(aa_num_rat(numeral_numeral_rat,K2)),aa_num_rat(numeral_numeral_rat,L)) ).

% Frct_code_post(6)
tff(fact_8800_Sum__Ico__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kr),set_or4665077453230672383an_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_8801_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or4665077453230672383an_nat(L,U)) ).

% finite_atLeastLessThan
tff(fact_8802_atLeastLessThan__singleton,axiom,
    ! [Mb: nat] : set_or4665077453230672383an_nat(Mb,aa_nat_nat(suc,Mb)) = insert_nat(Mb,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_8803_all__nat__less__eq,axiom,
    ! [Nb: nat,P2: fun_nat_o] :
      ( ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
         => aa_nat_o(P2,M4) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
         => aa_nat_o(P2,X4) ) ) ).

% all_nat_less_eq
tff(fact_8804_ex__nat__less__eq,axiom,
    ! [Nb: nat,P2: fun_nat_o] :
      ( ? [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
          & aa_nat_o(P2,M4) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
          & aa_nat_o(P2,X4) ) ) ).

% ex_nat_less_eq
tff(fact_8805_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(L,aa_nat_nat(suc,U)) = set_or1269000886237332187st_nat(L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_8806_lessThan__atLeast0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Nb) = set_or4665077453230672383an_nat(zero_zero_nat,Nb) ).

% lessThan_atLeast0
tff(fact_8807_atLeastLessThan0,axiom,
    ! [Mb: nat] : set_or4665077453230672383an_nat(Mb,zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_8808_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,Nb)) = insert_nat(Nb,set_or4665077453230672383an_nat(zero_zero_nat,Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_8809_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set_nat,Nb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N4),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
     => aa_set_nat_o(finite_finite_nat,N4) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_8810_atLeastLessThanSuc,axiom,
    ! [Mb: nat,Nb: nat] :
      set_or4665077453230672383an_nat(Mb,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),insert_nat(Nb,set_or4665077453230672383an_nat(Mb,Nb)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_8811_prod__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(zero_zero_nat,Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_fact
tff(fact_8812_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_8813_atLeastLessThan__nat__numeral,axiom,
    ! [Mb: nat,K2: num] :
      set_or4665077453230672383an_nat(Mb,aa_num_nat(numeral_numeral_nat,K2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_num_nat(pred_numeral,K2)),insert_nat(aa_num_nat(pred_numeral,K2),set_or4665077453230672383an_nat(Mb,aa_num_nat(pred_numeral,K2))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_8814_atLeast1__lessThan__eq__remove0,axiom,
    ! [Nb: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_8815_sum__power2,axiom,
    ! [K2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),set_or4665077453230672383an_nat(zero_zero_nat,K2)) = 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))),K2)),one_one_nat) ).

% sum_power2
tff(fact_8816_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I2)),aa_nat_nat(A,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I2)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aal(A),B)),set_or4665077453230672383an_nat(zero_zero_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),set_or4665077453230672383an_nat(zero_zero_nat,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(B),set_or4665077453230672383an_nat(zero_zero_nat,Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_8817_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(vEBT_size_VEBT,X13)),aa_VEBT_VEBT_nat(vEBT_size_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size_gen(1)
tff(fact_8818_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(size_size_VEBT_VEBT,X13)),aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size(3)
tff(fact_8819_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U)) ).

% finite_atLeastLessThan_int
tff(fact_8820_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa_set_int_o(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,zero_zero_int),U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_8821_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,U),one_one_int)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_8822_Cauchy__iff2,axiom,
    ! [X5: fun_nat_real] :
      ( topolo4055970368930404560y_real(X5)
    <=> ! [J3: nat] :
        ? [M7: nat] :
        ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),M4)
         => ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),N3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X5,M4)),aa_nat_real(X5,N3)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_8823_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_ej(L),Qr) ).

% divmod_step_integer_def
tff(fact_8824_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Z)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% csqrt.simps(1)
tff(fact_8825_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),L) = zero_z3403309356797280102nteger ).

% times_integer_code(2)
tff(fact_8826_times__integer__code_I1_J,axiom,
    ! [K2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_8827_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).

% minus_integer_code(2)
tff(fact_8828_minus__integer__code_I1_J,axiom,
    ! [K2: code_integer] : minus_8373710615458151222nteger(K2,zero_z3403309356797280102nteger) = K2 ).

% minus_integer_code(1)
tff(fact_8829_full__exhaustive__integer_H_Ocases,axiom,
    ! [Xb: produc2453742332425399446nteger] :
      ~ ! [F2: fun_Pr5098669352780829690e_term,D2: code_integer,I2: code_integer] : Xb != produc565448451787809096nteger(F2,produc1086072967326762835nteger(D2,I2)) ).

% full_exhaustive_integer'.cases
tff(fact_8830_exhaustive__integer_H_Ocases,axiom,
    ! [Xb: produc7509114972854644512nteger] :
      ~ ! [F2: fun_Co3460408610320954864e_term,D2: code_integer,I2: code_integer] : Xb != produc7264323995540650962nteger(F2,produc1086072967326762835nteger(D2,I2)) ).

% exhaustive_integer'.cases
tff(fact_8831_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,L) = L ).

% plus_integer_code(2)
tff(fact_8832_plus__integer__code_I1_J,axiom,
    ! [K2: code_integer] : plus_p5714425477246183910nteger(K2,zero_z3403309356797280102nteger) = K2 ).

% plus_integer_code(1)
tff(fact_8833_less__eq__integer__code_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_eq_integer_code(1)
tff(fact_8834_sgn__integer__code,axiom,
    ! [K2: code_integer] :
      aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K2) = $ite(
        K2 = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) ) ).

% sgn_integer_code
tff(fact_8835_divmod__integer_H__def,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(Mb,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Nb)),modulo364778990260209775nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Nb))) ).

% divmod_integer'_def
tff(fact_8836_imaginary__unit_Osimps_I1_J,axiom,
    aa_complex_real(re,imaginary_unit) = zero_zero_real ).

% imaginary_unit.simps(1)
tff(fact_8837_complex__Re__le__cmod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xb)),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% complex_Re_le_cmod
tff(fact_8838_zero__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,zero_zero_complex) = zero_zero_real ).

% zero_complex.simps(1)
tff(fact_8839_plus__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_real(re,aa_complex_complex(plus_plus_complex(Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,Xb)),aa_complex_real(re,Yb)) ).

% plus_complex.simps(1)
tff(fact_8840_abs__Re__le__cmod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,Xb))),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% abs_Re_le_cmod
tff(fact_8841_Re__csqrt,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z))) ).

% Re_csqrt
tff(fact_8842_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_8843_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
tff(fact_8844_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),zero_zero_real)
    <=> ( aa_complex_real(re,Z) = aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_8845_csqrt_Ocode,axiom,
    ! [Z: complex] :
      aa_complex_complex(csqrt,Z) = aa_real_complex(complex2(aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
            $ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_8846_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      aa_complex_real(im,aa_complex_complex(csqrt,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_8847_integer__of__int__code,axiom,
    ! [K2: int] :
      aa_int_Code_integer(code_integer_of_int,K2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K2))),
        $ite(
          K2 = zero_zero_int,
          zero_z3403309356797280102nteger,
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(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,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_8848_csqrt__of__real__nonpos,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xb)),zero_zero_real)
       => ( aa_complex_complex(csqrt,Xb) = aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_real_real(abs_abs_real,aa_complex_real(re,Xb))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_8849_complex__Im__of__int,axiom,
    ! [Z: int] : aa_complex_real(im,aa_int_complex(ring_17405671764205052669omplex,Z)) = zero_zero_real ).

% complex_Im_of_int
tff(fact_8850_complex__Im__fact,axiom,
    ! [Nb: nat] : aa_complex_real(im,aa_nat_complex(semiri5044797733671781792omplex,Nb)) = zero_zero_real ).

% complex_Im_fact
tff(fact_8851_complex__Im__of__nat,axiom,
    ! [Nb: nat] : aa_complex_real(im,aa_nat_complex(semiri8010041392384452111omplex,Nb)) = zero_zero_real ).

% complex_Im_of_nat
tff(fact_8852_Im__complex__of__real,axiom,
    ! [Z: real] : aa_complex_real(im,aa_real_complex(real_V4546457046886955230omplex,Z)) = zero_zero_real ).

% Im_complex_of_real
tff(fact_8853_Im__power__real,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_complex_real(im,aa_nat_complex(power_power_complex(Xb),Nb)) = zero_zero_real ) ) ).

% Im_power_real
tff(fact_8854_complex__Im__numeral,axiom,
    ! [V: num] : aa_complex_real(im,numera6690914467698888265omplex(V)) = zero_zero_real ).

% complex_Im_numeral
tff(fact_8855_Re__power__real,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_complex_real(re,aa_nat_complex(power_power_complex(Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),Nb) ) ) ).

% Re_power_real
tff(fact_8856_csqrt__of__real__nonneg,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xb))
       => ( aa_complex_complex(csqrt,Xb) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_complex_real(re,Xb))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_8857_csqrt__minus,axiom,
    ! [Xb: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,Xb)),zero_zero_real)
        | ( ( aa_complex_real(im,Xb) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xb)) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,Xb)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,Xb)) ) ) ).

% csqrt_minus
tff(fact_8858_uminus__integer__code_I1_J,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% uminus_integer_code(1)
tff(fact_8859_zero__integer__def,axiom,
    zero_z3403309356797280102nteger = aa_int_Code_integer(code_integer_of_int,zero_zero_int) ).

% zero_integer_def
tff(fact_8860_less__integer__code_I1_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_integer_code(1)
tff(fact_8861_abs__integer__code,axiom,
    ! [K2: code_integer] :
      abs_abs_Code_integer(K2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2),K2) ).

% abs_integer_code
tff(fact_8862_zero__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,zero_zero_complex) = zero_zero_real ).

% zero_complex.simps(2)
tff(fact_8863_one__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,one_one_complex) = zero_zero_real ).

% one_complex.simps(2)
tff(fact_8864_plus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : plus_p5714425477246183910nteger(aa_int_Code_integer(code_integer_of_int,Xaa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xaa),Xb)) ).

% plus_integer.abs_eq
tff(fact_8865_less__eq__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xaa),Xb) ) ).

% less_eq_integer.abs_eq
tff(fact_8866_plus__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_real(im,aa_complex_complex(plus_plus_complex(Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,Xb)),aa_complex_real(im,Yb)) ).

% plus_complex.simps(2)
tff(fact_8867_abs__Im__le__cmod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,Xb))),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% abs_Im_le_cmod
tff(fact_8868_times__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Yb))) ).

% times_complex.simps(2)
tff(fact_8869_cmod__eq__Re,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(im,Z) = zero_zero_real )
     => ( aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(abs_abs_real,aa_complex_real(re,Z)) ) ) ).

% cmod_eq_Re
tff(fact_8870_cmod__eq__Im,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(re,Z) = zero_zero_real )
     => ( aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(abs_abs_real,aa_complex_real(im,Z)) ) ) ).

% cmod_eq_Im
tff(fact_8871_Im__eq__0,axiom,
    ! [Z: complex] :
      ( ( aa_real_real(abs_abs_real,aa_complex_real(re,Z)) = aa_complex_real(real_V1022390504157884413omplex,Z) )
     => ( aa_complex_real(im,Z) = zero_zero_real ) ) ).

% Im_eq_0
tff(fact_8872_cmod__Im__le__iff,axiom,
    ! [Xb: complex,Yb: complex] :
      ( ( aa_complex_real(re,Xb) = aa_complex_real(re,Yb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,Xb))),aa_real_real(abs_abs_real,aa_complex_real(im,Yb))) ) ) ).

% cmod_Im_le_iff
tff(fact_8873_cmod__Re__le__iff,axiom,
    ! [Xb: complex,Yb: complex] :
      ( ( aa_complex_real(im,Xb) = aa_complex_real(im,Yb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,Xb))),aa_real_real(abs_abs_real,aa_complex_real(re,Yb))) ) ) ).

% cmod_Re_le_iff
tff(fact_8874_plus__complex_Ocode,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_complex(plus_plus_complex(Xb),Yb) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,Xb)),aa_complex_real(re,Yb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,Xb)),aa_complex_real(im,Yb))) ).

% plus_complex.code
tff(fact_8875_csqrt__principal,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z)))
      | ( ( aa_complex_real(re,aa_complex_complex(csqrt,Z)) = zero_zero_real )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(csqrt,Z))) ) ) ).

% csqrt_principal
tff(fact_8876_cmod__le,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))) ).

% cmod_le
tff(fact_8877_complex__eq,axiom,
    ! [A: complex] : A = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(re,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(im,A)))) ).

% complex_eq
tff(fact_8878_times__complex_Ocode,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_complex(times_times_complex(Xb),Yb) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(re,Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Yb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Yb)))) ).

% times_complex.code
tff(fact_8879_cmod__power2,axiom,
    ! [Z: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cmod_power2
tff(fact_8880_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_8881_norm__complex__def,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% norm_complex_def
tff(fact_8882_inverse__complex_Osimps_I1_J,axiom,
    ! [Xb: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(1)
tff(fact_8883_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% complex_neq_0
tff(fact_8884_Re__divide,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(re,Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Yb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Re_divide
tff(fact_8885_csqrt__square,axiom,
    ! [B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,B))
        | ( ( aa_complex_real(re,B) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B)) ) )
     => ( aa_complex_complex(csqrt,aa_nat_complex(power_power_complex(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = B ) ) ).

% csqrt_square
tff(fact_8886_csqrt__unique,axiom,
    ! [W2: complex,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,W2))
          | ( ( aa_complex_real(re,W2) = zero_zero_real )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W2)) ) )
       => ( aa_complex_complex(csqrt,Z) = W2 ) ) ) ).

% csqrt_unique
tff(fact_8887_inverse__complex_Osimps_I2_J,axiom,
    ! [Xb: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(2)
tff(fact_8888_Im__divide,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Xb),Yb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Yb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Im_divide
tff(fact_8889_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z))) ).

% complex_abs_le_norm
tff(fact_8890_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_8891_inverse__complex_Ocode,axiom,
    ! [Xb: complex] : aa_complex_complex(invers8013647133539491842omplex,Xb) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% inverse_complex.code
tff(fact_8892_Complex__divide,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),Yb) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(re,Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Yb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Yb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Yb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% Complex_divide
tff(fact_8893_cos__Arg__i__mult__zero,axiom,
    ! [Yb: complex] :
      ( ( Yb != zero_zero_complex )
     => ( ( aa_complex_real(re,Yb) = zero_zero_real )
       => ( aa_real_real(cos_real,aa_complex_real(arg,Yb)) = zero_zero_real ) ) ) ).

% cos_Arg_i_mult_zero
tff(fact_8894_complex__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_mult_cnj
tff(fact_8895_complex__cnj__zero,axiom,
    aa_complex_complex(cnj,zero_zero_complex) = zero_zero_complex ).

% complex_cnj_zero
tff(fact_8896_complex__cnj__zero__iff,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(cnj,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_complex ) ) ).

% complex_cnj_zero_iff
tff(fact_8897_complex__cnj__add,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_complex(cnj,aa_complex_complex(plus_plus_complex(Xb),Yb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(cnj,Xb)),aa_complex_complex(cnj,Yb)) ).

% complex_cnj_add
tff(fact_8898_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = zero_zero_real ).

% complex_In_mult_cnj_zero
tff(fact_8899_Arg__zero,axiom,
    aa_complex_real(arg,zero_zero_complex) = zero_zero_real ).

% Arg_zero
tff(fact_8900_Re__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)) = zero_zero_real )
    <=> ( aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Re_complex_div_eq_0
tff(fact_8901_Im__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)) = zero_zero_real )
    <=> ( aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Im_complex_div_eq_0
tff(fact_8902_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Re_complex_div_lt_0
tff(fact_8903_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Re_complex_div_gt_0
tff(fact_8904_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Re_complex_div_ge_0
tff(fact_8905_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Re_complex_div_le_0
tff(fact_8906_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Im_complex_div_gt_0
tff(fact_8907_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Im_complex_div_lt_0
tff(fact_8908_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Im_complex_div_le_0
tff(fact_8909_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Im_complex_div_ge_0
tff(fact_8910_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) )
      & ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ) ).

% complex_div_gt_0
tff(fact_8911_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ).

% Arg_bounded
tff(fact_8912_complex__add__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Z))) ).

% complex_add_cnj
tff(fact_8913_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W2))),aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Z)),W2)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W2))))) ).

% cnj_add_mult_eq_Re
tff(fact_8914_integer__of__num_I3_J,axiom,
    ! [Nb: num] :
      code_integer_of_num(aa_num_num(bit1,Nb)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Nb),
        plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(k,k),one_one_Code_integer) ) ).

% integer_of_num(3)
tff(fact_8915_int__of__integer__code,axiom,
    ! [K2: code_integer] :
      code_int_of_integer(K2) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2))),
        $ite(K2 = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_aam,code_divmod_integer(K2,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_8916_bit__cut__integer__def,axiom,
    ! [K2: code_integer] : code_bit_cut_integer(K2) = produc6677183202524767010eger_o(divide6298287555418463151nteger(K2,numera6620942414471956472nteger(aa_num_num(bit0,one))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),K2)) ).

% bit_cut_integer_def
tff(fact_8917_num__of__integer__code,axiom,
    ! [K2: code_integer] :
      code_num_of_integer(K2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_aan,code_divmod_integer(K2,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_8918_zero__integer_Orep__eq,axiom,
    code_int_of_integer(zero_z3403309356797280102nteger) = zero_zero_int ).

% zero_integer.rep_eq
tff(fact_8919_plus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(plus_p5714425477246183910nteger(Xb,Xaa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% plus_integer.rep_eq
tff(fact_8920_less__eq__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),Xaa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ) ).

% less_eq_integer.rep_eq
tff(fact_8921_integer__less__eq__iff,axiom,
    ! [K2: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,code_int_of_integer(K2)),code_int_of_integer(L)) ) ).

% integer_less_eq_iff
tff(fact_8922_integer__of__num_I2_J,axiom,
    ! [Nb: num] :
      code_integer_of_num(aa_num_num(bit0,Nb)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Nb),
        plus_p5714425477246183910nteger(k,k) ) ).

% integer_of_num(2)
tff(fact_8923_divmod__integer__def,axiom,
    ! [K2: code_integer,L: code_integer] : code_divmod_integer(K2,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(K2,L),modulo364778990260209775nteger(K2,L)) ).

% divmod_integer_def
tff(fact_8924_bit__cut__integer__code,axiom,
    ! [K2: code_integer] :
      code_bit_cut_integer(K2) = $ite(K2 = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_aao(K2),code_divmod_abs(K2,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_8925_nat__of__integer__code,axiom,
    ! [K2: code_integer] :
      code_nat_of_integer(K2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_aap,code_divmod_integer(K2,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_8926_card__lessThan,axiom,
    ! [U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_ord_lessThan_nat,U)) = U ).

% card_lessThan
tff(fact_8927_card__Collect__less__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bb,Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_8928_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_8929_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,set_or4665077453230672383an_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_atLeastLessThan
tff(fact_8930_card__Collect__le__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bc,Nb))) = aa_nat_nat(suc,Nb) ).

% card_Collect_le_nat
tff(fact_8931_of__nat__of__integer,axiom,
    ! [K2: code_integer] : aa_nat_Code_integer(semiri4939895301339042750nteger,code_nat_of_integer(K2)) = ord_max_Code_integer(zero_z3403309356797280102nteger,K2) ).

% of_nat_of_integer
tff(fact_8932_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,set_or1269000886237332187st_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_8933_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_atLeastLessThan_int
tff(fact_8934_nat__of__integer__non__positive,axiom,
    ! [K2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),zero_z3403309356797280102nteger)
     => ( code_nat_of_integer(K2) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_8935_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_8936_card__less,axiom,
    ! [M5: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M5)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aaq(M5),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_8937_card__less__Suc,axiom,
    ! [M5: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M5)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aar(M5),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aaq(M5),I))) ) ) ).

% card_less_Suc
tff(fact_8938_card__less__Suc2,axiom,
    ! [M5: set_nat,I: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M5)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aar(M5),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aaq(M5),I))) ) ) ).

% card_less_Suc2
tff(fact_8939_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_card_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,zero_zero_int),U)) = aa_int_nat(nat2,U) ).

% card_atLeastZeroLessThan_int
tff(fact_8940_nat__of__integer__code__post_I1_J,axiom,
    code_nat_of_integer(zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_8941_subset__card__intvl__is__intvl,axiom,
    ! [A4: set_nat,K2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),set_or4665077453230672383an_nat(K2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_set_nat_nat(finite_card_nat,A4))))
     => ( A4 = set_or4665077453230672383an_nat(K2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_set_nat_nat(finite_card_nat,A4))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_8942_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_z3403309356797280102nteger,J) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ).

% divmod_abs_code(6)
tff(fact_8943_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N4: set_nat,Nb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N4),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N4)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_8944_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kr),set_or4665077453230672383an_nat(zero_zero_nat,aa_set_nat_nat(finite_card_nat,S2)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kr),S2)) ).

% card_sum_le_nat_sum
tff(fact_8945_card__nth__roots,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_aas(C2),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_8946_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cr,Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_8947_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_z3403309356797280102nteger) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,abs_abs_Code_integer(J)) ).

% divmod_abs_code(5)
tff(fact_8948_divmod__abs__def,axiom,
    ! [K2: code_integer,L: code_integer] : code_divmod_abs(K2,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(abs_abs_Code_integer(K2),abs_abs_Code_integer(L)),modulo364778990260209775nteger(abs_abs_Code_integer(K2),abs_abs_Code_integer(L))) ).

% divmod_abs_def
tff(fact_8949_divmod__integer__code,axiom,
    ! [K2: code_integer,L: code_integer] :
      code_divmod_integer(K2,L) = $ite(
        K2 = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K2),code_divmod_abs(K2,L),produc6916734918728496179nteger(aTP_Lamm_aat(L),code_divmod_abs(K2,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K2),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),code_divmod_abs(K2,L),produc6916734918728496179nteger(aTP_Lamm_aau(L),code_divmod_abs(K2,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_8950_pair__leqI2,axiom,
    ! [A: nat,B: nat,S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S3),Ta)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Ta)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_8951_pair__leqI1,axiom,
    ! [A: nat,B: nat,S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Ta)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_8952_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_aav),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_8953_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_aaw),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_8954_less__eq__nat_Osimps_I2_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,Mb)),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_8955_max__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_aax,Nb)),Mb) ).

% max_Suc1
tff(fact_8956_max__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_aay,Nb)),Mb) ).

% max_Suc2
tff(fact_8957_diff__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_kr),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ).

% diff_Suc
tff(fact_8958_binomial__def,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_aaz(Nb),K2))) ).

% binomial_def
tff(fact_8959_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% push_bit_nonnegative_int_iff
tff(fact_8960_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_8961_concat__bit__of__zero__1,axiom,
    ! [Nb: nat,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),zero_zero_int),L) = aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),L) ).

% concat_bit_of_zero_1
tff(fact_8962_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% push_bit_of_Suc_0
tff(fact_8963_bit__push__bit__iff__int,axiom,
    ! [Mb: nat,K2: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Mb),K2)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_int
tff(fact_8964_bit__push__bit__iff__nat,axiom,
    ! [Mb: nat,Q3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),Q3)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_8965_concat__bit__eq,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K2)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),L)) ).

% concat_bit_eq
tff(fact_8966_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_kr),Nat) ).

% pred_def
tff(fact_8967_wmin__insertI,axiom,
    ! [Xb: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Yb: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xb),XS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(Xb,Yb),fun_pair_leq)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_min_weak)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Yb,YS)),fun_min_weak) ) ) ) ).

% wmin_insertI
tff(fact_8968_wmax__insertI,axiom,
    ! [Yb: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,Xb: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Yb),YS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(Xb,Yb),fun_pair_leq)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_max_weak)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(Xb,XS),YS),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_8969_bezw__0,axiom,
    ! [Xb: nat] : bezw(Xb,zero_zero_nat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_8970_prod__decode__aux_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Yb )
     => ( Yb = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xb)))) ) ) ).

% prod_decode_aux.elims
tff(fact_8971_wmax__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,X5)
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,X5),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_8972_wmin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] : member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,bot_bo2099793752762293965at_nat),fun_min_weak) ).

% wmin_emptyI
tff(fact_8973_prod__decode__aux_Osimps,axiom,
    ! [K2: nat,Mb: nat] :
      nat_prod_decode_aux(K2,Mb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),K2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),Mb)),nat_prod_decode_aux(aa_nat_nat(suc,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,K2)))) ).

% prod_decode_aux.simps
tff(fact_8974_prod__decode__aux_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Yb )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Yb = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xb)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_8975_smax__insertI,axiom,
    ! [Yb: product_prod_nat_nat,Y5: set_Pr1261947904930325089at_nat,Xb: product_prod_nat_nat,X5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Yb),Y5)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(Xb,Yb),fun_pair_less)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Y5),fun_max_strict)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(Xb,X5),Y5),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_8976_smin__insertI,axiom,
    ! [Xb: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Yb: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xb),XS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(Xb,Yb),fun_pair_less)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_min_strict)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Yb,YS)),fun_min_strict) ) ) ) ).

% smin_insertI
tff(fact_8977_Suc__0__div__numeral,axiom,
    ! [K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K2)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K2)) ).

% Suc_0_div_numeral
tff(fact_8978_smin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( ( X5 != bot_bo2099793752762293965at_nat )
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,bot_bo2099793752762293965at_nat),fun_min_strict) ) ).

% smin_emptyI
tff(fact_8979_smax__emptyI,axiom,
    ! [Y5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Y5)
     => ( ( Y5 != bot_bo2099793752762293965at_nat )
       => member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,Y5),fun_max_strict) ) ) ).

% smax_emptyI
tff(fact_8980_Suc__0__mod__numeral,axiom,
    ! [K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K2)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K2)) ).

% Suc_0_mod_numeral
tff(fact_8981_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ? [R3: fun_nat_nat] :
          ( aa_set_nat_o(strict1292158309912662752at_nat(R3),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S2)))
          & ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),aa_set_nat_nat(finite_card_nat,S2))
             => aa_set_nat_o(member_nat(aa_nat_nat(R3,N5)),S2) ) ) ) ).

% finite_enumerate
tff(fact_8982_divmod__integer__eq__cases,axiom,
    ! [K2: code_integer,L: code_integer] :
      code_divmod_integer(K2,L) = $ite(
        K2 = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K2),
          aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K2) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K2,L),produc6916734918728496179nteger(aTP_Lamm_aba(L),code_divmod_abs(K2,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_8983_rat__sgn__code,axiom,
    ! [P: rat] : quotient_of(aa_rat_rat(sgn_sgn_rat,P)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(sgn_sgn_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,quotient_of(P)))),one_one_int) ).

% rat_sgn_code
tff(fact_8984_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Yb )
     => ( Yb = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_8985_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_8986_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,quotient_of(R2))) ).

% quotient_of_denom_pos'
tff(fact_8987_bezw__non__0,axiom,
    ! [Yb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Yb)
     => ( bezw(Xb,Yb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Yb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Yb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Yb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Yb))))) ) ) ).

% bezw_non_0
tff(fact_8988_bezw_Osimps,axiom,
    ! [Xb: nat,Yb: nat] :
      bezw(Xb,Yb) = $ite(Yb = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Yb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Yb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Yb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Yb)))))) ).

% bezw.simps
tff(fact_8989_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Yb )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Yb = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_8990_normalize__def,axiom,
    ! [P: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(normalize,P) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),
        $let(
          a: int,
          a:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),a)) ),
        $ite(
          aa_Pro6799258434465214121nt_int(product_snd_int_int,P) = zero_zero_int,
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),
          $let(
            a: int,
            a:= aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P))),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),a)) ) ) ) ).

% normalize_def
tff(fact_8991_gcd__pos__int,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Mb),Nb))
    <=> ( ( Mb != zero_zero_int )
        | ( Nb != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_8992_gcd__0__left__int,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,zero_zero_int),Xb) = aa_int_int(abs_abs_int,Xb) ).

% gcd_0_left_int
tff(fact_8993_gcd__0__int,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),zero_zero_int) = aa_int_int(abs_abs_int,Xb) ).

% gcd_0_int
tff(fact_8994_gcd__ge__0__int,axiom,
    ! [Xb: int,Yb: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Yb)) ).

% gcd_ge_0_int
tff(fact_8995_bezout__int,axiom,
    ! [Xb: int,Yb: int] :
    ? [U2: int,V2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U2),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Yb)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Yb) ).

% bezout_int
tff(fact_8996_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),A) ) ).

% gcd_le1_int
tff(fact_8997_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),B) ) ).

% gcd_le2_int
tff(fact_8998_gcd__cases__int,axiom,
    ! [Xb: int,Yb: int,P2: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
         => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Yb)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),zero_zero_int)
           => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),aa_int_int(uminus_uminus_int,Yb))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
             => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,Xb)),Yb)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),zero_zero_int)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,Xb)),aa_int_int(uminus_uminus_int,Yb))) ) )
           => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Yb)) ) ) ) ) ).

% gcd_cases_int
tff(fact_8999_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),A)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
        & ! [E3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),A)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),B) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),D) ) )
    <=> ( D = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ) ) ).

% gcd_unique_int
tff(fact_9000_gcd__non__0__int,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Yb)
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Yb) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Yb),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Yb)) ) ) ).

% gcd_non_0_int
tff(fact_9001_gcd__code__int,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(gcd_gcd_int,K2),L) = aa_int_int(abs_abs_int,
        $ite(L = zero_zero_int,K2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,K2)),aa_int_int(abs_abs_int,L))))) ).

% gcd_code_int
tff(fact_9002_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,aa_complex_real(arg,Z)) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ) ).

% Arg_correct
tff(fact_9003_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_9004_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),A) = A ).

% gcd_nat.left_neutral
tff(fact_9005_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_9006_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),zero_zero_nat) = A ).

% gcd_nat.right_neutral
tff(fact_9007_gcd__0__nat,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),zero_zero_nat) = Xb ).

% gcd_0_nat
tff(fact_9008_gcd__0__left__nat,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),Xb) = Xb ).

% gcd_0_left_nat
tff(fact_9009_gcd__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9010_gcd__pos__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb))
    <=> ( ( Mb != zero_zero_nat )
        | ( Nb != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9011_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_9012_real__eq__imaginary__iff,axiom,
    ! [Yb: complex,Xb: complex] :
      ( aa_set_complex_o(member_complex(Yb),real_V2521375963428798218omplex)
     => ( aa_set_complex_o(member_complex(Xb),real_V2521375963428798218omplex)
       => ( ( Xb = aa_complex_complex(times_times_complex(imaginary_unit),Yb) )
        <=> ( ( Xb = zero_zero_complex )
            & ( Yb = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_9013_imaginary__eq__real__iff,axiom,
    ! [Yb: complex,Xb: complex] :
      ( aa_set_complex_o(member_complex(Yb),real_V2521375963428798218omplex)
     => ( aa_set_complex_o(member_complex(Xb),real_V2521375963428798218omplex)
       => ( ( aa_complex_complex(times_times_complex(imaginary_unit),Yb) = Xb )
        <=> ( ( Xb = zero_zero_complex )
            & ( Yb = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_9014_cis__neq__zero,axiom,
    ! [A: real] : aa_real_complex(cis,A) != zero_zero_complex ).

% cis_neq_zero
tff(fact_9015_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) ) ).

% gcd_le2_nat
tff(fact_9016_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A) ) ).

% gcd_le1_nat
tff(fact_9017_gcd__diff1__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff1_nat
tff(fact_9018_gcd__diff2__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff2_nat
tff(fact_9019_gcd__non__0__nat,axiom,
    ! [Yb: nat,Xb: nat] :
      ( ( Yb != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Yb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Yb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb)) ) ) ).

% gcd_non_0_nat
tff(fact_9020_gcd__nat_Osimps,axiom,
    ! [Xb: nat,Yb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Yb) = $ite(Yb = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Yb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb))) ).

% gcd_nat.simps
tff(fact_9021_gcd__nat_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Yb )
     => ( Yb = $ite(Xaa = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))) ) ) ).

% gcd_nat.elims
tff(fact_9022_complex__is__Real__iff,axiom,
    ! [Z: complex] :
      ( aa_set_complex_o(member_complex(Z),real_V2521375963428798218omplex)
    <=> ( aa_complex_real(im,Z) = zero_zero_real ) ) ).

% complex_is_Real_iff
tff(fact_9023_Complex__in__Reals,axiom,
    ! [Xb: real] : aa_set_complex_o(member_complex(aa_real_complex(complex2(Xb),zero_zero_real)),real_V2521375963428798218omplex) ).

% Complex_in_Reals
tff(fact_9024_cis__mult,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ).

% cis_mult
tff(fact_9025_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X3: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_9026_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X3: nat,Y: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9027_cis__Arg,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_complex(cis,aa_complex_real(arg,Z)) = aa_complex_complex(sgn_sgn_complex,Z) ) ) ).

% cis_Arg
tff(fact_9028_gcd__code__integer,axiom,
    ! [K2: code_integer,L: code_integer] :
      gcd_gcd_Code_integer(K2,L) = abs_abs_Code_integer($ite(L = zero_z3403309356797280102nteger,K2,gcd_gcd_Code_integer(L,modulo364778990260209775nteger(abs_abs_Code_integer(K2),abs_abs_Code_integer(L))))) ).

% gcd_code_integer
tff(fact_9029_cis__Arg__unique,axiom,
    ! [Z: complex,Xb: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,Xb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => ( aa_complex_real(arg,Z) = Xb ) ) ) ) ).

% cis_Arg_unique
tff(fact_9030_bezw__aux,axiom,
    ! [Xb: nat,Yb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Yb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xb,Yb))),aa_nat_int(semiri1314217659103216013at_int,Xb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xb,Yb))),aa_nat_int(semiri1314217659103216013at_int,Yb))) ).

% bezw_aux
tff(fact_9031_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Yb )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Yb = $ite(Xaa = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9032_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_complex_o(bij_betw_nat_complex(aTP_Lamm_abb(Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cr,Nb))) ) ).

% bij_betw_roots_unity
tff(fact_9033_Arg__def,axiom,
    ! [Z: complex] :
      aa_complex_real(arg,Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_abc(Z))) ).

% Arg_def
tff(fact_9034_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9035_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_9036_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9037_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9038_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9039_root__powr__inverse,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_9040_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,L),U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_9041_real__root__zero,axiom,
    ! [Nb: nat] : aa_real_real(aa_nat_fun_real_real(root,Nb),zero_zero_real) = zero_zero_real ).

% real_root_zero
tff(fact_9042_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,L),U)) ).

% finite_greaterThanLessThan_int
tff(fact_9043_real__root__Suc__0,axiom,
    ! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),Xb) = Xb ).

% real_root_Suc_0
tff(fact_9044_root__0,axiom,
    ! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),Xb) = zero_zero_real ).

% root_0
tff(fact_9045_real__root__eq__iff,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(aa_nat_fun_real_real(root,Nb),Yb) )
      <=> ( Xb = Yb ) ) ) ).

% real_root_eq_iff
tff(fact_9046_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = zero_zero_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9047_real__root__less__iff,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ).

% real_root_less_iff
tff(fact_9048_real__root__le__iff,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ).

% real_root_le_iff
tff(fact_9049_real__root__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9050_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = one_one_real )
      <=> ( Xb = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9051_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb) ) ) ).

% real_root_gt_0_iff
tff(fact_9052_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9053_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9054_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb) ) ) ).

% real_root_ge_0_iff
tff(fact_9055_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Yb) ) ) ).

% real_root_gt_1_iff
tff(fact_9056_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9057_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9058_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Yb) ) ) ).

% real_root_ge_1_iff
tff(fact_9059_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_9060_real__root__pos__pos__le,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ).

% real_root_pos_pos_le
tff(fact_9061_real__root__less__mono,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb)) ) ) ).

% real_root_less_mono
tff(fact_9062_real__root__le__mono,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb)) ) ) ).

% real_root_le_mono
tff(fact_9063_real__root__power,axiom,
    ! [Nb: nat,Xb: real,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),K2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),K2) ) ) ).

% real_root_power
tff(fact_9064_real__root__abs,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(abs_abs_real,Xb)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_abs
tff(fact_9065_sgn__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) = aa_real_real(sgn_sgn_real,Xb) ) ) ).

% sgn_root
tff(fact_9066_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int)),U) = aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,L),U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_9067_real__root__gt__zero,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_9068_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N4: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N4),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9069_root__abs__power,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),Nb))) = aa_real_real(abs_abs_real,Yb) ) ) ).

% root_abs_power
tff(fact_9070_real__root__pos__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_9071_real__root__strict__increasing,axiom,
    ! [Nb: nat,N4: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N4),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9072_real__root__decreasing,axiom,
    ! [Nb: nat,N4: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N4),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_9073_real__root__pow__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_9074_real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_9075_real__root__pos__unique,axiom,
    ! [Nb: nat,Yb: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),Nb) = Xb )
         => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = Yb ) ) ) ) ).

% real_root_pos_unique
tff(fact_9076_real__root__increasing,axiom,
    ! [Nb: nat,N4: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N4),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_9077_root__sgn__power,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Yb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Yb)),Nb))) = Yb ) ) ).

% root_sgn_power
tff(fact_9078_sgn__power__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),Nb)) = Xb ) ) ).

% sgn_power_root
tff(fact_9079_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_set_complex_o(bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_complex_real(real_V1022390504157884413omplex,C2)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C2)),aa_nat_real(semiri5074537144036343181t_real,Nb))))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cr,Nb))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_aas(C2),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9080_ln__root,axiom,
    ! [Nb: nat,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Nb),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% ln_root
tff(fact_9081_log__root,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_nat_fun_real_real(root,Nb),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),A)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_root
tff(fact_9082_log__base__root,axiom,
    ! [Nb: nat,B: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,Nb),B)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ).

% log_base_root
tff(fact_9083_split__root,axiom,
    ! [P2: fun_real_o,Nb: nat,Xb: real] :
      ( aa_real_o(P2,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_real_o(P2,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ! [Y4: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y4)),Nb)) = Xb )
             => aa_real_o(P2,Y4) ) ) ) ) ).

% split_root
tff(fact_9084_sin__integer__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9085_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or5834768355832116004an_nat(L,U)) ).

% finite_greaterThanLessThan
tff(fact_9086_Suc__funpow,axiom,
    ! [Nb: nat] : compow_fun_nat_nat(Nb,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Nb) ).

% Suc_funpow
tff(fact_9087_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,set_or5834768355832116004an_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_9088_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,L),U) = set_or5834768355832116004an_nat(L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_9089_sin__times__pi__eq__0,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),pi)) = zero_zero_real )
    <=> member_real(Xb,ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_9090_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( aa_set_complex_o(member_complex(Z),ring_1_Ints_complex)
    <=> ( ( aa_complex_real(im,Z) = zero_zero_real )
        & ? [I3: int] : aa_complex_real(re,Z) = aa_int_real(ring_1_of_int_real,I3) ) ) ).

% complex_is_Int_iff
tff(fact_9091_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_bc,aTP_Lamm_bb) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9092_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_abd) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9093_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9094_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Yb: list_int] :
      ( ( upto(Xb,Xaa) = Yb )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Yb = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_9095_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I),J))
     => ( upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_9096_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil
tff(fact_9097_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil2
tff(fact_9098_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I)
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_9099_upto__single,axiom,
    ! [I: int] : upto(I,I) = cons_int(I,nil_int) ).

% upto_single
tff(fact_9100_nth__upto,axiom,
    ! [I: int,K2: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K2))),J)
     => ( aa_nat_int(nth_int(upto(I,J)),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K2)) ) ) ).

% nth_upto
tff(fact_9101_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_9102_upto__rec__numeral_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9103_upto__rec__numeral_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9104_upto__rec__numeral_I3_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9105_upto__rec__numeral_I4_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9106_upto__split1,axiom,
    ! [I: int,J: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K2)
       => ( upto(I,K2) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K2)) ) ) ) ).

% upto_split1
tff(fact_9107_upto__split2,axiom,
    ! [I: int,J: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K2)
       => ( upto(I,K2) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K2)) ) ) ) ).

% upto_split2
tff(fact_9108_upto__aux__def,axiom,
    ! [I: int,J: int,Js: list_int] : upto_aux(I,J,Js) = append_int(upto(I,J),Js) ).

% upto_aux_def
tff(fact_9109_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,I),J) = set_int2(upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_9110_distinct__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(distinct_int,upto(I,J)) ).

% distinct_upto
tff(fact_9111_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil_int) ).

% upto_code
tff(fact_9112_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9113_upto__split3,axiom,
    ! [I: int,J: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K2)
       => ( upto(I,K2) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K2))) ) ) ) ).

% upto_split3
tff(fact_9114_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,I),J) = set_int2(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9115_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9116_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Yb: list_int] :
      ( ( upto(Xb,Xaa) = Yb )
     => ( Yb = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) ) ) ).

% upto.elims
tff(fact_9117_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_9118_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,I),J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_9119_take__bit__numeral__minus__numeral__int,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,aTP_Lamm_abe(Mb),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,Mb),Nb)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9120_and__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(3)
tff(fact_9121_and__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: 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,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(7)
tff(fact_9122_take__bit__num__simps_I1_J,axiom,
    ! [Mb: num] : bit_take_bit_num(zero_zero_nat,Mb) = none_num ).

% take_bit_num_simps(1)
tff(fact_9123_take__bit__num__simps_I2_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(aa_nat_nat(suc,Nb),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_9124_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_abf,bit_take_bit_num(Nb,Mb)) ).

% take_bit_num_simps(3)
tff(fact_9125_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Nb,Mb))) ).

% take_bit_num_simps(4)
tff(fact_9126_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_abf,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb)) ).

% take_bit_num_simps(6)
tff(fact_9127_and__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: 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,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(8)
tff(fact_9128_and__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(4)
tff(fact_9129_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_9130_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit0,Mb)) = case_nat_option_num(none_num,aTP_Lamm_abg(Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9131_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(Nb,one) = case_nat_option_num(none_num,aTP_Lamm_abh,Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9132_and__not__num_Osimps_I3_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit1,Nb)) = none_num ).

% and_not_num.simps(3)
tff(fact_9133_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit1,Mb)) = case_nat_option_num(none_num,aTP_Lamm_abi(Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9134_and__not__num__eq__None__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( bit_and_not_num(Mb,Nb) = none_num )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_9135_int__numeral__and__not__num,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,Nb)) ).

% int_numeral_and_not_num
tff(fact_9136_int__numeral__not__and__num,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Nb,Mb)) ).

% int_numeral_not_and_num
tff(fact_9137_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,Mb) = produc478579273971653890on_num(aTP_Lamm_abm,product_Pair_nat_num(Nb,Mb)) ).

% Bit_Operations.take_bit_num_code
tff(fact_9138_times__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_abo),Xaa),Xb)) ).

% times_int.abs_eq
tff(fact_9139_int_Oabs__induct,axiom,
    ! [P2: fun_int_o,Xb: int] :
      ( ! [Y: product_prod_nat_nat] : aa_int_o(P2,aa_Pro4240683299535347697at_int(abs_Integ,Y))
     => aa_int_o(P2,Xb) ) ).

% int.abs_induct
tff(fact_9140_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X3: nat,Y: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y)) ).

% eq_Abs_Integ
tff(fact_9141_nat_Oabs__eq,axiom,
    ! [Xb: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),Xb) ).

% nat.abs_eq
tff(fact_9142_zero__int__def,axiom,
    zero_zero_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int_def
tff(fact_9143_int__def,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Nb),zero_zero_nat)) ).

% int_def
tff(fact_9144_uminus__int_Oabs__eq,axiom,
    ! [Xb: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_abp),Xb)) ).

% uminus_int.abs_eq
tff(fact_9145_one__int__def,axiom,
    one_one_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_9146_less__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_abr),Xaa),Xb) ) ).

% less_int.abs_eq
tff(fact_9147_less__eq__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_abt),Xaa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_9148_plus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_abv),Xaa),Xb)) ).

% plus_int.abs_eq
tff(fact_9149_minus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_abx),Xaa),Xb)) ).

% minus_int.abs_eq
tff(fact_9150_take__bit__num__def,axiom,
    ! [Nb: nat,Mb: num] :
      bit_take_bit_num(Nb,Mb) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb)) = zero_zero_nat,none_num,some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb))))) ).

% take_bit_num_def
tff(fact_9151_and__not__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Yb: option_num] :
      ( ( bit_and_not_num(Xb,Xaa) = Yb )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Yb != none_num ) ) )
       => ( ( ( Xb = one )
           => ( ? [N2: num] : Xaa = aa_num_num(bit0,N2)
             => ( Yb != some_num(one) ) ) )
         => ( ( ( Xb = one )
             => ( ? [N2: num] : Xaa = aa_num_num(bit1,N2)
               => ( Yb != none_num ) ) )
           => ( ! [M: num] :
                  ( ( Xb = aa_num_num(bit0,M) )
                 => ( ( Xaa = one )
                   => ( Yb != some_num(aa_num_num(bit0,M)) ) ) )
             => ( ! [M: num] :
                    ( ( Xb = aa_num_num(bit0,M) )
                   => ! [N2: num] :
                        ( ( Xaa = aa_num_num(bit0,N2) )
                       => ( Yb != map_option_num_num(bit0,bit_and_not_num(M,N2)) ) ) )
               => ( ! [M: num] :
                      ( ( Xb = aa_num_num(bit0,M) )
                     => ! [N2: num] :
                          ( ( Xaa = aa_num_num(bit1,N2) )
                         => ( Yb != map_option_num_num(bit0,bit_and_not_num(M,N2)) ) ) )
                 => ( ! [M: num] :
                        ( ( Xb = aa_num_num(bit1,M) )
                       => ( ( Xaa = one )
                         => ( Yb != some_num(aa_num_num(bit0,M)) ) ) )
                   => ( ! [M: num] :
                          ( ( Xb = aa_num_num(bit1,M) )
                         => ! [N2: num] :
                              ( ( Xaa = aa_num_num(bit0,N2) )
                             => ( Yb != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_aby,bit_and_not_num(M,N2)) ) ) )
                     => ~ ! [M: num] :
                            ( ( Xb = aa_num_num(bit1,M) )
                           => ! [N2: num] :
                                ( ( Xaa = aa_num_num(bit1,N2) )
                               => ( Yb != map_option_num_num(bit0,bit_and_not_num(M,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_9152_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_max_strict,fun_max_weak)) ).

% max_rpair_set
tff(fact_9153_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9154_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,Nb)) = Nb ) ) ).

% numeral_num_of_nat
tff(fact_9155_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),one_one_nat)
     => ( aa_nat_num(num_of_nat,Nb) = one ) ) ).

% num_of_nat_One
tff(fact_9156_num__of__nat__double,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Nb)) = aa_num_num(bit0,aa_nat_num(num_of_nat,Nb)) ) ) ).

% num_of_nat_double
tff(fact_9157_num__of__nat__plus__distrib,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_num_num(plus_plus_num(aa_nat_num(num_of_nat,Mb)),aa_nat_num(num_of_nat,Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9158_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb),aa_num_num(inc,aa_nat_num(num_of_nat,Nb)),one) ).

% num_of_nat.simps(2)
tff(fact_9159_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_min_strict,fun_min_weak)) ).

% min_rpair_set
tff(fact_9160_and__not__num_Opelims,axiom,
    ! [Xb: num,Xaa: num,Yb: option_num] :
      ( ( bit_and_not_num(Xb,Xaa) = Yb )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(Xb,Xaa))
       => ( ( ( Xb = one )
           => ( ( Xaa = one )
             => ( ( Yb = none_num )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,one)) ) ) )
         => ( ( ( Xb = one )
             => ! [N2: num] :
                  ( ( Xaa = aa_num_num(bit0,N2) )
                 => ( ( Yb = some_num(one) )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
           => ( ( ( Xb = one )
               => ! [N2: num] :
                    ( ( Xaa = aa_num_num(bit1,N2) )
                   => ( ( Yb = none_num )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
             => ( ! [M: num] :
                    ( ( Xb = aa_num_num(bit0,M) )
                   => ( ( Xaa = one )
                     => ( ( Yb = some_num(aa_num_num(bit0,M)) )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M),one)) ) ) )
               => ( ! [M: num] :
                      ( ( Xb = aa_num_num(bit0,M) )
                     => ! [N2: num] :
                          ( ( Xaa = aa_num_num(bit0,N2) )
                         => ( ( Yb = map_option_num_num(bit0,bit_and_not_num(M,N2)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit0,N2))) ) ) )
                 => ( ! [M: num] :
                        ( ( Xb = aa_num_num(bit0,M) )
                       => ! [N2: num] :
                            ( ( Xaa = aa_num_num(bit1,N2) )
                           => ( ( Yb = map_option_num_num(bit0,bit_and_not_num(M,N2)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit1,N2))) ) ) )
                   => ( ! [M: num] :
                          ( ( Xb = aa_num_num(bit1,M) )
                         => ( ( Xaa = one )
                           => ( ( Yb = some_num(aa_num_num(bit0,M)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M),one)) ) ) )
                     => ( ! [M: num] :
                            ( ( Xb = aa_num_num(bit1,M) )
                           => ! [N2: num] :
                                ( ( Xaa = aa_num_num(bit0,N2) )
                               => ( ( Yb = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_aby,bit_and_not_num(M,N2)) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit0,N2))) ) ) )
                       => ~ ! [M: num] :
                              ( ( Xb = aa_num_num(bit1,M) )
                             => ! [N2: num] :
                                  ( ( Xaa = aa_num_num(bit1,N2) )
                                 => ( ( Yb = map_option_num_num(bit0,bit_and_not_num(M,N2)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_9161_and__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Yb: option_num] :
      ( ( bit_un7362597486090784418nd_num(Xb,Xaa) = Yb )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Yb != some_num(one) ) ) )
       => ( ( ( Xb = one )
           => ( ? [N2: num] : Xaa = aa_num_num(bit0,N2)
             => ( Yb != none_num ) ) )
         => ( ( ( Xb = one )
             => ( ? [N2: num] : Xaa = aa_num_num(bit1,N2)
               => ( Yb != some_num(one) ) ) )
           => ( ( ? [M: num] : Xb = aa_num_num(bit0,M)
               => ( ( Xaa = one )
                 => ( Yb != none_num ) ) )
             => ( ! [M: num] :
                    ( ( Xb = aa_num_num(bit0,M) )
                   => ! [N2: num] :
                        ( ( Xaa = aa_num_num(bit0,N2) )
                       => ( Yb != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M,N2)) ) ) )
               => ( ! [M: num] :
                      ( ( Xb = aa_num_num(bit0,M) )
                     => ! [N2: num] :
                          ( ( Xaa = aa_num_num(bit1,N2) )
                         => ( Yb != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M,N2)) ) ) )
                 => ( ( ? [M: num] : Xb = aa_num_num(bit1,M)
                     => ( ( Xaa = one )
                       => ( Yb != some_num(one) ) ) )
                   => ( ! [M: num] :
                          ( ( Xb = aa_num_num(bit1,M) )
                         => ! [N2: num] :
                              ( ( Xaa = aa_num_num(bit0,N2) )
                             => ( Yb != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M,N2)) ) ) )
                     => ~ ! [M: num] :
                            ( ( Xb = aa_num_num(bit1,M) )
                           => ! [N2: num] :
                                ( ( Xaa = aa_num_num(bit1,N2) )
                               => ( Yb != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_aby,bit_un7362597486090784418nd_num(M,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_9162_and__num_Osimps_I2_J,axiom,
    ! [Nb: num] : bit_un7362597486090784418nd_num(one,aa_num_num(bit0,Nb)) = none_num ).

% and_num.simps(2)
tff(fact_9163_and__num_Osimps_I4_J,axiom,
    ! [Mb: num] : bit_un7362597486090784418nd_num(aa_num_num(bit0,Mb),one) = none_num ).

% and_num.simps(4)
tff(fact_9164_and__num_Opelims,axiom,
    ! [Xb: num,Xaa: num,Yb: option_num] :
      ( ( bit_un7362597486090784418nd_num(Xb,Xaa) = Yb )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(Xb,Xaa))
       => ( ( ( Xb = one )
           => ( ( Xaa = one )
             => ( ( Yb = some_num(one) )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,one)) ) ) )
         => ( ( ( Xb = one )
             => ! [N2: num] :
                  ( ( Xaa = aa_num_num(bit0,N2) )
                 => ( ( Yb = none_num )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
           => ( ( ( Xb = one )
               => ! [N2: num] :
                    ( ( Xaa = aa_num_num(bit1,N2) )
                   => ( ( Yb = some_num(one) )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
             => ( ! [M: num] :
                    ( ( Xb = aa_num_num(bit0,M) )
                   => ( ( Xaa = one )
                     => ( ( Yb = none_num )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M),one)) ) ) )
               => ( ! [M: num] :
                      ( ( Xb = aa_num_num(bit0,M) )
                     => ! [N2: num] :
                          ( ( Xaa = aa_num_num(bit0,N2) )
                         => ( ( Yb = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M,N2)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit0,N2))) ) ) )
                 => ( ! [M: num] :
                        ( ( Xb = aa_num_num(bit0,M) )
                       => ! [N2: num] :
                            ( ( Xaa = aa_num_num(bit1,N2) )
                           => ( ( Yb = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M,N2)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit1,N2))) ) ) )
                   => ( ! [M: num] :
                          ( ( Xb = aa_num_num(bit1,M) )
                         => ( ( Xaa = one )
                           => ( ( Yb = some_num(one) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M),one)) ) ) )
                     => ( ! [M: num] :
                            ( ( Xb = aa_num_num(bit1,M) )
                           => ! [N2: num] :
                                ( ( Xaa = aa_num_num(bit0,N2) )
                               => ( ( Yb = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M,N2)) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit0,N2))) ) ) )
                       => ~ ! [M: num] :
                              ( ( Xb = aa_num_num(bit1,M) )
                             => ! [N2: num] :
                                  ( ( Xaa = aa_num_num(bit1,N2) )
                                 => ( ( Yb = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_aby,bit_un7362597486090784418nd_num(M,N2)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_9165_xor__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Yb: option_num] :
      ( ( bit_un2480387367778600638or_num(Xb,Xaa) = Yb )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Yb != none_num ) ) )
       => ( ( ( Xb = one )
           => ! [N2: num] :
                ( ( Xaa = aa_num_num(bit0,N2) )
               => ( Yb != some_num(aa_num_num(bit1,N2)) ) ) )
         => ( ( ( Xb = one )
             => ! [N2: num] :
                  ( ( Xaa = aa_num_num(bit1,N2) )
                 => ( Yb != some_num(aa_num_num(bit0,N2)) ) ) )
           => ( ! [M: num] :
                  ( ( Xb = aa_num_num(bit0,M) )
                 => ( ( Xaa = one )
                   => ( Yb != some_num(aa_num_num(bit1,M)) ) ) )
             => ( ! [M: num] :
                    ( ( Xb = aa_num_num(bit0,M) )
                   => ! [N2: num] :
                        ( ( Xaa = aa_num_num(bit0,N2) )
                       => ( Yb != map_option_num_num(bit0,bit_un2480387367778600638or_num(M,N2)) ) ) )
               => ( ! [M: num] :
                      ( ( Xb = aa_num_num(bit0,M) )
                     => ! [N2: num] :
                          ( ( Xaa = aa_num_num(bit1,N2) )
                         => ( Yb != some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M,N2))) ) ) )
                 => ( ! [M: num] :
                        ( ( Xb = aa_num_num(bit1,M) )
                       => ( ( Xaa = one )
                         => ( Yb != some_num(aa_num_num(bit0,M)) ) ) )
                   => ( ! [M: num] :
                          ( ( Xb = aa_num_num(bit1,M) )
                         => ! [N2: num] :
                              ( ( Xaa = aa_num_num(bit0,N2) )
                             => ( Yb != some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M,N2))) ) ) )
                     => ~ ! [M: num] :
                            ( ( Xb = aa_num_num(bit1,M) )
                           => ! [N2: num] :
                                ( ( Xaa = aa_num_num(bit1,N2) )
                               => ( Yb != map_option_num_num(bit0,bit_un2480387367778600638or_num(M,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_9166_less__eq__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_abt),aa_int1417691210696471063at_nat(rep_Integ,Xb)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_9167_xor__num_Osimps_I1_J,axiom,
    bit_un2480387367778600638or_num(one,one) = none_num ).

% xor_num.simps(1)
tff(fact_9168_nat_Orep__eq,axiom,
    ! [Xb: int] : aa_int_nat(nat2,Xb) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,Xb)) ).

% nat.rep_eq
tff(fact_9169_less__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_abr),aa_int1417691210696471063at_nat(rep_Integ,Xb)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_int.rep_eq
tff(fact_9170_xor__num_Opelims,axiom,
    ! [Xb: num,Xaa: num,Yb: option_num] :
      ( ( bit_un2480387367778600638or_num(Xb,Xaa) = Yb )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(Xb,Xaa))
       => ( ( ( Xb = one )
           => ( ( Xaa = one )
             => ( ( Yb = none_num )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,one)) ) ) )
         => ( ( ( Xb = one )
             => ! [N2: num] :
                  ( ( Xaa = aa_num_num(bit0,N2) )
                 => ( ( Yb = some_num(aa_num_num(bit1,N2)) )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
           => ( ( ( Xb = one )
               => ! [N2: num] :
                    ( ( Xaa = aa_num_num(bit1,N2) )
                   => ( ( Yb = some_num(aa_num_num(bit0,N2)) )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
             => ( ! [M: num] :
                    ( ( Xb = aa_num_num(bit0,M) )
                   => ( ( Xaa = one )
                     => ( ( Yb = some_num(aa_num_num(bit1,M)) )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M),one)) ) ) )
               => ( ! [M: num] :
                      ( ( Xb = aa_num_num(bit0,M) )
                     => ! [N2: num] :
                          ( ( Xaa = aa_num_num(bit0,N2) )
                         => ( ( Yb = map_option_num_num(bit0,bit_un2480387367778600638or_num(M,N2)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit0,N2))) ) ) )
                 => ( ! [M: num] :
                        ( ( Xb = aa_num_num(bit0,M) )
                       => ! [N2: num] :
                            ( ( Xaa = aa_num_num(bit1,N2) )
                           => ( ( Yb = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M,N2))) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit1,N2))) ) ) )
                   => ( ! [M: num] :
                          ( ( Xb = aa_num_num(bit1,M) )
                         => ( ( Xaa = one )
                           => ( ( Yb = some_num(aa_num_num(bit0,M)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M),one)) ) ) )
                     => ( ! [M: num] :
                            ( ( Xb = aa_num_num(bit1,M) )
                           => ! [N2: num] :
                                ( ( Xaa = aa_num_num(bit0,N2) )
                               => ( ( Yb = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M,N2))) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit0,N2))) ) ) )
                       => ~ ! [M: num] :
                              ( ( Xb = aa_num_num(bit1,M) )
                             => ! [N2: num] :
                                  ( ( Xaa = aa_num_num(bit1,N2) )
                                 => ( ( Yb = map_option_num_num(bit0,bit_un2480387367778600638or_num(M,N2)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_9171_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_abp)) ).

% uminus_int_def
tff(fact_9172_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_abo)) ).

% times_int_def
tff(fact_9173_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_abx)) ).

% minus_int_def
tff(fact_9174_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_abv)) ).

% plus_int_def
tff(fact_9175_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_abz) ).

% prod_encode_def
tff(fact_9176_Gcd__remove0__nat,axiom,
    ! [M5: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M5)
     => ( aa_set_nat_nat(gcd_Gcd_nat,M5) = aa_set_nat_nat(gcd_Gcd_nat,minus_minus_set_nat(M5,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9177_prod__encode__eq,axiom,
    ! [Xb: product_prod_nat_nat,Yb: product_prod_nat_nat] :
      ( ( aa_Pro4243173770044397973at_nat(nat_prod_encode,Xb) = aa_Pro4243173770044397973at_nat(nat_prod_encode,Yb) )
    <=> ( Xb = Yb ) ) ).

% prod_encode_eq
tff(fact_9178_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))) ).

% le_prod_encode_1
tff(fact_9179_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(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_9180_prod__encode__prod__decode__aux,axiom,
    ! [K2: nat,Mb: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K2,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K2)),Mb) ).

% prod_encode_prod_decode_aux
tff(fact_9181_list__encode_Oelims,axiom,
    ! [Xb: list_nat,Yb: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,Xb) = Yb )
     => ( ( ( Xb = nil_nat )
         => ( Yb != zero_zero_nat ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( Xb = cons_nat(X3,Xs2) )
             => ( Yb != 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,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_9182_list__encode__eq,axiom,
    ! [Xb: list_nat,Yb: list_nat] :
      ( ( aa_list_nat_nat(nat_list_encode,Xb) = aa_list_nat_nat(nat_list_encode,Yb) )
    <=> ( Xb = Yb ) ) ).

% list_encode_eq
tff(fact_9183_list__encode_Ocases,axiom,
    ! [Xb: list_nat] :
      ( ( Xb != nil_nat )
     => ~ ! [X3: nat,Xs2: list_nat] : Xb != cons_nat(X3,Xs2) ) ).

% list_encode.cases
tff(fact_9184_list__encode_Osimps_I1_J,axiom,
    aa_list_nat_nat(nat_list_encode,nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_9185_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K5)) ).

% Gcd_int_greater_eq_0
tff(fact_9186_list__encode_Osimps_I2_J,axiom,
    ! [Xb: nat,Xs: list_nat] : aa_list_nat_nat(nat_list_encode,cons_nat(Xb,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),aa_list_nat_nat(nat_list_encode,Xs)))) ).

% list_encode.simps(2)
tff(fact_9187_sub__BitM__One__eq,axiom,
    ! [Nb: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,Nb)),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(neg_numeral_sub_int(Nb),one)) ).

% sub_BitM_One_eq
tff(fact_9188_list__encode_Opelims,axiom,
    ! [Xb: list_nat,Yb: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,Xb) = Yb )
     => ( aa_list_nat_o(accp_list_nat(nat_list_encode_rel),Xb)
       => ( ( ( Xb = nil_nat )
           => ( ( Yb = zero_zero_nat )
             => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),nil_nat) ) )
         => ~ ! [X3: nat,Xs2: list_nat] :
                ( ( Xb = cons_nat(X3,Xs2) )
               => ( ( Yb = 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,Xs2)))) )
                 => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),cons_nat(X3,Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_9189_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aca)) ).

% pred_nat_def
tff(fact_9190_sorted__list__of__set__atMost__Suc,axiom,
    ! [K2: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K2))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K2)),cons_nat(aa_nat_nat(suc,K2),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_9191_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K2: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K2)),cons_nat(K2,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_9192_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9193_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9194_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Xb: nat,Yb: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acb,C2),set_or4665077453230672383an_nat(Xb,Yb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),Yb),
        set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Yb),C2)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9195_bij__betw__Suc,axiom,
    ! [M5: set_nat,N4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(bij_betw_nat_nat(suc),M5),N4)
    <=> ( image_nat_nat(suc,M5) = N4 ) ) ).

% bij_betw_Suc
tff(fact_9196_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,set_or1269000886237332187st_nat(I,J)) = set_or1269000886237332187st_nat(aa_nat_nat(suc,I),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9197_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,set_or4665077453230672383an_nat(I,J)) = set_or4665077453230672383an_nat(aa_nat_nat(suc,I),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9198_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A4)) ).

% zero_notin_Suc_image
tff(fact_9199_image__Suc__lessThan,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = set_or1269000886237332187st_nat(one_one_nat,Nb) ).

% image_Suc_lessThan
tff(fact_9200_image__Suc__atMost,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = set_or1269000886237332187st_nat(one_one_nat,aa_nat_nat(suc,Nb)) ).

% image_Suc_atMost
tff(fact_9201_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or1269000886237332187st_nat(zero_zero_nat,Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9202_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9203_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9204_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_9205_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9206_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_rat(fract(A),B))
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int))) ) ).

% rat_floor_lemma
tff(fact_9207_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or6659071591806873216st_nat(L,U)) ).

% finite_greaterThanAtMost
tff(fact_9208_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,set_or6659071591806873216st_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_greaterThanAtMost
tff(fact_9209_less__rat,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C2),D))
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ).

% less_rat
tff(fact_9210_add__rat,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(plus_plus_rat(aa_int_rat(fract(A),B)),aa_int_rat(fract(C2),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% add_rat
tff(fact_9211_le__rat,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C2),D))
        <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ).

% le_rat
tff(fact_9212_diff__rat,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(minus_minus_rat(aa_int_rat(fract(A),B)),aa_int_rat(fract(C2),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% diff_rat
tff(fact_9213_eq__rat_I3_J,axiom,
    ! [A: int,C2: int] : aa_int_rat(fract(zero_zero_int),A) = aa_int_rat(fract(zero_zero_int),C2) ).

% eq_rat(3)
tff(fact_9214_eq__rat_I2_J,axiom,
    ! [A: int] : aa_int_rat(fract(A),zero_zero_int) = aa_int_rat(fract(zero_zero_int),one_one_int) ).

% eq_rat(2)
tff(fact_9215_Rat__induct__pos,axiom,
    ! [P2: fun_rat_o,Q3: rat] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
         => aa_rat_o(P2,aa_int_rat(fract(A3),B3)) )
     => aa_rat_o(P2,Q3) ) ).

% Rat_induct_pos
tff(fact_9216_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( aa_int_rat(fract(A),B) = aa_int_rat(fract(C2),D) )
        <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),D) = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) ) ) ) ) ).

% eq_rat(1)
tff(fact_9217_mult__rat__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) = aa_int_rat(fract(A),B) ) ) ).

% mult_rat_cancel
tff(fact_9218_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,L),U) = set_or6659071591806873216st_nat(L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_9219_rat__number__collapse_I1_J,axiom,
    ! [K2: int] : aa_int_rat(fract(zero_zero_int),K2) = zero_zero_rat ).

% rat_number_collapse(1)
tff(fact_9220_rat__number__collapse_I6_J,axiom,
    ! [K2: int] : aa_int_rat(fract(K2),zero_zero_int) = zero_zero_rat ).

% rat_number_collapse(6)
tff(fact_9221_quotient__of__eq,axiom,
    ! [A: int,B: int,P: int,Q3: int] :
      ( ( quotient_of(aa_int_rat(fract(A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q3) )
     => ( aa_int_rat(fract(P),Q3) = aa_int_rat(fract(A),B) ) ) ).

% quotient_of_eq
tff(fact_9222_normalize__eq,axiom,
    ! [A: int,B: int,P: int,Q3: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q3) )
     => ( aa_int_rat(fract(P),Q3) = aa_int_rat(fract(A),B) ) ) ).

% normalize_eq
tff(fact_9223_finite__int__iff__bounded,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
    <=> ? [K3: int] : aa_set_int_o(ord_less_eq_set_int(image_int_int(abs_abs_int,S2)),set_ord_lessThan_int(K3)) ) ).

% finite_int_iff_bounded
tff(fact_9224_finite__int__iff__bounded__le,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
    <=> ? [K3: int] : aa_set_int_o(ord_less_eq_set_int(image_int_int(abs_abs_int,S2)),set_ord_atMost_int(K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_9225_Zero__rat__def,axiom,
    zero_zero_rat = aa_int_rat(fract(zero_zero_int),one_one_int) ).

% Zero_rat_def
tff(fact_9226_image__int__atLeastAtMost,axiom,
    ! [A: nat,B: nat] : image_nat_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(A,B)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% image_int_atLeastAtMost
tff(fact_9227_image__int__atLeastLessThan,axiom,
    ! [A: nat,B: nat] : image_nat_int(semiri1314217659103216013at_int,set_or4665077453230672383an_nat(A,B)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% image_int_atLeastLessThan
tff(fact_9228_quotient__of__Fract,axiom,
    ! [A: int,B: int] : quotient_of(aa_int_rat(fract(A),B)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ).

% quotient_of_Fract
tff(fact_9229_Fract__less__zero__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% Fract_less_zero_iff
tff(fact_9230_zero__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% zero_less_Fract_iff
tff(fact_9231_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% one_less_Fract_iff
tff(fact_9232_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% Fract_less_one_iff
tff(fact_9233_Fract__add__one,axiom,
    ! [Nb: int,Mb: int] :
      ( ( Nb != zero_zero_int )
     => ( aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(plus_plus_int,Mb),Nb)),Nb) = aa_rat_rat(plus_plus_rat(aa_int_rat(fract(Mb),Nb)),one_one_rat) ) ) ).

% Fract_add_one
tff(fact_9234_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
    <=> ~ aa_set_nat_o(finite_finite_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),S2)) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_9235_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9236_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_acc,L),aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U) ).

% image_add_int_atLeastLessThan
tff(fact_9237_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_Fract_iff
tff(fact_9238_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% Fract_le_zero_iff
tff(fact_9239_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% one_le_Fract_iff
tff(fact_9240_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% Fract_le_one_iff
tff(fact_9241_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,zero_zero_int),U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_9242_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or6656581121297822940st_int,L),U)) ).

% finite_greaterThanAtMost_int
tff(fact_9243_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or6656581121297822940st_int,L),U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_greaterThanAtMost_int
tff(fact_9244_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int)),U) = aa_int_set_int(aa_int8639817059373041269et_int(set_or6656581121297822940st_int,L),U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_9245_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or6656581121297822940st_int,I),J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_9246_UN__lessThan__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_lessThan_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_lessThan_UNIV
tff(fact_9247_UN__atMost__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atMost_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atMost_UNIV
tff(fact_9248_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_9249_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9250_infinite__UNIV__nat,axiom,
    ~ aa_set_nat_o(finite_finite_nat,top_top_set_nat) ).

% infinite_UNIV_nat
tff(fact_9251_nat__not__finite,axiom,
    ~ aa_set_nat_o(finite_finite_nat,top_top_set_nat) ).

% nat_not_finite
tff(fact_9252_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( set_or4665077453230672383an_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) = sup_sup_set_nat(set_or4665077453230672383an_nat(I,J),set_or4665077453230672383an_nat(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9253_range__mod,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acd,Nb),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,Nb) ) ) ).

% range_mod
tff(fact_9254_suminf__eq__SUP__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(X5,I2))
       => ( suminf_real(X5) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_ace(X5),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_9255_UNIV__nat__eq,axiom,
    top_top_set_nat = insert_nat(zero_zero_nat,image_nat_nat(suc,top_top_set_nat)) ).

% UNIV_nat_eq
tff(fact_9256_min__weak__def,axiom,
    fun_min_weak = sup_su5525570899277871387at_nat(min_ex6901939911449802026at_nat(fun_pair_leq),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% min_weak_def
tff(fact_9257_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_9258_range__mult,axiom,
    ! [A: real] :
      image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = $ite(A = zero_zero_real,insert_real(zero_zero_real,bot_bot_set_real),top_top_set_real) ).

% range_mult
tff(fact_9259_infinite__UNIV__int,axiom,
    ~ aa_set_int_o(finite_finite_int,top_top_set_int) ).

% infinite_UNIV_int
tff(fact_9260_bij__list__encode,axiom,
    aa_set_nat_o(bij_be8532844293280997160at_nat(nat_list_encode,top_top_set_list_nat),top_top_set_nat) ).

% bij_list_encode
tff(fact_9261_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_9262_bij__prod__encode,axiom,
    aa_set_nat_o(bij_be5333170631980326235at_nat(nat_prod_encode,top_to4669805908274784177at_nat),top_top_set_nat) ).

% bij_prod_encode
tff(fact_9263_surj__prod__encode,axiom,
    image_2486076414777270412at_nat(nat_prod_encode,top_to4669805908274784177at_nat) = top_top_set_nat ).

% surj_prod_encode
tff(fact_9264_int__in__range__abs,axiom,
    ! [Nb: nat] : aa_set_int_o(member_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),image_int_int(abs_abs_int,top_top_set_int)) ).

% int_in_range_abs
tff(fact_9265_max__weak__def,axiom,
    fun_max_weak = sup_su5525570899277871387at_nat(max_ex8135407076693332796at_nat(fun_pair_leq),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% max_weak_def
tff(fact_9266_root__def,axiom,
    ! [Nb: nat,Xb: real] :
      aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_acf,Nb)),Xb)) ).

% root_def
tff(fact_9267_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xb: real,D4: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( Xb != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
               => ( D4 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
               => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),D4,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9268_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9269_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9270_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9271_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9272_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9273_DERIV__isconst__all,axiom,
    ! [F: fun_real_real,Xb: real,Yb: real] :
      ( ! [X3: real] : has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real))
     => ( aa_real_real(F,Xb) = aa_real_real(F,Yb) ) ) ).

% DERIV_isconst_all
tff(fact_9274_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9275_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9276_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9277_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9278_DERIV__isconst3,axiom,
    ! [A: real,B: real,Xb: real,Yb: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( member_real(Xb,set_or1633881224788618240n_real(A,B))
       => ( member_real(Yb,set_or1633881224788618240n_real(A,B))
         => ( ! [X3: real] :
                ( member_real(X3,set_or1633881224788618240n_real(A,B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) )
           => ( aa_real_real(F,Xb) = aa_real_real(F,Yb) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_9279_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9280_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9281_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_9282_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_9283_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,set_or1222579329274155063t_real(A,B))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
     => ( ! [X3: real] :
            ( member_real(X3,set_or1222579329274155063t_real(A,B))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X3)) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(G,A)),aa_real_real(G,B)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_9284_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
             => has_fi5821293074295781190e_real(F,aa_real_real(F4,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
       => ? [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
            & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_real_real(F4,Z2)) ) ) ) ) ).

% MVT2
tff(fact_9285_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),D)
             => ( aa_real_real(F,Xb) = aa_real_real(F,Y) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9286_DERIV__ln,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_9287_DERIV__const__average,axiom,
    ! [A: real,B: real,V: fun_real_real,K2: real] :
      ( ( A != B )
     => ( ! [X3: real] : has_fi5821293074295781190e_real(V,K2,topolo2177554685111907308n_real(X3,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(V,A)),aa_real_real(V,B))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_9288_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y)),aa_real_real(F,Xb)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9289_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Xb)),aa_real_real(F,Y)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9290_DERIV__ln__divide,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9291_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S3: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_acg,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xb,S3)) ).

% DERIV_pow
tff(fact_9292_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_ach,R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9293_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F4: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L5: fun_nat_real] :
      ( ! [N2: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_hf(F),N2),aa_nat_real(aa_real_fun_nat_real(F4,X0),N2),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X3: real] :
            ( member_real(X3,set_or1633881224788618240n_real(A,B))
           => summable_real(aa_real_fun_nat_real(F,X3)) )
       => ( member_real(X0,set_or1633881224788618240n_real(A,B))
         => ( summable_real(aa_real_fun_nat_real(F4,X0))
           => ( summable_real(L5)
             => ( ! [N2: nat,X3: real,Y: real] :
                    ( member_real(X3,set_or1633881224788618240n_real(A,B))
                   => ( member_real(Y,set_or1633881224788618240n_real(A,B))
                     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(F,X3),N2)),aa_nat_real(aa_real_fun_nat_real(F,Y),N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L5,N2)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),Y)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_aci(F),suminf_real(aa_real_fun_nat_real(F4,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9294_DERIV__log,axiom,
    ! [Xb: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(ln_ln_real,B)),Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9295_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_acj(G),R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),Mb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9296_DERIV__powr,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(Xb,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_ack(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,Xb)),aa_real_real(F,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(ln_ln_real,aa_real_real(G,Xb)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Mb),aa_real_real(F,Xb))),aa_real_real(G,Xb)))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9297_DERIV__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9298_DERIV__arctan,axiom,
    ! [Xb: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ).

% DERIV_arctan
tff(fact_9299_arsinh__real__has__field__derivative,axiom,
    ! [Xb: real,A4: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xb,A4)) ).

% arsinh_real_has_field_derivative
tff(fact_9300_DERIV__real__sqrt__generic,axiom,
    ! [Xb: real,D4: real] :
      ( ( Xb != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( D4 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
           => ( D4 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D4,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9301_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X3: real] :
          ( member_real(X3,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_acl(F),X3)) )
     => ( member_real(X0,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
         => has_fi5821293074295781190e_real(aTP_Lamm_acn(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_acl(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9302_DERIV__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9303_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aco(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9304_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
          & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aco(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9305_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( Xb != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9306_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M: nat,T4: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),H)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ps(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9307_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ? [T4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H)
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ps(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9308_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M: nat,T4: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ps(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9309_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( Xb != zero_zero_real )
         => ( ! [M: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T4))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
                & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aco(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9310_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M: nat,T4: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aco(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9311_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B)
             => ? [T4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),C2)
                  & ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_acp(Diff),A),C2)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9312_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B)
             => ? [T4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),B)
                  & ( aa_real_real(F,B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_acp(Diff),B),C2)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C2)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9313_Taylor,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C2: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),B)
                 => ( ( Xb != C2 )
                   => ? [T4: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),C2),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T4)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),C2) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),T4)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xb) ) )
                        & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_acq(Diff),C2),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),C2)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9314_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun_na6724076748692969250l_real,K2: nat,B4: real] :
      ( ! [M: nat,T4: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
     => ( ( Nb = aa_nat_nat(suc,K2) )
       => ! [M2: nat,T5: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_acs(Nb),Diff),B4),M2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_act(Diff),M2),T5)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,T5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T5,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9315_DERIV__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_acu,suminf_real(aa_real_fun_nat_real(aTP_Lamm_acv,Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9316_surj__int__encode,axiom,
    image_int_nat(nat_int_encode,top_top_set_int) = top_top_set_nat ).

% surj_int_encode
tff(fact_9317_int__encode__eq,axiom,
    ! [Xb: int,Yb: int] :
      ( ( aa_int_nat(nat_int_encode,Xb) = aa_int_nat(nat_int_encode,Yb) )
    <=> ( Xb = Yb ) ) ).

% int_encode_eq
tff(fact_9318_isCont__Lb__Ub,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X3: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),F) )
       => ? [L6: real,M8: real] :
            ( ! [X: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B) )
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),aa_real_real(F,X))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X)),M8) ) )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),M8) )
               => ? [X3: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
                    & ( aa_real_real(F,X3) = Y3 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_9319_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X: real] :
                ( ( ( X != C2 )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),X))),R3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),zero_zero_real) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_9320_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X: real] :
                ( ( ( X != C2 )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),X))),R3) )
               => ( aa_real_real(F,X) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_9321_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X: real] :
                ( ( ( X != C2 )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),X))),R3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,X)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_9322_isCont__inverse__function2,axiom,
    ! [A: real,Xb: real,B: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
               => ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xb),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9323_isCont__ln,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero_real )
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),ln_ln_real) ) ).

% isCont_ln
tff(fact_9324_bij__int__encode,axiom,
    aa_set_nat_o(bij_betw_int_nat(nat_int_encode,top_top_set_int),top_top_set_nat) ).

% bij_int_encode
tff(fact_9325_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_acw,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_9326_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D4: real,G: fun_real_real,Xb: real,A: real,B: real] :
      ( has_fi5821293074295781190e_real(F,D4,topolo2177554685111907308n_real(aa_real_real(G,Xb),top_top_set_real))
     => ( ( D4 != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
           => ( ! [Y: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),B)
                   => ( aa_real_real(F,aa_real_real(G,Y)) = Y ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D4),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9327_LIM__less__bound,axiom,
    ! [B: real,Xb: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Xb)
     => ( ! [X3: real] :
            ( member_real(X3,set_or1633881224788618240n_real(B,Xb))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,Xb)) ) ) ) ).

% LIM_less_bound
tff(fact_9328_isCont__inverse__function,axiom,
    ! [D: real,Xb: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
     => ( ! [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),Xb))),D)
           => ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),Xb))),D)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xb),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_9329_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),G) ) )
         => ( ! [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
           => ( ! [Z2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                   => has_fi5821293074295781190e_real(F,aa_real_real(F4,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
             => ? [C: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(G2,C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),aa_real_real(F4,C)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9330_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat))
         => ! [N5: nat] : member_real(suminf_real(aTP_Lamm_acx(A)),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_acx(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_acx(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9331_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real)
         => ! [N5: nat] : member_real(suminf_real(aTP_Lamm_acx(A)),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_acx(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)),one_one_nat))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_acx(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9332_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => filterlim_nat_real(aTP_Lamm_acy(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_acx(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9333_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_9334_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_9335_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acz,C2),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9336_monoseq__convergent,axiom,
    ! [X5: fun_nat_real,B4: real] :
      ( topolo6980174941875973593q_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X5,I2))),B4)
       => ~ ! [L6: real] : ~ filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_9337_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2)))
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N2))),aa_nat_real(G,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ada(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N5)),L3)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_nat_real(G,N5))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9338_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun_nat_real] :
      ( ! [R3: real] :
        ? [N6: nat] :
        ! [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R3),aa_nat_real(X5,N2)) )
     => filterlim_nat_real(aTP_Lamm_adb(X5),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9339_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_adc,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_9340_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_add,C2),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9341_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_ade,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_9342_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_adf,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9343_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2)))
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),L)
       => ( ! [E: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
             => ? [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N5)),E)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9344_LIMSEQ__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_9345_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_adg(Xb),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9346_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C2)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_9347_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C2)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C2)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_9348_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_adh,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9349_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_adi,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9350_tendsto__exp__limit__sequentially,axiom,
    ! [Xb: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_adj,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_9351_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_adk,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9352_summable__Leibniz_I1_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => summable_real(aTP_Lamm_acx(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_9353_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => summable_real(aTP_Lamm_acx(A)) ) ) ) ).

% summable
tff(fact_9354_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_adl(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_adm(Theta),K),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_9355_summable__Leibniz_I4_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_adn(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_acx(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_9356_zeroseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_lu,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9357_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => filterlim_nat_real(aTP_Lamm_adn(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_acx(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9358_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_acx(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),suminf_real(aTP_Lamm_acx(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9359_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L3: real] :
              ( ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_acx(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)))),L3)
              & filterlim_nat_real(aTP_Lamm_adn(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_acx(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_acy(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9360_summable__Leibniz_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_acy(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_acx(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9361_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_acx(A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_acx(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9362_tendsto__exp__limit__at__right,axiom,
    ! [Xb: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ado,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_9363_tendsto__arcosh__at__left__1,axiom,
    filterlim_real_real(arcosh_real,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(one_one_real,set_or5849166863359141190n_real(one_one_real))) ).

% tendsto_arcosh_at_left_1
tff(fact_9364_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_9365_atLeast__Suc__greaterThan,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K2)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K2) ).

% atLeast_Suc_greaterThan
tff(fact_9366_INT__greaterThan__UNIV,axiom,
    comple7806235888213564991et_nat(image_nat_set_nat(set_or1210151606488870762an_nat,top_top_set_nat)) = bot_bot_set_nat ).

% INT_greaterThan_UNIV
tff(fact_9367_greaterThan__0,axiom,
    aa_nat_set_nat(set_or1210151606488870762an_nat,zero_zero_nat) = image_nat_nat(suc,top_top_set_nat) ).

% greaterThan_0
tff(fact_9368_greaterThan__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K2)) = minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K2),insert_nat(aa_nat_nat(suc,K2),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_9369_UN__atLeast__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atLeast_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atLeast_UNIV
tff(fact_9370_atLeast__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K2)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K2),insert_nat(K2,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_9371_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_adp(Nb),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9372_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_bot_real,at_infinity_real) ).

% at_bot_le_at_infinity
tff(fact_9373_exp__at__bot,axiom,
    filterlim_real_real(exp_real,topolo2815343760600316023s_real(zero_zero_real),at_bot_real) ).

% exp_at_bot
tff(fact_9374_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real(inverse_inverse_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5984915006950818249n_real(zero_zero_real))) ).

% filterlim_inverse_at_bot_neg
tff(fact_9375_ln__at__0,axiom,
    filterlim_real_real(ln_ln_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% ln_at_0
tff(fact_9376_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_9377_Gcd__eq__Max,axiom,
    ! [M5: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M5)
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M5)
         => ( aa_set_nat_nat(gcd_Gcd_nat,M5) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_adq,M5))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9378_eventually__sequentially__Suc,axiom,
    ! [P2: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_adr(P2),at_top_nat)
    <=> eventually_nat(P2,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9379_eventually__sequentially__seg,axiom,
    ! [P2: fun_nat_o,K2: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_ads(P2),K2),at_top_nat)
    <=> eventually_nat(P2,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9380_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ca,Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_9381_eventually__sequentiallyI,axiom,
    ! [C2: nat,P2: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),X3)
         => aa_nat_o(P2,X3) )
     => eventually_nat(P2,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9382_eventually__sequentially,axiom,
    ! [P2: fun_nat_o] :
      ( eventually_nat(P2,at_top_nat)
    <=> ? [N8: nat] :
        ! [N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N3)
         => aa_nat_o(P2,N3) ) ) ).

% eventually_sequentially
tff(fact_9383_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N8: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N8),F3) ) ).

% le_sequentially
tff(fact_9384_sequentially__offset,axiom,
    ! [P2: fun_nat_o,K2: nat] :
      ( eventually_nat(P2,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_ads(P2),K2),at_top_nat) ) ).

% sequentially_offset
tff(fact_9385_eventually__at__right__to__0,axiom,
    ! [P2: fun_real_o,A: real] :
      ( eventually_real(P2,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
    <=> eventually_real(aa_real_fun_real_o(aTP_Lamm_adt(P2),A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_9386_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S2)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S2))) ) ).

% card_le_Suc_Max
tff(fact_9387_Sup__nat__def,axiom,
    ! [X5: set_nat] :
      aa_set_nat_nat(complete_Sup_Sup_nat,X5) = $ite(X5 = bot_bot_set_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,X5)) ).

% Sup_nat_def
tff(fact_9388_divide__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_adu(Mb),Nb)))) ).

% divide_nat_def
tff(fact_9389_gcd__is__Max__divisors__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_adv(Nb),Mb))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9390_lhopital__left,axiom,
    ! [F: fun_real_real,Xb: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
       => ( eventually_real(aTP_Lamm_adw(G),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
         => ( eventually_real(aTP_Lamm_adw(G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(F),F4),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(G),G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(G2),F4),F3,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_adz(F),G),F3,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_9391_lhopital,axiom,
    ! [F: fun_real_real,Xb: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,top_top_set_real))
       => ( eventually_real(aTP_Lamm_adw(G),topolo2177554685111907308n_real(Xb,top_top_set_real))
         => ( eventually_real(aTP_Lamm_adw(G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(F),F4),topolo2177554685111907308n_real(Xb,top_top_set_real))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(G),G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(G2),F4),F3,topolo2177554685111907308n_real(Xb,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_adz(F),G),F3,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_9392_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( filterlim_real_real(G0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aTP_Lamm_adw(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_adw(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(F0),F4),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(G2),F4),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_adz(F0),G0),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_9393_lhopital__right,axiom,
    ! [F: fun_real_real,Xb: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
       => ( eventually_real(aTP_Lamm_adw(G),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
         => ( eventually_real(aTP_Lamm_adw(G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(F),F4),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(G),G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(G2),F4),F3,topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_adz(F),G),F3,topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_9394_Max__divisors__self__int,axiom,
    ! [Nb: int] :
      ( ( Nb != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_bd,Nb))) = aa_int_int(abs_abs_int,Nb) ) ) ).

% Max_divisors_self_int
tff(fact_9395_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_top_real,at_infinity_real) ).

% at_top_le_at_infinity
tff(fact_9396_gcd__is__Max__divisors__int,axiom,
    ! [Nb: int,Mb: int] :
      ( ( Nb != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Mb),Nb) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_aea(Nb),Mb))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_9397_eventually__at__top__to__right,axiom,
    ! [P2: fun_real_o] :
      ( eventually_real(P2,at_top_real)
    <=> eventually_real(aTP_Lamm_aeb(P2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_top_to_right
tff(fact_9398_eventually__at__right__to__top,axiom,
    ! [P2: fun_real_o] :
      ( eventually_real(P2,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
    <=> eventually_real(aTP_Lamm_aeb(P2),at_top_real) ) ).

% eventually_at_right_to_top
tff(fact_9399_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_aec,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_9400_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real(inverse_inverse_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)),at_top_real) ).

% filterlim_inverse_at_right_top
tff(fact_9401_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real(inverse_inverse_real,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% filterlim_inverse_at_top_right
tff(fact_9402_tendsto__power__div__exp__0,axiom,
    ! [K2: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_aed,K2),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_9403_lhopital__at__top,axiom,
    ! [G: fun_real_real,Xb: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Yb: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( eventually_real(aTP_Lamm_adw(G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(F),F4),topolo2177554685111907308n_real(Xb,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(G),G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(G2),F4),topolo2815343760600316023s_real(Yb),topolo2177554685111907308n_real(Xb,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(G),F),topolo2815343760600316023s_real(Yb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_9404_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Xb: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_adw(G2),at_top_real)
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(F),F4),at_top_real)
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(G2),F4),topolo2815343760600316023s_real(Xb),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(G),F),topolo2815343760600316023s_real(Xb),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_9405_tendsto__exp__limit__at__top,axiom,
    ! [Xb: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_aee,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_9406_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),X3)
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),zero_zero_real) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_9407_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Xb: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( eventually_real(aTP_Lamm_adw(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(F),F4),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(G2),F4),topolo2815343760600316023s_real(Xb),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(G),F),topolo2815343760600316023s_real(Xb),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_9408_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,Xb: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Yb: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
     => ( eventually_real(aTP_Lamm_adw(G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(F),F4),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(G),G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(G2),F4),topolo2815343760600316023s_real(Yb),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(G),F),topolo2815343760600316023s_real(Yb),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_9409_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,Xb: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Yb: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
     => ( eventually_real(aTP_Lamm_adw(G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(F),F4),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_adx(G),G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(G2),F4),topolo2815343760600316023s_real(Yb),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(G),F),topolo2815343760600316023s_real(Yb),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_9410_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_adp(Nb),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9411_GreatestI__nat,axiom,
    ! [P2: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(P2,K2)
     => ( ! [Y: nat] :
            ( aa_nat_o(P2,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B) )
       => aa_nat_o(P2,order_Greatest_nat(P2)) ) ) ).

% GreatestI_nat
tff(fact_9412_Greatest__le__nat,axiom,
    ! [P2: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(P2,K2)
     => ( ! [Y: nat] :
            ( aa_nat_o(P2,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),order_Greatest_nat(P2)) ) ) ).

% Greatest_le_nat
tff(fact_9413_GreatestI__ex__nat,axiom,
    ! [P2: fun_nat_o,B: nat] :
      ( ? [X_1: nat] : aa_nat_o(P2,X_1)
     => ( ! [Y: nat] :
            ( aa_nat_o(P2,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B) )
       => aa_nat_o(P2,order_Greatest_nat(P2)) ) ) ).

% GreatestI_ex_nat
tff(fact_9414_Bseq__eq__bounded,axiom,
    ! [F: fun_nat_real,A: real,B: real] :
      ( ord_less_eq_set_real(image_nat_real(F,top_top_set_nat),set_or1222579329274155063t_real(A,B))
     => bfun_nat_real(F,at_top_nat) ) ).

% Bseq_eq_bounded
tff(fact_9415_Bseq__realpow,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,Xb),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_9416_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
      <=> (Yb) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( (Yb)
          <=> ( Xaa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( (Yb)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X4: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                          & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                                & ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_aef(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9417_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ~ ( ( Deg = Xaa )
                  & $let(
                      n: nat,
                      n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m2: nat,
                        m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                        ( ! [X4: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                        & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                        & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                        & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                              & ! [X4: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_aef(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9418_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Deg3: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Node(Mima2,Dega,TreeLista,Summarya)),Deg3)
    <=> ( ( Dega = Deg3 )
        & $let(
            n: nat,
            n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            $let(
              m2: nat,
              m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Dega),n),
              ( ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                 => aa_nat_o(vEBT_VEBT_valid(X4),n) )
              & aa_nat_o(vEBT_VEBT_valid(Summarya),m2)
              & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
              & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X7)
                    & ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_aef(Dega),TreeLista),Summarya),n),m2))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9419_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( ( Deg = Xaa )
                & $let(
                    n: nat,
                    n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m2: nat,
                      m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                      ( ! [X4: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                      & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                      & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                      & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                            & ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_aef(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9420_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa))
                 => ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X4: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                          & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                                & ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_aef(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9421_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa))
                 => ~ ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                            & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                                  & ! [X4: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_aef(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9422_Sup__int__def,axiom,
    ! [X5: set_int] : complete_Sup_Sup_int(X5) = the_int(aTP_Lamm_aeg(X5)) ).

% Sup_int_def
tff(fact_9423_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
      <=> (Yb) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Yb)
                <=> ( Xaa = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( ( (Yb)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                            & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                                  & ! [X4: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_aef(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9424_GMVT,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),F) )
       => ( ! [X3: real] :
              ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) )
         => ( ! [X3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),G) )
           => ( ! [X3: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X3,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,A),C)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_9425_less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),Nb)),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% less_eq
tff(fact_9426_remdups__upt,axiom,
    ! [Mb: nat,Nb: nat] : remdups_nat(upt(Mb,Nb)) = upt(Mb,Nb) ).

% remdups_upt
tff(fact_9427_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_9428_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9429_sorted__list__of__set__range,axiom,
    ! [Mb: nat,Nb: nat] : linord2614967742042102400et_nat(set_or4665077453230672383an_nat(Mb,Nb)) = upt(Mb,Nb) ).

% sorted_list_of_set_range
tff(fact_9430_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_9431_nth__upt,axiom,
    ! [I: nat,K2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),J)
     => ( aa_nat_nat(nth_nat(upt(I,J)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) ) ) ).

% nth_upt
tff(fact_9432_upt__rec__numeral,axiom,
    ! [Mb: num,Nb: num] :
      upt(aa_num_nat(numeral_numeral_nat,Mb),aa_num_nat(numeral_numeral_nat,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)),cons_nat(aa_num_nat(numeral_numeral_nat,Mb),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),nil_nat) ).

% upt_rec_numeral
tff(fact_9433_greaterThanLessThan__upt,axiom,
    ! [Nb: nat,Mb: nat] : set_or5834768355832116004an_nat(Nb,Mb) = set_nat2(upt(aa_nat_nat(suc,Nb),Mb)) ).

% greaterThanLessThan_upt
tff(fact_9434_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_9435_upt__conv__Cons__Cons,axiom,
    ! [Mb: nat,Nb: nat,Ns: list_nat,Q3: nat] :
      ( ( cons_nat(Mb,cons_nat(Nb,Ns)) = upt(Mb,Q3) )
    <=> ( cons_nat(Nb,Ns) = upt(aa_nat_nat(suc,Mb),Q3) ) ) ).

% upt_conv_Cons_Cons
tff(fact_9436_greaterThanAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : set_or6659071591806873216st_nat(Nb,Mb) = set_nat2(upt(aa_nat_nat(suc,Nb),aa_nat_nat(suc,Mb))) ).

% greaterThanAtMost_upt
tff(fact_9437_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : set_or4665077453230672383an_nat(I,J) = set_nat2(upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_9438_atLeastAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : set_or1269000886237332187st_nat(Nb,Mb) = set_nat2(upt(Nb,aa_nat_nat(suc,Mb))) ).

% atLeastAtMost_upt
tff(fact_9439_atLeast__upt,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Nb) = set_nat2(upt(zero_zero_nat,Nb)) ).

% atLeast_upt
tff(fact_9440_distinct__upt,axiom,
    ! [I: nat,J: nat] : aa_list_nat_o(distinct_nat,upt(I,J)) ).

% distinct_upt
tff(fact_9441_map__Suc__upt,axiom,
    ! [Mb: nat,Nb: nat] : map_nat_nat(suc,upt(Mb,Nb)) = upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb)) ).

% map_Suc_upt
tff(fact_9442_map__add__upt,axiom,
    ! [Nb: nat,Mb: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aeh,Nb),upt(zero_zero_nat,Mb)) = upt(Nb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% map_add_upt
tff(fact_9443_atMost__upto,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,Nb) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,Nb))) ).

% atMost_upto
tff(fact_9444_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_9445_map__decr__upt,axiom,
    ! [Mb: nat,Nb: nat] : map_nat_nat(aTP_Lamm_aei,upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = upt(Mb,Nb) ).

% map_decr_upt
tff(fact_9446_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))) ) ) ).

% upt_add_eq_append
tff(fact_9447_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xb: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(Xb,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = Xb )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9448_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J),cons_nat(I,upt(aa_nat_nat(suc,I),J)),nil_nat) ).

% upt_rec
tff(fact_9449_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J),append_nat(upt(I,J),cons_nat(J,nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_9450_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9451_tl__upt,axiom,
    ! [Mb: nat,Nb: nat] : tl_nat(upt(Mb,Nb)) = upt(aa_nat_nat(suc,Mb),Nb) ).

% tl_upt
tff(fact_9452_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_9453_sum__list__upt,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups4561878855575611511st_nat(upt(Mb,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kr),set_or4665077453230672383an_nat(Mb,Nb)) ) ) ).

% sum_list_upt
tff(fact_9454_card__length__sum__list__rec,axiom,
    ! [Mb: nat,N4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aej(Mb),N4))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aek(Mb),N4)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_ael(Mb),N4)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9455_card__length__sum__list,axiom,
    ! [Mb: nat,N4: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aej(Mb),N4))) = 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,N4),Mb)),one_one_nat)),N4) ).

% card_length_sum_list
tff(fact_9456_sorted__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(Mb,Nb)) ).

% sorted_upt
tff(fact_9457_sorted__wrt__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(Mb,Nb)) ).

% sorted_wrt_upt
tff(fact_9458_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_9459_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(sorted_wrt_int(ord_less_int),upto(I,J)) ).

% sorted_wrt_upto
tff(fact_9460_sorted__upto,axiom,
    ! [Mb: int,Nb: int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),upto(Mb,Nb)) ).

% sorted_upto
tff(fact_9461_range__abs__Nats,axiom,
    image_int_int(abs_abs_int,top_top_set_int) = semiring_1_Nats_int ).

% range_abs_Nats
tff(fact_9462_min__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)) ).

% min_Suc_Suc
tff(fact_9463_min__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% min_0L
tff(fact_9464_min__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9465_min__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),aa_num_nat(pred_numeral,K2))) ).

% min_Suc_numeral
tff(fact_9466_min__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K2)),Nb)) ).

% min_numeral_Suc
tff(fact_9467_nat__mult__min__left,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ).

% nat_mult_min_left
tff(fact_9468_nat__mult__min__right,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)) ).

% nat_mult_min_right
tff(fact_9469_min__diff,axiom,
    ! [Mb: nat,I: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)),I) ).

% min_diff
tff(fact_9470_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9471_min__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_aem,Nb)),Mb) ).

% min_Suc2
tff(fact_9472_min__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_aen,Nb)),Mb) ).

% min_Suc1
tff(fact_9473_complex__is__Nat__iff,axiom,
    ! [Z: complex] :
      ( aa_set_complex_o(member_complex(Z),semiri3842193898606819883omplex)
    <=> ( ( aa_complex_real(im,Z) = zero_zero_real )
        & ? [I3: nat] : aa_complex_real(re,Z) = aa_nat_real(semiri5074537144036343181t_real,I3) ) ) ).

% complex_is_Nat_iff
tff(fact_9474_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_aeo) ).

% Rats_eq_int_div_nat
tff(fact_9475_min__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : ord_mi8085742599997312461d_enat(Q3,zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% min_enat_simps(2)
tff(fact_9476_min__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : ord_mi8085742599997312461d_enat(zero_z5237406670263579293d_enat,Q3) = zero_z5237406670263579293d_enat ).

% min_enat_simps(3)
tff(fact_9477_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
tff(fact_9478_Rats__no__top__le,axiom,
    ! [Xb: real] :
    ? [X3: real] :
      ( member_real(X3,field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),X3) ) ).

% Rats_no_top_le
tff(fact_9479_Rats__eq__int__div__int,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_aep) ).

% Rats_eq_int_div_int
tff(fact_9480_Rolle,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z2,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_9481_continuous__on__arcosh_H,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A4,F)
     => ( ! [X3: real] :
            ( member_real(X3,A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X3)) )
       => topolo5044208981011980120l_real(A4,aTP_Lamm_aeq(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_9482_continuous__image__closed__interval,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ? [C: real,D2: real] :
            ( ( image_real_real(F,set_or1222579329274155063t_real(A,B)) = set_or1222579329274155063t_real(C,D2) )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D2) ) ) ) ).

% continuous_image_closed_interval
tff(fact_9483_continuous__on__arcosh,axiom,
    ! [A4: set_real] :
      ( ord_less_eq_set_real(A4,set_ord_atLeast_real(one_one_real))
     => topolo5044208981011980120l_real(A4,arcosh_real) ) ).

% continuous_on_arcosh
tff(fact_9484_continuous__on__artanh,axiom,
    ! [A4: set_real] :
      ( ord_less_eq_set_real(A4,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))
     => topolo5044208981011980120l_real(A4,artanh_real) ) ).

% continuous_on_artanh
tff(fact_9485_Rolle__deriv,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F4,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                & ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(F4,Z2),X) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_9486_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ( aa_real_real(F,B) = aa_real_real(F,A) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_9487_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),zero_zero_real) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_9488_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_9489_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: fun_real_real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),B)
             => ( aa_real_real(F,Xb) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_9490_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_acf,Nb),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_9491_pred__nat__trancl__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),Nb)),transi2905341329935302413cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% pred_nat_trancl_eq_le
tff(fact_9492_log__inj,axiom,
    ! [B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => inj_on_real_real(aa_rea7340605660300375871l_real(log,B),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_9493_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K2: nat] :
      ( ! [N2: nat] :
          ( aa_set_nat_o(member_nat(N2),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acb,K2)),N4) ) ).

% inj_on_diff_nat
tff(fact_9494_inj__on__set__encode,axiom,
    inj_on_set_nat_nat(nat_set_encode,collect_set_nat(finite_finite_nat)) ).

% inj_on_set_encode
tff(fact_9495_inj__int__encode,axiom,
    ! [A4: set_int] : aa_set_int_o(inj_on_int_nat(nat_int_encode),A4) ).

% inj_int_encode
tff(fact_9496_inj__prod__encode,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(inj_on2178005380612969504at_nat(nat_prod_encode),A4) ).

% inj_prod_encode
tff(fact_9497_inj__Suc,axiom,
    ! [N4: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N4) ).

% inj_Suc
tff(fact_9498_inj__list__encode,axiom,
    ! [A4: set_list_nat] : inj_on_list_nat_nat(nat_list_encode,A4) ).

% inj_list_encode
tff(fact_9499_summable__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3))
         => summable_real(comp_nat_real_nat(F,G)) ) ) ) ).

% summable_reindex
tff(fact_9500_suminf__reindex__mono,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(comp_nat_real_nat(F,G))),suminf_real(F)) ) ) ) ).

% suminf_reindex_mono
tff(fact_9501_suminf__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3))
         => ( ! [X3: nat] :
                ( ~ aa_set_nat_o(member_nat(X3),image_nat_nat(G,top_top_set_nat))
               => ( aa_nat_real(F,X3) = zero_zero_real ) )
           => ( suminf_real(comp_nat_real_nat(F,G)) = suminf_real(F) ) ) ) ) ) ).

% suminf_reindex
tff(fact_9502_measure__function__int,axiom,
    fun_is_measure_int(comp_int_nat_int(nat2,abs_abs_int)) ).

% measure_function_int
tff(fact_9503_surj__int__decode,axiom,
    image_nat_int(nat_int_decode,top_top_set_nat) = top_top_set_int ).

% surj_int_decode
tff(fact_9504_int__encode__inverse,axiom,
    ! [Xb: int] : aa_nat_int(nat_int_decode,aa_int_nat(nat_int_encode,Xb)) = Xb ).

% int_encode_inverse
tff(fact_9505_int__decode__inverse,axiom,
    ! [Nb: nat] : aa_int_nat(nat_int_encode,aa_nat_int(nat_int_decode,Nb)) = Nb ).

% int_decode_inverse
tff(fact_9506_int__decode__eq,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( aa_nat_int(nat_int_decode,Xb) = aa_nat_int(nat_int_decode,Yb) )
    <=> ( Xb = Yb ) ) ).

% int_decode_eq
tff(fact_9507_inj__int__decode,axiom,
    ! [A4: set_nat] : aa_set_nat_o(inj_on_nat_int(nat_int_decode),A4) ).

% inj_int_decode
tff(fact_9508_bij__int__decode,axiom,
    aa_set_int_o(bij_betw_nat_int(nat_int_decode,top_top_set_nat),top_top_set_int) ).

% bij_int_decode
tff(fact_9509_powr__real__of__int_H,axiom,
    ! [Xb: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( ( Xb != zero_zero_real )
          | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,Nb)) = aa_int_real(power_int_real(Xb),Nb) ) ) ) ).

% powr_real_of_int'
tff(fact_9510_pairs__le__eq__Sigma,axiom,
    ! [Mb: nat] : collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_vy(Mb))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,Mb),aTP_Lamm_aer(Mb)) ).

% pairs_le_eq_Sigma
tff(fact_9511_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_aet,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_9512_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_aev,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_9513_take__upt,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb)),Nb)
     => ( take_nat(Mb,upt(I,Nb)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb)) ) ) ).

% take_upt
tff(fact_9514_eventually__prod__sequentially,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o] :
      ( eventu1038000079068216329at_nat(P2,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N8: nat] :
        ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),M4)
         => ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N3)
             => aa_Pro6150637604445964947_nat_o(P2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N3),M4)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_9515_drop__upt,axiom,
    ! [Mb: nat,I: nat,J: nat] : drop_nat(Mb,upt(I,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb),J) ).

% drop_upt
tff(fact_9516_at__right__to__0,axiom,
    ! [A: real] : topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)) = filtermap_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_aew,A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% at_right_to_0
tff(fact_9517_at__right__to__top,axiom,
    topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)) = filtermap_real_real(inverse_inverse_real,at_top_real) ).

% at_right_to_top
tff(fact_9518_at__top__to__right,axiom,
    at_top_real = filtermap_real_real(inverse_inverse_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% at_top_to_right
tff(fact_9519_filtermap__ln__at__right,axiom,
    filtermap_real_real(ln_ln_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) = at_bot_real ).

% filtermap_ln_at_right
tff(fact_9520_pos__deriv__imp__strict__mono,axiom,
    ! [F: fun_real_real,F4: fun_real_real] :
      ( ! [X3: real] : has_fi5821293074295781190e_real(F,aa_real_real(F4,X3),topolo2177554685111907308n_real(X3,top_top_set_real))
     => ( ! [X3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F4,X3))
       => order_7092887310737990675l_real(F) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_9521_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_9522_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,Nb: nat] :
      ( order_5726023648592871131at_nat(F)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(F,Nb)) ) ).

% strict_mono_imp_increasing
tff(fact_9523_infinite__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ? [R3: fun_nat_nat] :
          ( order_5726023648592871131at_nat(R3)
          & ! [N5: nat] : aa_set_nat_o(member_nat(aa_nat_nat(R3,N5)),S2) ) ) ).

% infinite_enumerate
tff(fact_9524_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_abt)) ).

% less_eq_int_def
tff(fact_9525_less__int__def,axiom,
    ord_less_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_abr)) ).

% less_int_def
tff(fact_9526_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_9527_vimage__Suc__insert__Suc,axiom,
    ! [Nb: nat,A4: set_nat] : vimage_nat_nat(suc,insert_nat(aa_nat_nat(suc,Nb),A4)) = insert_nat(Nb,vimage_nat_nat(suc,A4)) ).

% vimage_Suc_insert_Suc
tff(fact_9528_vimage__Suc__insert__0,axiom,
    ! [A4: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A4)) = vimage_nat_nat(suc,A4) ).

% vimage_Suc_insert_0
tff(fact_9529_finite__vimage__Suc__iff,axiom,
    ! [F3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,vimage_nat_nat(suc,F3))
    <=> aa_set_nat_o(finite_finite_nat,F3) ) ).

% finite_vimage_Suc_iff
tff(fact_9530_set__decode__div__2,axiom,
    ! [Xb: nat] : nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = vimage_nat_nat(suc,nat_set_decode(Xb)) ).

% set_decode_div_2
tff(fact_9531_set__encode__vimage__Suc,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(nat_set_encode,vimage_nat_nat(suc,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(nat_set_encode,A4)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% set_encode_vimage_Suc
tff(fact_9532_positive__rat,axiom,
    ! [A: int,B: int] :
      ( aa_rat_o(positive,aa_int_rat(fract(A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% positive_rat
tff(fact_9533_Rat_Opositive__add,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(positive,Xb)
     => ( aa_rat_o(positive,Yb)
       => aa_rat_o(positive,aa_rat_rat(plus_plus_rat(Xb),Yb)) ) ) ).

% Rat.positive_add
tff(fact_9534_Rat_Opositive__zero,axiom,
    ~ aa_rat_o(positive,zero_zero_rat) ).

% Rat.positive_zero
tff(fact_9535_Rat_Opositive__minus,axiom,
    ! [Xb: rat] :
      ( ~ aa_rat_o(positive,Xb)
     => ( ( Xb != zero_zero_rat )
       => aa_rat_o(positive,aa_rat_rat(uminus_uminus_rat,Xb)) ) ) ).

% Rat.positive_minus
tff(fact_9536_Rat_Opositive_Orep__eq,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(positive,Xb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_rat7754553326175492643nt_int(rep_Rat,Xb))),aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_rat7754553326175492643nt_int(rep_Rat,Xb)))) ) ).

% Rat.positive.rep_eq
tff(fact_9537_Rat_Opositive__def,axiom,
    positive = map_fu898904425404107465nt_o_o(rep_Rat,id_o,aTP_Lamm_aex) ).

% Rat.positive_def
tff(fact_9538_Restr__natLeq,axiom,
    ! [Nb: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bb,Nb)),aTP_Lamm_aey(Nb))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aez(Nb))) ).

% Restr_natLeq
tff(fact_9539_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_9540_Restr__natLeq2,axiom,
    ! [Nb: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq,Nb),aTP_Lamm_afa(Nb))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aez(Nb))) ).

% Restr_natLeq2
tff(fact_9541_Field__natLeq__on,axiom,
    ! [Nb: nat] : field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aez(Nb)))) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bb,Nb)) ).

% Field_natLeq_on
tff(fact_9542_wf__int__ge__less__than2,axiom,
    ! [D: int] : wf_int(int_ge_less_than2(D)) ).

% wf_int_ge_less_than2
tff(fact_9543_wf__int__ge__less__than,axiom,
    ! [D: int] : wf_int(int_ge_less_than(D)) ).

% wf_int_ge_less_than
tff(fact_9544_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_9545_mono__times__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb)) ) ).

% mono_times_nat
tff(fact_9546_mono__ge2__power__minus__self,axiom,
    ! [K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_afb,K2)) ) ).

% mono_ge2_power_minus_self
tff(fact_9547_incseq__bounded,axiom,
    ! [X5: fun_nat_real,B4: real] :
      ( order_mono_nat_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I2)),B4)
       => bfun_nat_real(X5,at_top_nat) ) ) ).

% incseq_bounded
tff(fact_9548_incseq__convergent,axiom,
    ! [X5: fun_nat_real,B4: real] :
      ( order_mono_nat_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I2)),B4)
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I4)),L6) ) ) ) ).

% incseq_convergent
tff(fact_9549_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( ! [X3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3))
     => ( order_mono_nat_real(F)
       => ( order_5726023648592871131at_nat(G)
         => ( bfun_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_afc(F),G),at_top_nat)
          <=> bfun_nat_real(F,at_top_nat) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
tff(fact_9550_quotient__of__def,axiom,
    ! [Xb: rat] : quotient_of(Xb) = the_Pr4378521158711661632nt_int(aTP_Lamm_afd(Xb)) ).

% quotient_of_def
tff(fact_9551_normalize__stable,axiom,
    ! [Q3: int,P: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3)
     => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P),Q3)
       => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q3)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q3) ) ) ) ).

% normalize_stable
tff(fact_9552_quotient__of__coprime,axiom,
    ! [R2: rat,P: int,Q3: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q3) )
     => aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P),Q3) ) ).

% quotient_of_coprime
tff(fact_9553_normalize__coprime,axiom,
    ! [R2: product_prod_int_int,P: int,Q3: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q3) )
     => aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P),Q3) ) ).

% normalize_coprime
tff(fact_9554_Rat__cases,axiom,
    ! [Q3: rat] :
      ~ ! [A3: int,B3: int] :
          ( ( Q3 = aa_int_rat(fract(A3),B3) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
           => ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A3),B3) ) ) ).

% Rat_cases
tff(fact_9555_Rat__induct,axiom,
    ! [P2: fun_rat_o,Q3: rat] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
         => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A3),B3)
           => aa_rat_o(P2,aa_int_rat(fract(A3),B3)) ) )
     => aa_rat_o(P2,Q3) ) ).

% Rat_induct
tff(fact_9556_Rat__cases__nonzero,axiom,
    ! [Q3: rat] :
      ( ! [A3: int,B3: int] :
          ( ( Q3 = aa_int_rat(fract(A3),B3) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
           => ( ( A3 != zero_zero_int )
             => ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A3),B3) ) ) )
     => ( Q3 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
tff(fact_9557_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X3: product_prod_int_int] :
      ( ( R2 = aa_int_rat(fract(aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3)) )
      & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3))
      & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3))
      & ! [Y3: product_prod_int_int] :
          ( ( ( R2 = aa_int_rat(fract(aa_Pro6799258434465214121nt_int(product_fst_int_int,Y3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Y3)) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,Y3))
            & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Y3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Y3)) )
         => ( Y3 = X3 ) ) ) ).

% quotient_of_unique
tff(fact_9558_coprime__Suc__left__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,Nb)),Nb) ).

% coprime_Suc_left_nat
tff(fact_9559_coprime__Suc__right__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(suc,Nb)) ).

% coprime_Suc_right_nat
tff(fact_9560_coprime__Suc__0__right,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) ).

% coprime_Suc_0_right
tff(fact_9561_coprime__Suc__0__left,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ).

% coprime_Suc_0_left
tff(fact_9562_coprime__diff__one__right__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ).

% coprime_diff_one_right_nat
tff(fact_9563_coprime__diff__one__left__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),Nb) ) ).

% coprime_diff_one_left_nat
tff(fact_9564_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : aa_set_nat_nat(gcd_Gcd_nat,set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_gcd_nat,Xs),zero_zero_nat) ).

% Gcd_nat_set_eq_fold
tff(fact_9565_Rats__abs__nat__div__natE,axiom,
    ! [Xb: real] :
      ( member_real(Xb,field_5140801741446780682s_real)
     => ~ ! [M: nat,N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( ( aa_real_real(abs_abs_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N2)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,M),N2) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_9566_Gcd__int__set__eq__fold,axiom,
    ! [Xs: list_int] : gcd_Gcd_int(set_int2(Xs)) = aa_int_int(fold_int_int(gcd_gcd_int,Xs),zero_zero_int) ).

% Gcd_int_set_eq_fold
tff(fact_9567_sort__upt,axiom,
    ! [Mb: nat,Nb: nat] : linord738340561235409698at_nat(aTP_Lamm_kr,upt(Mb,Nb)) = upt(Mb,Nb) ).

% sort_upt
tff(fact_9568_sort__upto,axiom,
    ! [I: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_eg,upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_9569_vanishes__mult__bounded,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( ? [A7: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A7)
          & ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N2))),A7) )
     => ( vanishes(Y5)
       => vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_afe(X5),Y5)) ) ) ).

% vanishes_mult_bounded
tff(fact_9570_vanishes__const,axiom,
    ! [C2: rat] :
      ( vanishes(aTP_Lamm_aff(C2))
    <=> ( C2 = zero_zero_rat ) ) ).

% vanishes_const
tff(fact_9571_vanishes__add,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( vanishes(X5)
     => ( vanishes(Y5)
       => vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_afg(X5),Y5)) ) ) ).

% vanishes_add
tff(fact_9572_vanishesD,axiom,
    ! [X5: fun_nat_rat,R2: rat] :
      ( vanishes(X5)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
       => ? [K: nat] :
          ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N5))),R2) ) ) ) ).

% vanishesD
tff(fact_9573_vanishesI,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [R3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R3)
         => ? [K4: nat] :
            ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),N2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N2))),R3) ) )
     => vanishes(X5) ) ).

% vanishesI
tff(fact_9574_vanishes__def,axiom,
    ! [X5: fun_nat_rat] :
      ( vanishes(X5)
    <=> ! [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
         => ? [K3: nat] :
            ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N3))),R5) ) ) ) ).

% vanishes_def
tff(fact_9575_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_nq,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_9576_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_pj,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_9577_ATP_Olambda__3,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_zv,Uu)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_3
tff(fact_9578_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_np,Uu) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_4
tff(fact_9579_ATP_Olambda__5,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_acw,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_5
tff(fact_9580_ATP_Olambda__6,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_aec,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Uu)),Uu) ).

% ATP.lambda_6
tff(fact_9581_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_aei,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_7
tff(fact_9582_ATP_Olambda__8,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_qh,Uu) = plus_p5714425477246183910nteger(Uu,one_one_Code_integer) ).

% ATP.lambda_8
tff(fact_9583_ATP_Olambda__9,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_qg,Uu) = aa_complex_complex(plus_plus_complex(Uu),one_one_complex) ).

% ATP.lambda_9
tff(fact_9584_ATP_Olambda__10,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_qe,Uu) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),one_one_real) ).

% ATP.lambda_10
tff(fact_9585_ATP_Olambda__11,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_qc,Uu) = aa_rat_rat(plus_plus_rat(Uu),one_one_rat) ).

% ATP.lambda_11
tff(fact_9586_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_qf,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_12
tff(fact_9587_ATP_Olambda__13,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_qd,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_13
tff(fact_9588_ATP_Olambda__14,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_aex,Uu)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu))) ) ).

% ATP.lambda_14
tff(fact_9589_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_adc,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_15
tff(fact_9590_ATP_Olambda__16,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_acu,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_no,Uu)) ).

% ATP.lambda_16
tff(fact_9591_ATP_Olambda__17,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_adq,Uu) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ca,Uu)) ).

% ATP.lambda_17
tff(fact_9592_ATP_Olambda__18,axiom,
    ! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_aev,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5414030515140494994real_o(aTP_Lamm_aeu(Uu)))) ).

% ATP.lambda_18
tff(fact_9593_ATP_Olambda__19,axiom,
    ! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_aet,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc6771430404735790350plex_o(aTP_Lamm_aes(Uu)))) ).

% ATP.lambda_19
tff(fact_9594_ATP_Olambda__20,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ade,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_20
tff(fact_9595_ATP_Olambda__21,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_aby,Uu) = some_num(aa_num_num(bit1,Uu)) ).

% ATP.lambda_21
tff(fact_9596_ATP_Olambda__22,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_abf,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_22
tff(fact_9597_ATP_Olambda__23,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_zz,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_23
tff(fact_9598_ATP_Olambda__24,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_aaa,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_24
tff(fact_9599_ATP_Olambda__25,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_ij,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_25
tff(fact_9600_ATP_Olambda__26,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_aeo,Uu)
    <=> ? [I3: int,N3: nat] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I3)),aa_nat_real(semiri5074537144036343181t_real,N3)) )
          & ( N3 != zero_zero_nat ) ) ) ).

% ATP.lambda_26
tff(fact_9601_ATP_Olambda__27,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_aep,Uu)
    <=> ? [I3: int,J3: int] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I3)),aa_int_real(ring_1_of_int_real,J3)) )
          & ( J3 != zero_zero_int ) ) ) ).

% ATP.lambda_27
tff(fact_9602_ATP_Olambda__28,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_abh,Uu) = some_num(one) ).

% ATP.lambda_28
tff(fact_9603_ATP_Olambda__29,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_abl(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_abj(Uua),aTP_Lamm_abk(Uua),Uu) ).

% ATP.lambda_29
tff(fact_9604_ATP_Olambda__30,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aTP_Lamm_ye(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_z3403309356797280102nteger) ).

% ATP.lambda_30
tff(fact_9605_ATP_Olambda__31,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aTP_Lamm_yd(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_complex) ).

% ATP.lambda_31
tff(fact_9606_ATP_Olambda__32,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_yf,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_real) ).

% ATP.lambda_32
tff(fact_9607_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_yb(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_rat) ).

% ATP.lambda_33
tff(fact_9608_ATP_Olambda__34,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_yc,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_int) ).

% ATP.lambda_34
tff(fact_9609_ATP_Olambda__35,axiom,
    ! [Uu: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_pk(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real,aa_nat_real(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_35
tff(fact_9610_ATP_Olambda__36,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_zy,Uu),Uua) = $ite(Uu = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_36
tff(fact_9611_ATP_Olambda__37,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aTP_Lamm_xz(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_z3403309356797280102nteger) ).

% ATP.lambda_37
tff(fact_9612_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aTP_Lamm_xy(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_complex) ).

% ATP.lambda_38
tff(fact_9613_ATP_Olambda__39,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ya,Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_real) ).

% ATP.lambda_39
tff(fact_9614_ATP_Olambda__40,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_xw(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_rat) ).

% ATP.lambda_40
tff(fact_9615_ATP_Olambda__41,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_xx,Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_int) ).

% ATP.lambda_41
tff(fact_9616_ATP_Olambda__42,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_abm,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_abl(Uua),Uu) ).

% ATP.lambda_42
tff(fact_9617_ATP_Olambda__43,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_abj(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_abf,bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_43
tff(fact_9618_ATP_Olambda__44,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_abg(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_abf,bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_44
tff(fact_9619_ATP_Olambda__45,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_wc(Uu),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_wb(Uu),Uua),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_45
tff(fact_9620_ATP_Olambda__46,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_wa(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vz(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_46
tff(fact_9621_ATP_Olambda__47,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_xv(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_xu(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_47
tff(fact_9622_ATP_Olambda__48,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_xt(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vz(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_48
tff(fact_9623_ATP_Olambda__49,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_49
tff(fact_9624_ATP_Olambda__50,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_px,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,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_50
tff(fact_9625_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_ws(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_51
tff(fact_9626_ATP_Olambda__52,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_wr(Uu),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,Uua))),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_52
tff(fact_9627_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_wt,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_53
tff(fact_9628_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_wp(Uu),Uua) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,Uua))),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_54
tff(fact_9629_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_wq,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_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)),Uua)),aa_nat_int(semiri1314217659103216013at_int,Uua))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_55
tff(fact_9630_ATP_Olambda__56,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_zs,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_56
tff(fact_9631_ATP_Olambda__57,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_aam,Uu),Uua) = $let(
        l2: int,
        l2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ).

% ATP.lambda_57
tff(fact_9632_ATP_Olambda__58,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_lt,Uu),Uua) = $let(
        m3: code_integer,
        m3:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,Uu)),
        $ite(Uua = zero_zero_nat,m3,plus_p5714425477246183910nteger(m3,one_one_Code_integer)) ) ).

% ATP.lambda_58
tff(fact_9633_ATP_Olambda__59,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ls,Uu),Uua) = $let(
        m3: complex,
        m3:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_complex_complex(plus_plus_complex(m3),one_one_complex)) ) ).

% ATP.lambda_59
tff(fact_9634_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_lq,Uu),Uua) = $let(
        m3: real,
        m3:= 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,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m3),one_one_real)) ) ).

% ATP.lambda_60
tff(fact_9635_ATP_Olambda__61,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_lo,Uu),Uua) = $let(
        m3: rat,
        m3:= aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_rat_rat(plus_plus_rat(m3),one_one_rat)) ) ).

% ATP.lambda_61
tff(fact_9636_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lr,Uu),Uua) = $let(
        m3: nat,
        m3:= 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(semiri1316708129612266289at_nat,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m3),one_one_nat)) ) ).

% ATP.lambda_62
tff(fact_9637_ATP_Olambda__63,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_lp,Uu),Uua) = $let(
        m3: int,
        m3:= 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(semiri1314217659103216013at_int,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_int_int(aa_int_fun_int_int(plus_plus_int,m3),one_one_int)) ) ).

% ATP.lambda_63
tff(fact_9638_ATP_Olambda__64,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_abc(Uu),Uua)
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi) ) ) ).

% ATP.lambda_64
tff(fact_9639_ATP_Olambda__65,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aa_real_fun_int_o(aTP_Lamm_zw,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu)
        & aa_real_o(aa_real_fun_real_o(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_65
tff(fact_9640_ATP_Olambda__66,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_zx(Uu),Uua)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu)
        & aa_rat_o(aa_rat_fun_rat_o(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_66
tff(fact_9641_ATP_Olambda__67,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_no,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_67
tff(fact_9642_ATP_Olambda__68,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_acv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_68
tff(fact_9643_ATP_Olambda__69,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_acx(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_69
tff(fact_9644_ATP_Olambda__70,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_wx(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_70
tff(fact_9645_ATP_Olambda__71,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ww(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_71
tff(fact_9646_ATP_Olambda__72,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_wy,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_72
tff(fact_9647_ATP_Olambda__73,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_wu(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_73
tff(fact_9648_ATP_Olambda__74,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_wv,Uu),Uua) = 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)),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_74
tff(fact_9649_ATP_Olambda__75,axiom,
    ! [Uu: rat,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_afd(Uu),Uua)
    <=> ( ( Uu = aa_int_rat(fract(aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua)) )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua))
        & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua)) ) ) ).

% ATP.lambda_75
tff(fact_9650_ATP_Olambda__76,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_xr(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))),Uua) ).

% ATP.lambda_76
tff(fact_9651_ATP_Olambda__77,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xs,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))),Uua) ).

% ATP.lambda_77
tff(fact_9652_ATP_Olambda__78,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_xq(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))),Uua) ).

% ATP.lambda_78
tff(fact_9653_ATP_Olambda__79,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_qw(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_79
tff(fact_9654_ATP_Olambda__80,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qx,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu),Uua)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_80
tff(fact_9655_ATP_Olambda__81,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qv(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Uu),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_81
tff(fact_9656_ATP_Olambda__82,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_abd,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_82
tff(fact_9657_ATP_Olambda__83,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_aeg(Uu),Uua)
    <=> ( aa_set_int_o(member_int(Uua),Uu)
        & ! [X4: int] :
            ( aa_set_int_o(member_int(X4),Uu)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Uua) ) ) ) ).

% ATP.lambda_83
tff(fact_9658_ATP_Olambda__84,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qb,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_84
tff(fact_9659_ATP_Olambda__85,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_aed,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_85
tff(fact_9660_ATP_Olambda__86,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_ql,Uu),Uua)
    <=> ( aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(Uu),Uua)
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Uua) ) ) ).

% ATP.lambda_86
tff(fact_9661_ATP_Olambda__87,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aa_set8643480048548153067plex_o(aTP_Lamm_qk,Uu),Uua)
    <=> ( aa_set_complex_o(ord_less_set_complex(Uu),Uua)
        & aa_set_complex_o(finite3207457112153483333omplex,Uua) ) ) ).

% ATP.lambda_87
tff(fact_9662_ATP_Olambda__88,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_qi,Uu),Uua)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Uu),Uua)
        & aa_set_nat_o(finite_finite_nat,Uua) ) ) ).

% ATP.lambda_88
tff(fact_9663_ATP_Olambda__89,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_qj,Uu),Uua)
    <=> ( aa_set_int_o(ord_less_set_int(Uu),Uua)
        & aa_set_int_o(finite_finite_int,Uua) ) ) ).

% ATP.lambda_89
tff(fact_9664_ATP_Olambda__90,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_afb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uua)),Uua) ).

% ATP.lambda_90
tff(fact_9665_ATP_Olambda__91,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vv,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_91
tff(fact_9666_ATP_Olambda__92,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vu,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_92
tff(fact_9667_ATP_Olambda__93,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_cr,Uu),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_93
tff(fact_9668_ATP_Olambda__94,axiom,
    ! [Uu: nat,Uua: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_cq,Uu),Uua)
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_94
tff(fact_9669_ATP_Olambda__95,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adj,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_95
tff(fact_9670_ATP_Olambda__96,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ado,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),Uua))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Uua)) ).

% ATP.lambda_96
tff(fact_9671_ATP_Olambda__97,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_aee,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),Uua))),Uua) ).

% ATP.lambda_97
tff(fact_9672_ATP_Olambda__98,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lu,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_98
tff(fact_9673_ATP_Olambda__99,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_zu,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi)
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_99
tff(fact_9674_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vf(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_100
tff(fact_9675_ATP_Olambda__101,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_vg(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_101
tff(fact_9676_ATP_Olambda__102,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_vh(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_102
tff(fact_9677_ATP_Olambda__103,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_vi(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_103
tff(fact_9678_ATP_Olambda__104,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_kq(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_104
tff(fact_9679_ATP_Olambda__105,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kn(Uu),Uua) = aa_rat_rat(plus_plus_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_105
tff(fact_9680_ATP_Olambda__106,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_kp(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_106
tff(fact_9681_ATP_Olambda__107,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ko(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_107
tff(fact_9682_ATP_Olambda__108,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jp(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_108
tff(fact_9683_ATP_Olambda__109,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_jn(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_109
tff(fact_9684_ATP_Olambda__110,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_jo(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,aa_nat_nat(suc,Uua))),aa_nat_int(Uu,Uua)) ).

% ATP.lambda_110
tff(fact_9685_ATP_Olambda__111,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_im(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).

% ATP.lambda_111
tff(fact_9686_ATP_Olambda__112,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_io(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_112
tff(fact_9687_ATP_Olambda__113,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_in(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_113
tff(fact_9688_ATP_Olambda__114,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_oo(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_114
tff(fact_9689_ATP_Olambda__115,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_km(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_115
tff(fact_9690_ATP_Olambda__116,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kk(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_116
tff(fact_9691_ATP_Olambda__117,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_kl(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_117
tff(fact_9692_ATP_Olambda__118,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_kg(Uu),Uua) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_118
tff(fact_9693_ATP_Olambda__119,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_kj(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_119
tff(fact_9694_ATP_Olambda__120,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kh(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_120
tff(fact_9695_ATP_Olambda__121,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ki(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_121
tff(fact_9696_ATP_Olambda__122,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ml(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_122
tff(fact_9697_ATP_Olambda__123,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mj(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_123
tff(fact_9698_ATP_Olambda__124,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mk(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_124
tff(fact_9699_ATP_Olambda__125,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_xf(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_125
tff(fact_9700_ATP_Olambda__126,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_xh(Uu),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_126
tff(fact_9701_ATP_Olambda__127,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_xc(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_127
tff(fact_9702_ATP_Olambda__128,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_wz(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_128
tff(fact_9703_ATP_Olambda__129,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_complex(aTP_Lamm_qa(Uu),Uua) = aa_real_complex(complex2(aa_complex_real(Uu,Uua)),zero_zero_real) ).

% ATP.lambda_129
tff(fact_9704_ATP_Olambda__130,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_complex(aTP_Lamm_pz(Uu),Uua) = aa_real_complex(complex2(aa_nat_real(Uu,Uua)),zero_zero_real) ).

% ATP.lambda_130
tff(fact_9705_ATP_Olambda__131,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ia(Uu),Uua)
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_131
tff(fact_9706_ATP_Olambda__132,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_id(Uu),Uua)
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_132
tff(fact_9707_ATP_Olambda__133,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_hw(Uu),Uua)
    <=> ( aa_real_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_133
tff(fact_9708_ATP_Olambda__134,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_hy(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_134
tff(fact_9709_ATP_Olambda__135,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( aa_real_o(aTP_Lamm_ib(Uu),Uua)
    <=> ( aa_real_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_135
tff(fact_9710_ATP_Olambda__136,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( aa_real_o(aTP_Lamm_ie(Uu),Uua)
    <=> ( aa_real_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_136
tff(fact_9711_ATP_Olambda__137,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( aa_int_o(aTP_Lamm_hx(Uu),Uua)
    <=> ( aa_int_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_137
tff(fact_9712_ATP_Olambda__138,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( aa_int_o(aTP_Lamm_hz(Uu),Uua)
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_138
tff(fact_9713_ATP_Olambda__139,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_ic(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_139
tff(fact_9714_ATP_Olambda__140,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_if(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_140
tff(fact_9715_ATP_Olambda__141,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_tj(Uu),Uua)
    <=> ( aa_complex_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_141
tff(fact_9716_ATP_Olambda__142,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_tm(Uu),Uua)
    <=> ( aa_complex_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_142
tff(fact_9717_ATP_Olambda__143,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_tp(Uu),Uua)
    <=> ( aa_complex_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_143
tff(fact_9718_ATP_Olambda__144,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_th(Uu),Uua)
    <=> ( aa_real_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_144
tff(fact_9719_ATP_Olambda__145,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_tk(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_145
tff(fact_9720_ATP_Olambda__146,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( aa_real_o(aTP_Lamm_tn(Uu),Uua)
    <=> ( aa_real_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_146
tff(fact_9721_ATP_Olambda__147,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( aa_real_o(aTP_Lamm_tq(Uu),Uua)
    <=> ( aa_real_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_147
tff(fact_9722_ATP_Olambda__148,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( aa_int_o(aTP_Lamm_ti(Uu),Uua)
    <=> ( aa_int_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_148
tff(fact_9723_ATP_Olambda__149,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( aa_int_o(aTP_Lamm_tl(Uu),Uua)
    <=> ( aa_int_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_149
tff(fact_9724_ATP_Olambda__150,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_to(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_150
tff(fact_9725_ATP_Olambda__151,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_acy(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_acx(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_151
tff(fact_9726_ATP_Olambda__152,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_adn(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_acx(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_152
tff(fact_9727_ATP_Olambda__153,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abz,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_153
tff(fact_9728_ATP_Olambda__154,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_oy(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_154
tff(fact_9729_ATP_Olambda__155,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ox,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_155
tff(fact_9730_ATP_Olambda__156,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_abe(Uu),Uua) = aa_int_int(aa_nat_fun_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_156
tff(fact_9731_ATP_Olambda__157,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_acf,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_157
tff(fact_9732_ATP_Olambda__158,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_py,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_158
tff(fact_9733_ATP_Olambda__159,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pm,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_159
tff(fact_9734_ATP_Olambda__160,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_aan,Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= aa_num_num(plus_plus_num(l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(plus_plus_num(l3),one)) ) ) ).

% ATP.lambda_160
tff(fact_9735_ATP_Olambda__161,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_aap,Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ).

% ATP.lambda_161
tff(fact_9736_ATP_Olambda__162,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_vk(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_162
tff(fact_9737_ATP_Olambda__163,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_vl,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_163
tff(fact_9738_ATP_Olambda__164,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_vj(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_164
tff(fact_9739_ATP_Olambda__165,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_iu,Uu),Uua) = produc1086072967326762835nteger(Uu,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer)) ).

% ATP.lambda_165
tff(fact_9740_ATP_Olambda__166,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_it,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(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_166
tff(fact_9741_ATP_Olambda__167,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_is,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)) ).

% ATP.lambda_167
tff(fact_9742_ATP_Olambda__168,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ii,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_168
tff(fact_9743_ATP_Olambda__169,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ih,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_169
tff(fact_9744_ATP_Olambda__170,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ig,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_170
tff(fact_9745_ATP_Olambda__171,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_qu(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).

% ATP.lambda_171
tff(fact_9746_ATP_Olambda__172,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qt,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% ATP.lambda_172
tff(fact_9747_ATP_Olambda__173,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qs(Uu),Uua) = aa_rat_rat(plus_plus_rat(Uu),aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% ATP.lambda_173
tff(fact_9748_ATP_Olambda__174,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adk,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_174
tff(fact_9749_ATP_Olambda__175,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_jv,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_o_int(zero_n2684676970156552555ol_int,Uua != zero_zero_int)) ).

% ATP.lambda_175
tff(fact_9750_ATP_Olambda__176,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adi,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_176
tff(fact_9751_ATP_Olambda__177,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adf,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_177
tff(fact_9752_ATP_Olambda__178,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ace(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(Uu),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_178
tff(fact_9753_ATP_Olambda__179,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_tz(Uu),Uua) = minus_8373710615458151222nteger(Uu,aa_nat_Code_integer(semiri4939895301339042750nteger,Uua)) ).

% ATP.lambda_179
tff(fact_9754_ATP_Olambda__180,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ub(Uu),Uua) = aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_180
tff(fact_9755_ATP_Olambda__181,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ua,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_181
tff(fact_9756_ATP_Olambda__182,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ty(Uu),Uua) = aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_182
tff(fact_9757_ATP_Olambda__183,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uc,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_183
tff(fact_9758_ATP_Olambda__184,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_ud(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_184
tff(fact_9759_ATP_Olambda__185,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_uw(Uu),Uua) = plus_p5714425477246183910nteger(Uu,aa_nat_Code_integer(semiri4939895301339042750nteger,Uua)) ).

% ATP.lambda_185
tff(fact_9760_ATP_Olambda__186,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_uv(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_186
tff(fact_9761_ATP_Olambda__187,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uu,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_187
tff(fact_9762_ATP_Olambda__188,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ut(Uu),Uua) = aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_188
tff(fact_9763_ATP_Olambda__189,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ux,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_189
tff(fact_9764_ATP_Olambda__190,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_uy(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_190
tff(fact_9765_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aca,Uu),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_191
tff(fact_9766_ATP_Olambda__192,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_vq,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),Uu) ) ).

% ATP.lambda_192
tff(fact_9767_ATP_Olambda__193,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aTP_Lamm_ax(Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Uua),Uu) ) ).

% ATP.lambda_193
tff(fact_9768_ATP_Olambda__194,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aTP_Lamm_aw(Uu),Uua)
    <=> aa_set_complex_o(ord_le211207098394363844omplex(Uua),Uu) ) ).

% ATP.lambda_194
tff(fact_9769_ATP_Olambda__195,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_av,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Uua),Uu) ) ).

% ATP.lambda_195
tff(fact_9770_ATP_Olambda__196,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_ay(Uu),Uua)
    <=> aa_set_int_o(ord_less_eq_set_int(Uua),Uu) ) ).

% ATP.lambda_196
tff(fact_9771_ATP_Olambda__197,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_vr,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu) ) ).

% ATP.lambda_197
tff(fact_9772_ATP_Olambda__198,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_vs,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Uua),Uu) ) ).

% ATP.lambda_198
tff(fact_9773_ATP_Olambda__199,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bc,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_199
tff(fact_9774_ATP_Olambda__200,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_vt,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_200
tff(fact_9775_ATP_Olambda__201,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acd,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_201
tff(fact_9776_ATP_Olambda__202,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_mg,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),Uu) ) ).

% ATP.lambda_202
tff(fact_9777_ATP_Olambda__203,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_mc,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Uua),Uu) ) ).

% ATP.lambda_203
tff(fact_9778_ATP_Olambda__204,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_md,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu) ) ).

% ATP.lambda_204
tff(fact_9779_ATP_Olambda__205,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_me,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Uua),Uu) ) ).

% ATP.lambda_205
tff(fact_9780_ATP_Olambda__206,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bb,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_206
tff(fact_9781_ATP_Olambda__207,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_mf,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu) ) ).

% ATP.lambda_207
tff(fact_9782_ATP_Olambda__208,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acz,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_208
tff(fact_9783_ATP_Olambda__209,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_209
tff(fact_9784_ATP_Olambda__210,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_acg,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_210
tff(fact_9785_ATP_Olambda__211,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_aew,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu) ).

% ATP.lambda_211
tff(fact_9786_ATP_Olambda__212,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aeh,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_212
tff(fact_9787_ATP_Olambda__213,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_acc,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_213
tff(fact_9788_ATP_Olambda__214,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ach,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_214
tff(fact_9789_ATP_Olambda__215,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_cb,Uu),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu) ) ).

% ATP.lambda_215
tff(fact_9790_ATP_Olambda__216,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_bz,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Uua),Uu) ) ).

% ATP.lambda_216
tff(fact_9791_ATP_Olambda__217,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ca,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_217
tff(fact_9792_ATP_Olambda__218,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bd,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_218
tff(fact_9793_ATP_Olambda__219,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_abp,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_219
tff(fact_9794_ATP_Olambda__220,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vp,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_220
tff(fact_9795_ATP_Olambda__221,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_add,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_221
tff(fact_9796_ATP_Olambda__222,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_a(Uu),Uua)
    <=> aa_set_complex_o(member_complex(Uua),Uu) ) ).

% ATP.lambda_222
tff(fact_9797_ATP_Olambda__223,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_aa(Uu),Uua)
    <=> member_real(Uua,Uu) ) ).

% ATP.lambda_223
tff(fact_9798_ATP_Olambda__224,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ab(Uu),Uua)
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_224
tff(fact_9799_ATP_Olambda__225,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ac(Uu),Uua)
    <=> member_set_nat(Uua,Uu) ) ).

% ATP.lambda_225
tff(fact_9800_ATP_Olambda__226,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_226
tff(fact_9801_ATP_Olambda__227,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ae(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_227
tff(fact_9802_ATP_Olambda__228,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_afa(Uu),Uua) = order_underS_nat(bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_228
tff(fact_9803_ATP_Olambda__229,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ni(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_229
tff(fact_9804_ATP_Olambda__230,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nh(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_230
tff(fact_9805_ATP_Olambda__231,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_aeb(Uu),Uua)
    <=> aa_real_o(Uu,aa_real_real(inverse_inverse_real,Uua)) ) ).

% ATP.lambda_231
tff(fact_9806_ATP_Olambda__232,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_pb(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_232
tff(fact_9807_ATP_Olambda__233,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jb(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_233
tff(fact_9808_ATP_Olambda__234,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_adr(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_234
tff(fact_9809_ATP_Olambda__235,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_jl(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_235
tff(fact_9810_ATP_Olambda__236,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ja(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_236
tff(fact_9811_ATP_Olambda__237,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_jm(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_237
tff(fact_9812_ATP_Olambda__238,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_abk(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_238
tff(fact_9813_ATP_Olambda__239,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_abi(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_239
tff(fact_9814_ATP_Olambda__240,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_abx,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abw(Uu),Uua)) ).

% ATP.lambda_240
tff(fact_9815_ATP_Olambda__241,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_abv,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abu(Uu),Uua)) ).

% ATP.lambda_241
tff(fact_9816_ATP_Olambda__242,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_abt,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abs(Uu),Uua)) ).

% ATP.lambda_242
tff(fact_9817_ATP_Olambda__243,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_abr,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abq(Uu),Uua)) ).

% ATP.lambda_243
tff(fact_9818_ATP_Olambda__244,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_abo,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abn(Uu),Uua)) ).

% ATP.lambda_244
tff(fact_9819_ATP_Olambda__245,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_acn(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_acm(Uu),Uua)) ).

% ATP.lambda_245
tff(fact_9820_ATP_Olambda__246,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_abb(Uu),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_246
tff(fact_9821_ATP_Olambda__247,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_adw(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).

% ATP.lambda_247
tff(fact_9822_ATP_Olambda__248,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ce,Uu),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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_248
tff(fact_9823_ATP_Olambda__249,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_vo(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_249
tff(fact_9824_ATP_Olambda__250,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_vn(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_250
tff(fact_9825_ATP_Olambda__251,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_vm(Uu),Uua) = times_times_rat(aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_251
tff(fact_9826_ATP_Olambda__252,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_qq(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(Uu,aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))) ).

% ATP.lambda_252
tff(fact_9827_ATP_Olambda__253,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_qp(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_253
tff(fact_9828_ATP_Olambda__254,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_qo(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_254
tff(fact_9829_ATP_Olambda__255,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_qm(Uu),Uua) = times_times_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_255
tff(fact_9830_ATP_Olambda__256,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_qr(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_256
tff(fact_9831_ATP_Olambda__257,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_qn(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_257
tff(fact_9832_ATP_Olambda__258,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_aer(Uu),Uua) = aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)) ).

% ATP.lambda_258
tff(fact_9833_ATP_Olambda__259,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adh,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_259
tff(fact_9834_ATP_Olambda__260,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aen,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_260
tff(fact_9835_ATP_Olambda__261,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aem,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_261
tff(fact_9836_ATP_Olambda__262,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aax,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_262
tff(fact_9837_ATP_Olambda__263,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aay,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_263
tff(fact_9838_ATP_Olambda__264,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_real(aTP_Lamm_nm(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_264
tff(fact_9839_ATP_Olambda__265,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_nl(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_265
tff(fact_9840_ATP_Olambda__266,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nn(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_266
tff(fact_9841_ATP_Olambda__267,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_adb(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_267
tff(fact_9842_ATP_Olambda__268,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_kv(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_268
tff(fact_9843_ATP_Olambda__269,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_ky(Uu),Uua) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_269
tff(fact_9844_ATP_Olambda__270,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_complex(aTP_Lamm_kx(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_270
tff(fact_9845_ATP_Olambda__271,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_la(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_271
tff(fact_9846_ATP_Olambda__272,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_kz(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_272
tff(fact_9847_ATP_Olambda__273,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_kw(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_273
tff(fact_9848_ATP_Olambda__274,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_ku(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_274
tff(fact_9849_ATP_Olambda__275,axiom,
    ! [Uu: fun_Pr672294278902479040t_real,Uua: product_prod_nat_nat] : aa_Pro6168199777849092593t_real(aTP_Lamm_xo(Uu),Uua) = aa_real_real(ln_ln_real,aa_Pro6168199777849092593t_real(Uu,Uua)) ).

% ATP.lambda_275
tff(fact_9850_ATP_Olambda__276,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_real(aTP_Lamm_xn(Uu),Uua) = aa_real_real(ln_ln_real,aa_complex_real(Uu,Uua)) ).

% ATP.lambda_276
tff(fact_9851_ATP_Olambda__277,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_xk(Uu),Uua) = aa_real_real(ln_ln_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_277
tff(fact_9852_ATP_Olambda__278,axiom,
    ! [Uu: fun_set_nat_real,Uua: set_nat] : aa_set_nat_real(aTP_Lamm_xl(Uu),Uua) = aa_real_real(ln_ln_real,aa_set_nat_real(Uu,Uua)) ).

% ATP.lambda_278
tff(fact_9853_ATP_Olambda__279,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_xp(Uu),Uua) = aa_real_real(ln_ln_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_279
tff(fact_9854_ATP_Olambda__280,axiom,
    ! [Uu: fun_int_real,Uua: int] : aa_int_real(aTP_Lamm_xm(Uu),Uua) = aa_real_real(ln_ln_real,aa_int_real(Uu,Uua)) ).

% ATP.lambda_280
tff(fact_9855_ATP_Olambda__281,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_ue(Uu),Uua) = times_times_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_281
tff(fact_9856_ATP_Olambda__282,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_uf(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_282
tff(fact_9857_ATP_Olambda__283,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_ug(Uu),Uua) = times_times_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_283
tff(fact_9858_ATP_Olambda__284,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_uh(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_284
tff(fact_9859_ATP_Olambda__285,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_ui(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_285
tff(fact_9860_ATP_Olambda__286,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_lb(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_286
tff(fact_9861_ATP_Olambda__287,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_complex(aTP_Lamm_qz(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_287
tff(fact_9862_ATP_Olambda__288,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_lc(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_288
tff(fact_9863_ATP_Olambda__289,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_rat(aTP_Lamm_qy(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_289
tff(fact_9864_ATP_Olambda__290,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ra(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_290
tff(fact_9865_ATP_Olambda__291,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_complex(aTP_Lamm_lf(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_int_int(Uu,Uua)) ).

% ATP.lambda_291
tff(fact_9866_ATP_Olambda__292,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_ld(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_292
tff(fact_9867_ATP_Olambda__293,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_le(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_293
tff(fact_9868_ATP_Olambda__294,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_lg(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_294
tff(fact_9869_ATP_Olambda__295,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_jq(Uu),Uua) = plus_plus_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_295
tff(fact_9870_ATP_Olambda__296,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_ju(Uu),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_296
tff(fact_9871_ATP_Olambda__297,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_jr(Uu),Uua) = plus_plus_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_297
tff(fact_9872_ATP_Olambda__298,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_jt(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_298
tff(fact_9873_ATP_Olambda__299,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_js(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_299
tff(fact_9874_ATP_Olambda__300,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_aeq(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_300
tff(fact_9875_ATP_Olambda__301,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_il(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_301
tff(fact_9876_ATP_Olambda__302,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_ik(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_302
tff(fact_9877_ATP_Olambda__303,axiom,
    ! [Uu: fun_Pr672294278902479040t_real,Uua: product_prod_nat_nat] : aa_Pro6168199777849092593t_real(aTP_Lamm_tt(Uu),Uua) = aa_real_real(exp_real,aa_Pro6168199777849092593t_real(Uu,Uua)) ).

% ATP.lambda_303
tff(fact_9878_ATP_Olambda__304,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_complex(aTP_Lamm_tu(Uu),Uua) = aa_complex_complex(exp_complex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_304
tff(fact_9879_ATP_Olambda__305,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_real(aTP_Lamm_ts(Uu),Uua) = aa_real_real(exp_real,aa_complex_real(Uu,Uua)) ).

% ATP.lambda_305
tff(fact_9880_ATP_Olambda__306,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_tv(Uu),Uua) = aa_real_real(exp_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_306
tff(fact_9881_ATP_Olambda__307,axiom,
    ! [Uu: fun_int_real,Uua: int] : aa_int_real(aTP_Lamm_tr(Uu),Uua) = aa_real_real(exp_real,aa_int_real(Uu,Uua)) ).

% ATP.lambda_307
tff(fact_9882_ATP_Olambda__308,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_adl(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_308
tff(fact_9883_ATP_Olambda__309,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_aci(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_309
tff(fact_9884_ATP_Olambda__310,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_aey(Uu),Uua) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bb,Uu)) ).

% ATP.lambda_310
tff(fact_9885_ATP_Olambda__311,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pl(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(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_311
tff(fact_9886_ATP_Olambda__312,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ng(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uu,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_312
tff(fact_9887_ATP_Olambda__313,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ej(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_313
tff(fact_9888_ATP_Olambda__314,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_eh(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(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_314
tff(fact_9889_ATP_Olambda__315,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ei(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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(aa_int2565445551602696444nt_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_315
tff(fact_9890_ATP_Olambda__316,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_oh(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_316
tff(fact_9891_ATP_Olambda__317,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_oi(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_317
tff(fact_9892_ATP_Olambda__318,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_oj(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_318
tff(fact_9893_ATP_Olambda__319,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ok(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_319
tff(fact_9894_ATP_Olambda__320,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_fa(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_320
tff(fact_9895_ATP_Olambda__321,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fe(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_321
tff(fact_9896_ATP_Olambda__322,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_ev(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_322
tff(fact_9897_ATP_Olambda__323,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ey(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_323
tff(fact_9898_ATP_Olambda__324,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fb(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_324
tff(fact_9899_ATP_Olambda__325,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ew(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_325
tff(fact_9900_ATP_Olambda__326,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fc(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_326
tff(fact_9901_ATP_Olambda__327,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_ex(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_327
tff(fact_9902_ATP_Olambda__328,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ez(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_328
tff(fact_9903_ATP_Olambda__329,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fd(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_329
tff(fact_9904_ATP_Olambda__330,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] :
      aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_re(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_330
tff(fact_9905_ATP_Olambda__331,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ri(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_real(Uua,Uub),one_one_real) ).

% ATP.lambda_331
tff(fact_9906_ATP_Olambda__332,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_rb(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_332
tff(fact_9907_ATP_Olambda__333,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_rf(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_real(Uua,Uub),one_one_real) ).

% ATP.lambda_333
tff(fact_9908_ATP_Olambda__334,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_rj(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_334
tff(fact_9909_ATP_Olambda__335,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rc(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_335
tff(fact_9910_ATP_Olambda__336,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_rg(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_real(Uua,Uub),one_one_real) ).

% ATP.lambda_336
tff(fact_9911_ATP_Olambda__337,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_rk(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_337
tff(fact_9912_ATP_Olambda__338,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_rd(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_338
tff(fact_9913_ATP_Olambda__339,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_rh(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_real(Uua,Uub),one_one_real) ).

% ATP.lambda_339
tff(fact_9914_ATP_Olambda__340,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_eq(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_340
tff(fact_9915_ATP_Olambda__341,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_eu(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_341
tff(fact_9916_ATP_Olambda__342,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_el(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_342
tff(fact_9917_ATP_Olambda__343,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_eo(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_343
tff(fact_9918_ATP_Olambda__344,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_er(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_344
tff(fact_9919_ATP_Olambda__345,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_em(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_345
tff(fact_9920_ATP_Olambda__346,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_nw(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_346
tff(fact_9921_ATP_Olambda__347,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_es(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_347
tff(fact_9922_ATP_Olambda__348,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nx(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_348
tff(fact_9923_ATP_Olambda__349,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ny(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_349
tff(fact_9924_ATP_Olambda__350,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_en(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_350
tff(fact_9925_ATP_Olambda__351,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ep(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_351
tff(fact_9926_ATP_Olambda__352,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_et(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_352
tff(fact_9927_ATP_Olambda__353,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] :
      aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_ro(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_353
tff(fact_9928_ATP_Olambda__354,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_rs(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_real(Uua,Uub),one_one_real) ).

% ATP.lambda_354
tff(fact_9929_ATP_Olambda__355,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_rl(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_355
tff(fact_9930_ATP_Olambda__356,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_rp(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_real(Uua,Uub),one_one_real) ).

% ATP.lambda_356
tff(fact_9931_ATP_Olambda__357,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_rt(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_357
tff(fact_9932_ATP_Olambda__358,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rm(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_358
tff(fact_9933_ATP_Olambda__359,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_rq(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),one_one_real) ).

% ATP.lambda_359
tff(fact_9934_ATP_Olambda__360,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ru(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_360
tff(fact_9935_ATP_Olambda__361,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_rn(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_361
tff(fact_9936_ATP_Olambda__362,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_rr(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_real(Uua,Uub),one_one_real) ).

% ATP.lambda_362
tff(fact_9937_ATP_Olambda__363,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aau(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu),Uub))) ).

% ATP.lambda_363
tff(fact_9938_ATP_Olambda__364,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aba(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu),Uub))) ).

% ATP.lambda_364
tff(fact_9939_ATP_Olambda__365,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aat(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu,Uub))) ).

% ATP.lambda_365
tff(fact_9940_ATP_Olambda__366,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_od(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_366
tff(fact_9941_ATP_Olambda__367,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_oe(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_367
tff(fact_9942_ATP_Olambda__368,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_of(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_368
tff(fact_9943_ATP_Olambda__369,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_og(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_369
tff(fact_9944_ATP_Olambda__370,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
      aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_hr(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_370
tff(fact_9945_ATP_Olambda__371,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_o,Uub: complex] :
      aa_complex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_hv(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_371
tff(fact_9946_ATP_Olambda__372,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_o,Uub: real] :
      aa_real_complex(aa_fun1098856492951973088omplex(aTP_Lamm_hm(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_372
tff(fact_9947_ATP_Olambda__373,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_o,Uub: real] :
      aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_hp(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_373
tff(fact_9948_ATP_Olambda__374,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_o,Uub: real] :
      aa_real_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_hs(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_374
tff(fact_9949_ATP_Olambda__375,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_o,Uub: nat] :
      aa_nat_complex(aa_fun1400689748890728928omplex(aTP_Lamm_hn(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_375
tff(fact_9950_ATP_Olambda__376,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_ht(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_376
tff(fact_9951_ATP_Olambda__377,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_o,Uub: int] :
      aa_int_complex(aa_fun6439487526662521568omplex(aTP_Lamm_ho(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_377
tff(fact_9952_ATP_Olambda__378,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_o,Uub: int] :
      aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_hq(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_378
tff(fact_9953_ATP_Olambda__379,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] :
      aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_hu(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_379
tff(fact_9954_ATP_Olambda__380,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_o,Uub: complex] :
      aa_complex_complex(aa_fun3681620399087435360omplex(aTP_Lamm_sz(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_complex(Uu,Uub),one_one_complex) ).

% ATP.lambda_380
tff(fact_9955_ATP_Olambda__381,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
      aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_td(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu,Uub),one_one_real) ).

% ATP.lambda_381
tff(fact_9956_ATP_Olambda__382,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_o,Uub: real] :
      aa_real_complex(aa_fun1098856492951973088omplex(aTP_Lamm_sw(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_complex(Uu,Uub),one_one_complex) ).

% ATP.lambda_382
tff(fact_9957_ATP_Olambda__383,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_o,Uub: real] :
      aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_ta(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu,Uub),one_one_real) ).

% ATP.lambda_383
tff(fact_9958_ATP_Olambda__384,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_o,Uub: real] :
      aa_real_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_te(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_384
tff(fact_9959_ATP_Olambda__385,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_o,Uub: nat] :
      aa_nat_complex(aa_fun1400689748890728928omplex(aTP_Lamm_sx(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_complex(Uu,Uub),one_one_complex) ).

% ATP.lambda_385
tff(fact_9960_ATP_Olambda__386,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_o,Uub: nat] :
      aa_nat_real(aa_fun5510454223943551838t_real(aTP_Lamm_tb(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_real(Uu,Uub),one_one_real) ).

% ATP.lambda_386
tff(fact_9961_ATP_Olambda__387,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_tf(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_387
tff(fact_9962_ATP_Olambda__388,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_o,Uub: int] :
      aa_int_complex(aa_fun6439487526662521568omplex(aTP_Lamm_sy(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_complex(Uu,Uub),one_one_complex) ).

% ATP.lambda_388
tff(fact_9963_ATP_Olambda__389,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_o,Uub: int] :
      aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_tc(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu,Uub),one_one_real) ).

% ATP.lambda_389
tff(fact_9964_ATP_Olambda__390,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_adx(Uu),Uua),Uub)
    <=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_390
tff(fact_9965_ATP_Olambda__391,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_xu(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_391
tff(fact_9966_ATP_Olambda__392,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_392
tff(fact_9967_ATP_Olambda__393,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_wb(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_393
tff(fact_9968_ATP_Olambda__394,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_hc(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_394
tff(fact_9969_ATP_Olambda__395,axiom,
    ! [Uu: fun_co7370231399432144932at_int,Uua: nat,Uub: complex] : aa_complex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_sf(Uu),Uua),Uub) = aa_nat_int(aa_com2205869648058525157at_int(Uu,Uub),Uua) ).

% ATP.lambda_395
tff(fact_9970_ATP_Olambda__396,axiom,
    ! [Uu: fun_co333780937270390912nt_int,Uua: int,Uub: complex] : aa_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_gd(Uu),Uua),Uub) = aa_int_int(aa_com3205223359959579713nt_int(Uu,Uub),Uua) ).

% ATP.lambda_396
tff(fact_9971_ATP_Olambda__397,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_gi(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_397
tff(fact_9972_ATP_Olambda__398,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_hf(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_398
tff(fact_9973_ATP_Olambda__399,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_gu(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_399
tff(fact_9974_ATP_Olambda__400,axiom,
    ! [Uu: fun_real_fun_nat_int,Uua: nat,Uub: real] : aa_real_int(aa_nat_fun_real_int(aTP_Lamm_sc(Uu),Uua),Uub) = aa_nat_int(aa_real_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_400
tff(fact_9975_ATP_Olambda__401,axiom,
    ! [Uu: fun_real_fun_int_int,Uua: int,Uub: real] : aa_real_int(aa_int_fun_real_int(aTP_Lamm_fv(Uu),Uua),Uub) = aa_int_int(aa_real_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_401
tff(fact_9976_ATP_Olambda__402,axiom,
    ! [Uu: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_gm(Uu),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu,Uub),Uua) ).

% ATP.lambda_402
tff(fact_9977_ATP_Olambda__403,axiom,
    ! [Uu: fun_na8457291176974909128ex_nat,Uua: complex,Uub: nat] : aa_nat_nat(aa_com6383720667567721865at_nat(aTP_Lamm_su(Uu),Uua),Uub) = aa_complex_nat(aa_nat3391284541783913865ex_nat(Uu,Uub),Uua) ).

% ATP.lambda_403
tff(fact_9978_ATP_Olambda__404,axiom,
    ! [Uu: fun_nat_fun_real_nat,Uua: real,Uub: nat] : aa_nat_nat(aa_real_fun_nat_nat(aTP_Lamm_sn(Uu),Uua),Uub) = aa_real_nat(aa_nat_fun_real_nat(Uu,Uub),Uua) ).

% ATP.lambda_404
tff(fact_9979_ATP_Olambda__405,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_xd(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_405
tff(fact_9980_ATP_Olambda__406,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xa(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_406
tff(fact_9981_ATP_Olambda__407,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_xi(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_407
tff(fact_9982_ATP_Olambda__408,axiom,
    ! [Uu: fun_nat_fun_int_nat,Uua: int,Uub: nat] : aa_nat_nat(aa_int_fun_nat_nat(aTP_Lamm_sr(Uu),Uua),Uub) = aa_int_nat(aa_nat_fun_int_nat(Uu,Uub),Uua) ).

% ATP.lambda_408
tff(fact_9983_ATP_Olambda__409,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_fz(Uu),Uua),Uub) = aa_int_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_409
tff(fact_9984_ATP_Olambda__410,axiom,
    ! [Uu: fun_in1307110748760623106omplex,Uua: complex,Uub: int] : aa_int_complex(aa_com9156536935073639875omplex(aTP_Lamm_gq(Uu),Uua),Uub) = aa_complex_complex(aa_int3545933452204062147omplex(Uu,Uub),Uua) ).

% ATP.lambda_410
tff(fact_9985_ATP_Olambda__411,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_gy(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_411
tff(fact_9986_ATP_Olambda__412,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_pe(Uu),Uua),Uub) = aa_complex_complex(times_times_complex($ite(Uub = Uu,one_one_complex,zero_zero_complex)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_412
tff(fact_9987_ATP_Olambda__413,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(Uub = Uu,one_one_real,zero_zero_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_413
tff(fact_9988_ATP_Olambda__414,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] :
      aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_pg(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,
          $ite(Uub = Uu,one_one_int,zero_zero_int)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_414
tff(fact_9989_ATP_Olambda__415,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_aao(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_415
tff(fact_9990_ATP_Olambda__416,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_xj(Uu),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_xi(Uu),Uub),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_416
tff(fact_9991_ATP_Olambda__417,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xg(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xa(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_417
tff(fact_9992_ATP_Olambda__418,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_xe(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_xd(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_418
tff(fact_9993_ATP_Olambda__419,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xb(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xa(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_419
tff(fact_9994_ATP_Olambda__420,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aak(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_aaj(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_420
tff(fact_9995_ATP_Olambda__421,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aai(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_aah(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_421
tff(fact_9996_ATP_Olambda__422,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aag(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_aaf(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_422
tff(fact_9997_ATP_Olambda__423,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aae(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_aad(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_423
tff(fact_9998_ATP_Olambda__424,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aac(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_int2464330468366073224_int_o(aTP_Lamm_aab(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_424
tff(fact_9999_ATP_Olambda__425,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aco(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_425
tff(fact_10000_ATP_Olambda__426,axiom,
    ! [Uu: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_pr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(aa_nat480766339200631653x_real(Uua,Uub),zero_zero_complex)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_426
tff(fact_10001_ATP_Olambda__427,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_ps(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_427
tff(fact_10002_ATP_Olambda__428,axiom,
    ! [Uu: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_pt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_rat_real(aa_nat_fun_rat_real(Uua,Uub),zero_zero_rat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_428
tff(fact_10003_ATP_Olambda__429,axiom,
    ! [Uu: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_pu(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_429
tff(fact_10004_ATP_Olambda__430,axiom,
    ! [Uu: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_pv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_430
tff(fact_10005_ATP_Olambda__431,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_yh(Uu),Uua),Uub)
    <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_os(Uu),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_real ) ) ).

% ATP.lambda_431
tff(fact_10006_ATP_Olambda__432,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_yg(Uu),Uua),Uub)
    <=> ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oq(Uu),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_complex ) ) ).

% ATP.lambda_432
tff(fact_10007_ATP_Olambda__433,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ip(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_433
tff(fact_10008_ATP_Olambda__434,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ir(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_434
tff(fact_10009_ATP_Olambda__435,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_iq(Uu),Uua),Uub) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub))),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_435
tff(fact_10010_ATP_Olambda__436,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_acl(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_436
tff(fact_10011_ATP_Olambda__437,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_437
tff(fact_10012_ATP_Olambda__438,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aek(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) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_438
tff(fact_10013_ATP_Olambda__439,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_pq(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),aa_nat_complex(Uu,Uub))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_439
tff(fact_10014_ATP_Olambda__440,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_po(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_440
tff(fact_10015_ATP_Olambda__441,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_ael(Uu),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_441
tff(fact_10016_ATP_Olambda__442,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_cm(Uu),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Uub)),Uua) ) ) ).

% ATP.lambda_442
tff(fact_10017_ATP_Olambda__443,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( aa_lis2793599718760257805_nat_o(aa_nat1970911183850433186_nat_o(aTP_Lamm_cl(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s5460976970255530739at_nat(Uub)),Uua) ) ) ).

% ATP.lambda_443
tff(fact_10018_ATP_Olambda__444,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_ck(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua) ) ) ).

% ATP.lambda_444
tff(fact_10019_ATP_Olambda__445,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_cn(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Uub)),Uua) ) ) ).

% ATP.lambda_445
tff(fact_10020_ATP_Olambda__446,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_co(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Uub)),Uua) ) ) ).

% ATP.lambda_446
tff(fact_10021_ATP_Olambda__447,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_ch(Uu),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Uub) = Uua ) ) ) ).

% ATP.lambda_447
tff(fact_10022_ATP_Olambda__448,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( aa_lis2793599718760257805_nat_o(aa_nat1970911183850433186_nat_o(aTP_Lamm_cg(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Uub)),Uu)
        & ( size_s5460976970255530739at_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_448
tff(fact_10023_ATP_Olambda__449,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_cf(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_449
tff(fact_10024_ATP_Olambda__450,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_ci(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Uub)),Uu)
        & ( aa_list_nat_nat(size_size_list_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_450
tff(fact_10025_ATP_Olambda__451,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_cj(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & ( aa_list_int_nat(size_size_list_int,Uub) = Uua ) ) ) ).

% ATP.lambda_451
tff(fact_10026_ATP_Olambda__452,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aej(Uu),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_452
tff(fact_10027_ATP_Olambda__453,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aar(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_453
tff(fact_10028_ATP_Olambda__454,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_aaz(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,pow_nat(set_or4665077453230672383an_nat(zero_zero_nat,Uu)))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_454
tff(fact_10029_ATP_Olambda__455,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aez(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uub) ) ) ).

% ATP.lambda_455
tff(fact_10030_ATP_Olambda__456,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aaq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_456
tff(fact_10031_ATP_Olambda__457,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_pp(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uu),Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_457
tff(fact_10032_ATP_Olambda__458,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uu),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_458
tff(fact_10033_ATP_Olambda__459,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vw(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_459
tff(fact_10034_ATP_Olambda__460,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_zq(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uua)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_460
tff(fact_10035_ATP_Olambda__461,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_at(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_461
tff(fact_10036_ATP_Olambda__462,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_zr(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_462
tff(fact_10037_ATP_Olambda__463,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ba(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_463
tff(fact_10038_ATP_Olambda__464,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_az(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_464
tff(fact_10039_ATP_Olambda__465,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_au(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_465
tff(fact_10040_ATP_Olambda__466,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_adv(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uua)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uu) ) ) ).

% ATP.lambda_466
tff(fact_10041_ATP_Olambda__467,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aea(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uub),Uua)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uub),Uu) ) ) ).

% ATP.lambda_467
tff(fact_10042_ATP_Olambda__468,axiom,
    ! [Uu: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_be(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_468
tff(fact_10043_ATP_Olambda__469,axiom,
    ! [Uu: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_bf(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_469
tff(fact_10044_ATP_Olambda__470,axiom,
    ! [Uu: set_list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_bg(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_470
tff(fact_10045_ATP_Olambda__471,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_bh(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_471
tff(fact_10046_ATP_Olambda__472,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bi(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_472
tff(fact_10047_ATP_Olambda__473,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bj(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_473
tff(fact_10048_ATP_Olambda__474,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_ds(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_474
tff(fact_10049_ATP_Olambda__475,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_ea(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_475
tff(fact_10050_ATP_Olambda__476,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_dm(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_476
tff(fact_10051_ATP_Olambda__477,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_du(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_477
tff(fact_10052_ATP_Olambda__478,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_ec(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_478
tff(fact_10053_ATP_Olambda__479,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_do(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_479
tff(fact_10054_ATP_Olambda__480,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_dw(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_480
tff(fact_10055_ATP_Olambda__481,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_ee(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_481
tff(fact_10056_ATP_Olambda__482,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_dq(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_482
tff(fact_10057_ATP_Olambda__483,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_dy(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_483
tff(fact_10058_ATP_Olambda__484,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_cy(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_484
tff(fact_10059_ATP_Olambda__485,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_dg(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_485
tff(fact_10060_ATP_Olambda__486,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_cs(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_486
tff(fact_10061_ATP_Olambda__487,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_da(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_487
tff(fact_10062_ATP_Olambda__488,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_di(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_488
tff(fact_10063_ATP_Olambda__489,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_cu(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_489
tff(fact_10064_ATP_Olambda__490,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_dc(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_490
tff(fact_10065_ATP_Olambda__491,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_dk(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_491
tff(fact_10066_ATP_Olambda__492,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_cw(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_492
tff(fact_10067_ATP_Olambda__493,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_de(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_493
tff(fact_10068_ATP_Olambda__494,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_adu(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uu) ) ).

% ATP.lambda_494
tff(fact_10069_ATP_Olambda__495,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_vy(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_495
tff(fact_10070_ATP_Olambda__496,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_aes(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu) ) ).

% ATP.lambda_496
tff(fact_10071_ATP_Olambda__497,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_aeu(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Uua),Uub)),Uu) ) ).

% ATP.lambda_497
tff(fact_10072_ATP_Olambda__498,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_wo(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_498
tff(fact_10073_ATP_Olambda__499,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_nr(Uu),Uua),Uub)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub),Uu) ) ).

% ATP.lambda_499
tff(fact_10074_ATP_Olambda__500,axiom,
    ! [Uu: set_Pr1200682188532975830nteger,Uua: fun_Co3460408610320954864e_term,Uub: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(aTP_Lamm_ns(Uu),Uua),Uub)
    <=> member6901806533512420791nteger(produc7264323995540650962nteger(Uua,Uub),Uu) ) ).

% ATP.lambda_500
tff(fact_10075_ATP_Olambda__501,axiom,
    ! [Uu: set_Pr3486312109254157132nteger,Uua: fun_Pr5098669352780829690e_term,Uub: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(aTP_Lamm_nt(Uu),Uua),Uub)
    <=> member8978266922497061677nteger(produc565448451787809096nteger(Uua,Uub),Uu) ) ).

% ATP.lambda_501
tff(fact_10076_ATP_Olambda__502,axiom,
    ! [Uu: set_Pr6700895561065273913nt_int,Uua: fun_Pr7763680713840462443e_term,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(aTP_Lamm_nu(Uu),Uua),Uub)
    <=> member753276645541486746nt_int(produc7320969046608631221nt_int(Uua,Uub),Uu) ) ).

% ATP.lambda_502
tff(fact_10077_ATP_Olambda__503,axiom,
    ! [Uu: set_Pr4166680992137330533nt_int,Uua: fun_in5730379987311621695e_term,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(aTP_Lamm_nv(Uu),Uua),Uub)
    <=> member5890090881338784710nt_int(produc4729913460834754785nt_int(Uua,Uub),Uu) ) ).

% ATP.lambda_503
tff(fact_10078_ATP_Olambda__504,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_cp(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_504
tff(fact_10079_ATP_Olambda__505,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_aas(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_505
tff(fact_10080_ATP_Olambda__506,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_op(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(suc,Uub))),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_506
tff(fact_10081_ATP_Olambda__507,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_or(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_507
tff(fact_10082_ATP_Olambda__508,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_adm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_508
tff(fact_10083_ATP_Olambda__509,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_acm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_509
tff(fact_10084_ATP_Olambda__510,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_oq(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_510
tff(fact_10085_ATP_Olambda__511,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_os(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_511
tff(fact_10086_ATP_Olambda__512,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_yi(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_512
tff(fact_10087_ATP_Olambda__513,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ys(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_513
tff(fact_10088_ATP_Olambda__514,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_yj(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_514
tff(fact_10089_ATP_Olambda__515,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_cc(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_515
tff(fact_10090_ATP_Olambda__516,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_adz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_516
tff(fact_10091_ATP_Olambda__517,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ady(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_517
tff(fact_10092_ATP_Olambda__518,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_afe(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_518
tff(fact_10093_ATP_Olambda__519,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aal(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_519
tff(fact_10094_ATP_Olambda__520,axiom,
    ! [Uu: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_jw(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu,Uub)) ).

% ATP.lambda_520
tff(fact_10095_ATP_Olambda__521,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ka(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uub)),aa_complex_real(Uu,Uub)) ).

% ATP.lambda_521
tff(fact_10096_ATP_Olambda__522,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_kd(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_complex_rat(Uua,Uub)),aa_complex_rat(Uu,Uub)) ).

% ATP.lambda_522
tff(fact_10097_ATP_Olambda__523,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_jx(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_523
tff(fact_10098_ATP_Olambda__524,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_kb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_524
tff(fact_10099_ATP_Olambda__525,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_ke(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uub)),aa_real_rat(Uu,Uub)) ).

% ATP.lambda_525
tff(fact_10100_ATP_Olambda__526,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_jy(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_526
tff(fact_10101_ATP_Olambda__527,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kf(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_527
tff(fact_10102_ATP_Olambda__528,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_jz(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_528
tff(fact_10103_ATP_Olambda__529,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_kc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_529
tff(fact_10104_ATP_Olambda__530,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ada(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_530
tff(fact_10105_ATP_Olambda__531,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_iv(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uua,Uub)),aa_complex_nat(Uu,Uub)) ).

% ATP.lambda_531
tff(fact_10106_ATP_Olambda__532,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_iw(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_real_nat(Uua,Uub)),aa_real_nat(Uu,Uub)) ).

% ATP.lambda_532
tff(fact_10107_ATP_Olambda__533,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_ix(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(Uua,Uub)),aa_set_nat_nat(Uu,Uub)) ).

% ATP.lambda_533
tff(fact_10108_ATP_Olambda__534,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ph(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_534
tff(fact_10109_ATP_Olambda__535,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_iz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uu,Uub)) ).

% ATP.lambda_535
tff(fact_10110_ATP_Olambda__536,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_iy(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uua,Uub)),aa_int_nat(Uu,Uub)) ).

% ATP.lambda_536
tff(fact_10111_ATP_Olambda__537,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_fq(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_537
tff(fact_10112_ATP_Olambda__538,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fs(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_538
tff(fact_10113_ATP_Olambda__539,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_afg(Uu),Uua),Uub) = aa_rat_rat(plus_plus_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_539
tff(fact_10114_ATP_Olambda__540,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fr(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_540
tff(fact_10115_ATP_Olambda__541,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_on(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_541
tff(fact_10116_ATP_Olambda__542,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_fp(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_542
tff(fact_10117_ATP_Olambda__543,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ack(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_543
tff(fact_10118_ATP_Olambda__544,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_as(Uu),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        | aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_544
tff(fact_10119_ATP_Olambda__545,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_ar(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        | aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_545
tff(fact_10120_ATP_Olambda__546,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_am(Uu),Uua),Uub)
    <=> ( aa_real_o(Uu,Uub)
        | aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_546
tff(fact_10121_ATP_Olambda__547,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_an(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
        | aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_547
tff(fact_10122_ATP_Olambda__548,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ao(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        | aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_548
tff(fact_10123_ATP_Olambda__549,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ap(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_549
tff(fact_10124_ATP_Olambda__550,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_aq(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_550
tff(fact_10125_ATP_Olambda__551,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_al(Uu),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_551
tff(fact_10126_ATP_Olambda__552,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_ak(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_552
tff(fact_10127_ATP_Olambda__553,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_af(Uu),Uua),Uub)
    <=> ( aa_real_o(Uu,Uub)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_553
tff(fact_10128_ATP_Olambda__554,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_ag(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_554
tff(fact_10129_ATP_Olambda__555,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ah(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_555
tff(fact_10130_ATP_Olambda__556,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ai(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_556
tff(fact_10131_ATP_Olambda__557,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_aj(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_557
tff(fact_10132_ATP_Olambda__558,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_cd(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_558
tff(fact_10133_ATP_Olambda__559,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ob(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_559
tff(fact_10134_ATP_Olambda__560,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_oc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_560
tff(fact_10135_ATP_Olambda__561,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_oz(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_561
tff(fact_10136_ATP_Olambda__562,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pa(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_562
tff(fact_10137_ATP_Olambda__563,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_adp(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_563
tff(fact_10138_ATP_Olambda__564,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_acj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_564
tff(fact_10139_ATP_Olambda__565,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_ow(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_565
tff(fact_10140_ATP_Olambda__566,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_ll(Uu),Uua),Uub) = plus_p5714425477246183910nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Uub)),Uua)) ).

% ATP.lambda_566
tff(fact_10141_ATP_Olambda__567,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_lk(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_567
tff(fact_10142_ATP_Olambda__568,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ln(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_568
tff(fact_10143_ATP_Olambda__569,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_li(Uu),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_569
tff(fact_10144_ATP_Olambda__570,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lm(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Uub)),Uua)) ).

% ATP.lambda_570
tff(fact_10145_ATP_Olambda__571,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_lj(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).

% ATP.lambda_571
tff(fact_10146_ATP_Olambda__572,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_nc(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_572
tff(fact_10147_ATP_Olambda__573,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_nf(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_573
tff(fact_10148_ATP_Olambda__574,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_nd(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_574
tff(fact_10149_ATP_Olambda__575,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ne(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_575
tff(fact_10150_ATP_Olambda__576,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_576
tff(fact_10151_ATP_Olambda__577,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_je(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_577
tff(fact_10152_ATP_Olambda__578,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_jh(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_578
tff(fact_10153_ATP_Olambda__579,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_jf(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_579
tff(fact_10154_ATP_Olambda__580,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_jg(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_580
tff(fact_10155_ATP_Olambda__581,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ks(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_581
tff(fact_10156_ATP_Olambda__582,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_nz(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_582
tff(fact_10157_ATP_Olambda__583,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_oa(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_583
tff(fact_10158_ATP_Olambda__584,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_vc(Uu),Uua),Uub) = plus_p5714425477246183910nteger(Uu,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_584
tff(fact_10159_ATP_Olambda__585,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_vb(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_585
tff(fact_10160_ATP_Olambda__586,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_va(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_586
tff(fact_10161_ATP_Olambda__587,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_uz(Uu),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_587
tff(fact_10162_ATP_Olambda__588,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vd(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_588
tff(fact_10163_ATP_Olambda__589,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ve(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_589
tff(fact_10164_ATP_Olambda__590,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mi(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_590
tff(fact_10165_ATP_Olambda__591,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mh(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_591
tff(fact_10166_ATP_Olambda__592,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_tw(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_592
tff(fact_10167_ATP_Olambda__593,axiom,
    ! [Uu: fun_real_o,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_adt(Uu),Uua),Uub)
    <=> aa_real_o(Uu,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uua)) ) ).

% ATP.lambda_593
tff(fact_10168_ATP_Olambda__594,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ov(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_594
tff(fact_10169_ATP_Olambda__595,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_jd(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_595
tff(fact_10170_ATP_Olambda__596,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ads(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_596
tff(fact_10171_ATP_Olambda__597,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jc(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_597
tff(fact_10172_ATP_Olambda__598,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_tg(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_598
tff(fact_10173_ATP_Olambda__599,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_hh(Uu),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_599
tff(fact_10174_ATP_Olambda__600,axiom,
    ! [Uu: fun_Pr2028404973800837934_nat_o,Uua: nat,Uub: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_hi(Uu),Uua),Uub) = aa_Pro989595756709598047_nat_o(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_600
tff(fact_10175_ATP_Olambda__601,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_hj(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_601
tff(fact_10176_ATP_Olambda__602,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hk(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).

% ATP.lambda_602
tff(fact_10177_ATP_Olambda__603,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_hl(Uu),Uua),Uub) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_603
tff(fact_10178_ATP_Olambda__604,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_pc(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_604
tff(fact_10179_ATP_Olambda__605,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pd(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_605
tff(fact_10180_ATP_Olambda__606,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_afc(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_606
tff(fact_10181_ATP_Olambda__607,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_real(aa_fun1953249243402654830x_real(aTP_Lamm_un(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub))) ).

% ATP.lambda_607
tff(fact_10182_ATP_Olambda__608,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_uj(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(Uu,Uub)),aa_complex_real(Uua,Uub))) ).

% ATP.lambda_608
tff(fact_10183_ATP_Olambda__609,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_complex,Uub: real] : aa_real_real(aa_fun2786211375485122666l_real(aTP_Lamm_uo(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_real_complex(Uu,Uub)),aa_real_complex(Uua,Uub))) ).

% ATP.lambda_609
tff(fact_10184_ATP_Olambda__610,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_uk(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub))) ).

% ATP.lambda_610
tff(fact_10185_ATP_Olambda__611,axiom,
    ! [Uu: fun_set_nat_complex,Uua: fun_set_nat_complex,Uub: set_nat] : aa_set_nat_real(aa_fun5767801665285160478t_real(aTP_Lamm_up(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_set_nat_complex(Uu,Uub)),aa_set_nat_complex(Uua,Uub))) ).

% ATP.lambda_611
tff(fact_10186_ATP_Olambda__612,axiom,
    ! [Uu: fun_set_nat_real,Uua: fun_set_nat_real,Uub: set_nat] : aa_set_nat_real(aa_fun3569254276315353500t_real(aTP_Lamm_ul(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(Uu,Uub)),aa_set_nat_real(Uua,Uub))) ).

% ATP.lambda_612
tff(fact_10187_ATP_Olambda__613,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_us(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(Uua,Uub))) ).

% ATP.lambda_613
tff(fact_10188_ATP_Olambda__614,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ur(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub))) ).

% ATP.lambda_614
tff(fact_10189_ATP_Olambda__615,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_complex,Uub: int] : aa_int_real(aa_fun1985759105522019690t_real(aTP_Lamm_uq(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_int_complex(Uu,Uub)),aa_int_complex(Uua,Uub))) ).

% ATP.lambda_615
tff(fact_10190_ATP_Olambda__616,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_um(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(Uu,Uub)),aa_int_real(Uua,Uub))) ).

% ATP.lambda_616
tff(fact_10191_ATP_Olambda__617,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_yv(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_complex(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_complex,aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_617
tff(fact_10192_ATP_Olambda__618,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_zd(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_real(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_real,aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_618
tff(fact_10193_ATP_Olambda__619,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_yx(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_rat(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_rat,aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_619
tff(fact_10194_ATP_Olambda__620,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_zb(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_nat,aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_620
tff(fact_10195_ATP_Olambda__621,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_yz(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_int,aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_621
tff(fact_10196_ATP_Olambda__622,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_zf(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_complex(Uua,Uuc),
        $ite(Uuc = Uu,one_one_complex,aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_622
tff(fact_10197_ATP_Olambda__623,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_zg(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_real(Uua,Uuc),
        $ite(Uuc = Uu,one_one_real,aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_623
tff(fact_10198_ATP_Olambda__624,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_zh(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_rat(Uua,Uuc),
        $ite(Uuc = Uu,one_one_rat,aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_624
tff(fact_10199_ATP_Olambda__625,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_zi(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu,one_one_nat,aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_625
tff(fact_10200_ATP_Olambda__626,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_zj(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu,one_one_int,aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_626
tff(fact_10201_ATP_Olambda__627,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_yw(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc),aa_nat_complex(Uub,Uuc)) ).

% ATP.lambda_627
tff(fact_10202_ATP_Olambda__628,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_ze(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_real(Uua,Uuc),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_628
tff(fact_10203_ATP_Olambda__629,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_yy(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_629
tff(fact_10204_ATP_Olambda__630,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_zc(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_630
tff(fact_10205_ATP_Olambda__631,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_za(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_631
tff(fact_10206_ATP_Olambda__632,axiom,
    ! [Uu: fun_nat_real,Uua: set_nat,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_pi(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),Uua),aa_nat_real(Uub,Uuc),aa_nat_real(Uu,Uuc)) ).

% ATP.lambda_632
tff(fact_10207_ATP_Olambda__633,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mb(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ma(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_633
tff(fact_10208_ATP_Olambda__634,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_lz(Uu),Uua),Uub),Uuc) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ly(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_634
tff(fact_10209_ATP_Olambda__635,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_lx(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_lw(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_635
tff(fact_10210_ATP_Olambda__636,axiom,
    ! [Uu: set_nat,Uua: fun_na8457291176974909128ex_nat,Uub: fun_na1554462459066888140plex_o,Uuc: complex] : aa_complex_nat(aa_fun1649587401935388427ex_nat(aa_fun3336366540202158548ex_nat(aTP_Lamm_sv(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_com6383720667567721865at_nat(aTP_Lamm_su(Uua),Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_bp(Uu),Uub),Uuc))) ).

% ATP.lambda_636
tff(fact_10211_ATP_Olambda__637,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_nat,Uub: fun_nat_fun_int_o,Uuc: int] : aa_int_nat(aa_fun442955204921438347nt_nat(aa_fun5585961000942689878nt_nat(aTP_Lamm_ss(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_int_fun_nat_nat(aTP_Lamm_sr(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_bo(Uu),Uub),Uuc))) ).

% ATP.lambda_637
tff(fact_10212_ATP_Olambda__638,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_real_nat,Uub: fun_nat_fun_real_o,Uuc: real] : aa_real_nat(aa_fun1455215566758156043al_nat(aa_fun1180509458258223190al_nat(aTP_Lamm_sp(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_real_fun_nat_nat(aTP_Lamm_sn(Uua),Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun7270066119382671164_nat_o(aTP_Lamm_so(Uu),Uub),Uuc))) ).

% ATP.lambda_638
tff(fact_10213_ATP_Olambda__639,axiom,
    ! [Uu: set_complex,Uua: fun_co7370231399432144932at_int,Uub: fun_co5676208753217392844_nat_o,Uuc: nat] : aa_nat_int(aa_fun1349975166126329225at_int(aa_fun7503395680861046006at_int(aTP_Lamm_sg(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_sf(Uua),Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_bt(Uu),Uub),Uuc))) ).

% ATP.lambda_639
tff(fact_10214_ATP_Olambda__640,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_int,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_int(aa_fun3188195345082886411at_int(aa_fun8006801000864960506at_int(aTP_Lamm_sd(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_nat_fun_real_int(aTP_Lamm_sc(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_bk(Uu),Uub),Uuc))) ).

% ATP.lambda_640
tff(fact_10215_ATP_Olambda__641,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: nat] : aa_nat_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_sa(Uu),Uua),Uub),Uuc) = groups861055069439313189ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_hc(Uua),Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_bt(Uu),Uub),Uuc))) ).

% ATP.lambda_641
tff(fact_10216_ATP_Olambda__642,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_ry(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(aTP_Lamm_gy(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_bq(Uu),Uub),Uuc))) ).

% ATP.lambda_642
tff(fact_10217_ATP_Olambda__643,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_rw(Uu),Uua),Uub),Uuc) = groups4696554848551431203al_nat(aa_nat_fun_real_nat(aTP_Lamm_gu(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_bk(Uu),Uub),Uuc))) ).

% ATP.lambda_643
tff(fact_10218_ATP_Olambda__644,axiom,
    ! [Uu: set_complex,Uua: fun_co333780937270390912nt_int,Uub: fun_co3431063785546946928_int_o,Uuc: int] : aa_int_int(aa_fun2158622962795119753nt_int(aa_fun607866698265288346nt_int(aTP_Lamm_sk(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_gd(Uua),Uuc),collect_complex(aa_int_fun_complex_o(aa_fun7279530946696969788plex_o(aTP_Lamm_ge(Uu),Uub),Uuc))) ).

% ATP.lambda_644
tff(fact_10219_ATP_Olambda__645,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_int,Uub: fun_real_fun_int_o,Uuc: int] : aa_int_int(aa_fun3996843141751676939nt_int(aa_fun129049606478405534nt_int(aTP_Lamm_si(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_int_fun_real_int(aTP_Lamm_fv(Uua),Uuc),collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_bl(Uu),Uub),Uuc))) ).

% ATP.lambda_645
tff(fact_10220_ATP_Olambda__646,axiom,
    ! [Uu: set_real,Uua: fun_re4238438383274954274t_real,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_real(aa_fun6247686994378451851t_real(aa_fun7274880274332377722t_real(aTP_Lamm_hg(Uu),Uua),Uub),Uuc) = groups8097168146408367636l_real(aa_nat_fun_real_real(aTP_Lamm_hf(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_bk(Uu),Uub),Uuc))) ).

% ATP.lambda_646
tff(fact_10221_ATP_Olambda__647,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: nat] : aa_nat_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_hd(Uu),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_hc(Uua),Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_bt(Uu),Uub),Uuc))) ).

% ATP.lambda_647
tff(fact_10222_ATP_Olambda__648,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_gz(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_gy(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_bq(Uu),Uub),Uuc))) ).

% ATP.lambda_648
tff(fact_10223_ATP_Olambda__649,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_gv(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_gu(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_bk(Uu),Uub),Uuc))) ).

% ATP.lambda_649
tff(fact_10224_ATP_Olambda__650,axiom,
    ! [Uu: set_int,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: complex] : aa_complex_complex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_gr(Uu),Uua),Uub),Uuc) = groups3049146728041665814omplex(aa_com9156536935073639875omplex(aTP_Lamm_gq(Uua),Uuc),collect_int(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_bs(Uu),Uub),Uuc))) ).

% ATP.lambda_650
tff(fact_10225_ATP_Olambda__651,axiom,
    ! [Uu: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: complex] : aa_complex_complex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_gn(Uu),Uua),Uub),Uuc) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_gm(Uua),Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_bp(Uu),Uub),Uuc))) ).

% ATP.lambda_651
tff(fact_10226_ATP_Olambda__652,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: complex] : aa_complex_complex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_gj(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_gi(Uua),Uuc),collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_bm(Uu),Uub),Uuc))) ).

% ATP.lambda_652
tff(fact_10227_ATP_Olambda__653,axiom,
    ! [Uu: set_complex,Uua: fun_co333780937270390912nt_int,Uub: fun_co3431063785546946928_int_o,Uuc: int] : aa_int_int(aa_fun2158622962795119753nt_int(aa_fun607866698265288346nt_int(aTP_Lamm_gf(Uu),Uua),Uub),Uuc) = groups5690904116761175830ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_gd(Uua),Uuc),collect_complex(aa_int_fun_complex_o(aa_fun7279530946696969788plex_o(aTP_Lamm_ge(Uu),Uub),Uuc))) ).

% ATP.lambda_653
tff(fact_10228_ATP_Olambda__654,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_o,Uuc: int] : aa_int_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_ga(Uu),Uua),Uub),Uuc) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_fz(Uua),Uuc),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_bo(Uu),Uub),Uuc))) ).

% ATP.lambda_654
tff(fact_10229_ATP_Olambda__655,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_int,Uub: fun_real_fun_int_o,Uuc: int] : aa_int_int(aa_fun3996843141751676939nt_int(aa_fun129049606478405534nt_int(aTP_Lamm_fw(Uu),Uua),Uub),Uuc) = groups1932886352136224148al_int(aa_int_fun_real_int(aTP_Lamm_fv(Uua),Uuc),collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_bl(Uu),Uub),Uuc))) ).

% ATP.lambda_655
tff(fact_10230_ATP_Olambda__656,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_yu(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_yt(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_656
tff(fact_10231_ATP_Olambda__657,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_yr(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_yq(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_657
tff(fact_10232_ATP_Olambda__658,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_yp(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_yo(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(power_power_int(Uub),Uuc)) ).

% ATP.lambda_658
tff(fact_10233_ATP_Olambda__659,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_yn(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_ym(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_659
tff(fact_10234_ATP_Olambda__660,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_yl(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_yk(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_660
tff(fact_10235_ATP_Olambda__661,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_act(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_661
tff(fact_10236_ATP_Olambda__662,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_acr(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_662
tff(fact_10237_ATP_Olambda__663,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_acq(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_663
tff(fact_10238_ATP_Olambda__664,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_acp(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_664
tff(fact_10239_ATP_Olambda__665,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mm(Uu),Uua),Uub),Uuc) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uu),Uub)) ).

% ATP.lambda_665
tff(fact_10240_ATP_Olambda__666,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ms(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_666
tff(fact_10241_ATP_Olambda__667,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mo(Uu),Uua),Uub),Uuc) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc))),aa_nat_rat(power_power_rat(Uu),Uub)) ).

% ATP.lambda_667
tff(fact_10242_ATP_Olambda__668,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_mq(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu),Uuc))),aa_nat_int(power_power_int(Uu),Uub)) ).

% ATP.lambda_668
tff(fact_10243_ATP_Olambda__669,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abn(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).

% ATP.lambda_669
tff(fact_10244_ATP_Olambda__670,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ly(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uua),Uu)),Uuc)),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_670
tff(fact_10245_ATP_Olambda__671,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ma(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu)),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_671
tff(fact_10246_ATP_Olambda__672,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_lw(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uua),Uu)),Uuc)),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_672
tff(fact_10247_ATP_Olambda__673,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_wm(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Uu),Uuc))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_673
tff(fact_10248_ATP_Olambda__674,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_wl(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(comm_s2602460028002588243omplex(Uu),Uuc))),aa_nat_complex(comm_s2602460028002588243omplex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_674
tff(fact_10249_ATP_Olambda__675,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_wn(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_675
tff(fact_10250_ATP_Olambda__676,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_wj(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uu),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_676
tff(fact_10251_ATP_Olambda__677,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_wk(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uu),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_677
tff(fact_10252_ATP_Olambda__678,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_wh(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uu),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_678
tff(fact_10253_ATP_Olambda__679,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_wg(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_679
tff(fact_10254_ATP_Olambda__680,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_wi(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_680
tff(fact_10255_ATP_Olambda__681,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_we(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc))),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_681
tff(fact_10256_ATP_Olambda__682,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wd(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_682
tff(fact_10257_ATP_Olambda__683,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_wf(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(power_power_int(Uu),Uuc))),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_683
tff(fact_10258_ATP_Olambda__684,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_my(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_complex(power_power_complex(Uu),Uuc)) ).

% ATP.lambda_684
tff(fact_10259_ATP_Olambda__685,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_nb(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)) ).

% ATP.lambda_685
tff(fact_10260_ATP_Olambda__686,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_mz(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_rat(power_power_rat(Uu),Uuc)) ).

% ATP.lambda_686
tff(fact_10261_ATP_Olambda__687,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_na(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_int(power_power_int(Uu),Uuc)) ).

% ATP.lambda_687
tff(fact_10262_ATP_Olambda__688,axiom,
    ! [Uu: set_complex,Uua: fun_re7319081385389557872plex_o,Uub: real,Uuc: complex] :
      ( aa_complex_o(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_gg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_688
tff(fact_10263_ATP_Olambda__689,axiom,
    ! [Uu: set_complex,Uua: fun_na1554462459066888140plex_o,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_gk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_nat_fun_complex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_689
tff(fact_10264_ATP_Olambda__690,axiom,
    ! [Uu: set_complex,Uua: fun_in5065666251602127472plex_o,Uub: int,Uuc: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_go(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_int_fun_complex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_690
tff(fact_10265_ATP_Olambda__691,axiom,
    ! [Uu: set_real,Uua: fun_nat_fun_real_o,Uub: nat,Uuc: real] :
      ( aa_real_o(aa_nat_fun_real_o(aa_fun1719600354486425148real_o(aTP_Lamm_sl(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & aa_real_o(aa_nat_fun_real_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_691
tff(fact_10266_ATP_Olambda__692,axiom,
    ! [Uu: set_nat,Uua: fun_co5676208753217392844_nat_o,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_ha(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_complex_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_692
tff(fact_10267_ATP_Olambda__693,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_o,Uub: real,Uuc: nat] :
      ( aa_nat_o(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_gs(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_real_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_693
tff(fact_10268_ATP_Olambda__694,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_o,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_gw(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_694
tff(fact_10269_ATP_Olambda__695,axiom,
    ! [Uu: set_int,Uua: fun_co3431063785546946928_int_o,Uub: complex,Uuc: int] :
      ( aa_int_o(aa_complex_fun_int_o(aa_fun5644928480641789244_int_o(aTP_Lamm_gb(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_complex_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_695
tff(fact_10270_ATP_Olambda__696,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_o,Uub: real,Uuc: int] :
      ( aa_int_o(aa_real_fun_int_o(aa_fun2967211649596192572_int_o(aTP_Lamm_ft(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_real_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_696
tff(fact_10271_ATP_Olambda__697,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_o,Uub: nat,Uuc: int] :
      ( aa_int_o(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_fx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_697
tff(fact_10272_ATP_Olambda__698,axiom,
    ! [Uu: set_complex,Uua: fun_co5676208753217392844_nat_o,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_bt(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_nat_o(aa_complex_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_698
tff(fact_10273_ATP_Olambda__699,axiom,
    ! [Uu: set_complex,Uua: fun_co3431063785546946928_int_o,Uub: int,Uuc: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aa_fun7279530946696969788plex_o(aTP_Lamm_ge(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_int_o(aa_complex_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_699
tff(fact_10274_ATP_Olambda__700,axiom,
    ! [Uu: set_real,Uua: fun_re7319081385389557872plex_o,Uub: complex,Uuc: real] :
      ( aa_real_o(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_bm(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_700
tff(fact_10275_ATP_Olambda__701,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_o,Uub: nat,Uuc: real] :
      ( aa_real_o(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_bk(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & aa_nat_o(aa_real_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_701
tff(fact_10276_ATP_Olambda__702,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_o,Uub: int,Uuc: real] :
      ( aa_real_o(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_bl(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & aa_int_o(aa_real_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_702
tff(fact_10277_ATP_Olambda__703,axiom,
    ! [Uu: set_nat,Uua: fun_na1554462459066888140plex_o,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_bp(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_complex_o(aa_nat_fun_complex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_703
tff(fact_10278_ATP_Olambda__704,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_real_o,Uub: real,Uuc: nat] :
      ( aa_nat_o(aa_real_fun_nat_o(aa_fun7270066119382671164_nat_o(aTP_Lamm_so(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_real_o(aa_nat_fun_real_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_704
tff(fact_10279_ATP_Olambda__705,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_bn(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_705
tff(fact_10280_ATP_Olambda__706,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_o,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_bo(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_706
tff(fact_10281_ATP_Olambda__707,axiom,
    ! [Uu: set_int,Uua: fun_in5065666251602127472plex_o,Uub: complex,Uuc: int] :
      ( aa_int_o(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_bs(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_complex_o(aa_int_fun_complex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_707
tff(fact_10282_ATP_Olambda__708,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_o,Uub: nat,Uuc: int] :
      ( aa_int_o(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_bq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_708
tff(fact_10283_ATP_Olambda__709,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_br(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_709
tff(fact_10284_ATP_Olambda__710,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mn(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_710
tff(fact_10285_ATP_Olambda__711,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mt(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_711
tff(fact_10286_ATP_Olambda__712,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mp(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_712
tff(fact_10287_ATP_Olambda__713,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_mr(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu),Uuc)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_713
tff(fact_10288_ATP_Olambda__714,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_mu(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_714
tff(fact_10289_ATP_Olambda__715,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_mx(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_715
tff(fact_10290_ATP_Olambda__716,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_mv(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_716
tff(fact_10291_ATP_Olambda__717,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_mw(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu),Uuc)),aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_717
tff(fact_10292_ATP_Olambda__718,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vx(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_718
tff(fact_10293_ATP_Olambda__719,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aab(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_719
tff(fact_10294_ATP_Olambda__720,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abs(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_720
tff(fact_10295_ATP_Olambda__721,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abq(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_721
tff(fact_10296_ATP_Olambda__722,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abu(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).

% ATP.lambda_722
tff(fact_10297_ATP_Olambda__723,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abw(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_723
tff(fact_10298_ATP_Olambda__724,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_cz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_complex_complex(times_times_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_724
tff(fact_10299_ATP_Olambda__725,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_dh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(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_725
tff(fact_10300_ATP_Olambda__726,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_ct(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & ( aa_complex_complex(times_times_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_726
tff(fact_10301_ATP_Olambda__727,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_db(Uu),Uua),Uub),Uuc)
    <=> ( 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_727
tff(fact_10302_ATP_Olambda__728,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_dj(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & ( aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_728
tff(fact_10303_ATP_Olambda__729,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_cv(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_729
tff(fact_10304_ATP_Olambda__730,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_dd(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(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_730
tff(fact_10305_ATP_Olambda__731,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_dl(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_731
tff(fact_10306_ATP_Olambda__732,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_cx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_complex_complex(times_times_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_732
tff(fact_10307_ATP_Olambda__733,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_df(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(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_733
tff(fact_10308_ATP_Olambda__734,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_dt(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_complex_complex(plus_plus_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_734
tff(fact_10309_ATP_Olambda__735,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_eb(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_735
tff(fact_10310_ATP_Olambda__736,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_dn(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & ( aa_complex_complex(plus_plus_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_736
tff(fact_10311_ATP_Olambda__737,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_dv(Uu),Uua),Uub),Uuc)
    <=> ( 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_737
tff(fact_10312_ATP_Olambda__738,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_ed(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_738
tff(fact_10313_ATP_Olambda__739,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_dp(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_complex_complex(plus_plus_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_739
tff(fact_10314_ATP_Olambda__740,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_dx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_740
tff(fact_10315_ATP_Olambda__741,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_ef(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_741
tff(fact_10316_ATP_Olambda__742,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_dr(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_complex_complex(plus_plus_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_742
tff(fact_10317_ATP_Olambda__743,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_dz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_743
tff(fact_10318_ATP_Olambda__744,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ot(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_744
tff(fact_10319_ATP_Olambda__745,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ou(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_745
tff(fact_10320_ATP_Olambda__746,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_yk(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uuc)),aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_746
tff(fact_10321_ATP_Olambda__747,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_yq(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uuc)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_747
tff(fact_10322_ATP_Olambda__748,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_ym(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uuc)),aa_nat_rat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_748
tff(fact_10323_ATP_Olambda__749,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_yt(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_749
tff(fact_10324_ATP_Olambda__750,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_yo(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uuc)),aa_nat_int(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_750
tff(fact_10325_ATP_Olambda__751,axiom,
    ! [Uu: set_real,Uua: fun_nat_fun_real_nat,Uub: fun_nat_fun_real_o,Uuc: nat] : aa_nat_nat(aa_fun6477086565944264623at_nat(aa_fun888394026836741882at_nat(aTP_Lamm_sm(Uu),Uua),Uub),Uuc) = groups4696554848551431203al_nat(aa_nat_fun_real_nat(Uua,Uuc),collect_real(aa_nat_fun_real_o(aa_fun1719600354486425148real_o(aTP_Lamm_sl(Uu),Uub),Uuc))) ).

% ATP.lambda_751
tff(fact_10326_ATP_Olambda__752,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: complex] : aa_complex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_rz(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_ha(Uu),Uub),Uuc))) ).

% ATP.lambda_752
tff(fact_10327_ATP_Olambda__753,axiom,
    ! [Uu: set_nat,Uua: fun_co7370231399432144932at_int,Uub: fun_co5676208753217392844_nat_o,Uuc: complex] : aa_complex_int(aa_fun7310784931243472999ex_int(aa_fun7210183792663747796ex_int(aTP_Lamm_se(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_com2205869648058525157at_int(Uua,Uuc),collect_nat(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_ha(Uu),Uub),Uuc))) ).

% ATP.lambda_753
tff(fact_10328_ATP_Olambda__754,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: int] : aa_int_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_rx(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_gw(Uu),Uub),Uuc))) ).

% ATP.lambda_754
tff(fact_10329_ATP_Olambda__755,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_rv(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_gs(Uu),Uub),Uuc))) ).

% ATP.lambda_755
tff(fact_10330_ATP_Olambda__756,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_int,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_int(aa_fun7318160320247150695al_int(aa_fun4230461673125498710al_int(aTP_Lamm_sb(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_real_fun_nat_int(Uua,Uuc),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_gs(Uu),Uub),Uuc))) ).

% ATP.lambda_756
tff(fact_10331_ATP_Olambda__757,axiom,
    ! [Uu: set_complex,Uua: fun_na8457291176974909128ex_nat,Uub: fun_na1554462459066888140plex_o,Uuc: nat] : aa_nat_nat(aa_fun1031781741659083821at_nat(aa_fun7518049653624490486at_nat(aTP_Lamm_st(Uu),Uua),Uub),Uuc) = groups861055069439313189ex_nat(aa_nat3391284541783913865ex_nat(Uua,Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_gk(Uu),Uub),Uuc))) ).

% ATP.lambda_757
tff(fact_10332_ATP_Olambda__758,axiom,
    ! [Uu: set_int,Uua: fun_co333780937270390912nt_int,Uub: fun_co3431063785546946928_int_o,Uuc: complex] : aa_complex_int(aa_fun5055871773053899019ex_int(aa_fun7361901416012365340ex_int(aTP_Lamm_sj(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_complex_fun_int_o(aa_fun5644928480641789244_int_o(aTP_Lamm_gb(Uu),Uub),Uuc))) ).

% ATP.lambda_758
tff(fact_10333_ATP_Olambda__759,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_nat,Uub: fun_nat_fun_int_o,Uuc: nat] : aa_nat_nat(aa_fun8666973529875159599at_nat(aa_fun3399039426249668090at_nat(aTP_Lamm_sq(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(Uua,Uuc),collect_int(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_fx(Uu),Uub),Uuc))) ).

% ATP.lambda_759
tff(fact_10334_ATP_Olambda__760,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_int,Uub: fun_real_fun_int_o,Uuc: real] : aa_real_int(aa_fun7555737461897725707al_int(aa_fun1679075639499870878al_int(aTP_Lamm_sh(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_real_fun_int_int(Uua,Uuc),collect_int(aa_real_fun_int_o(aa_fun2967211649596192572_int_o(aTP_Lamm_ft(Uu),Uub),Uuc))) ).

% ATP.lambda_760
tff(fact_10335_ATP_Olambda__761,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: complex] : aa_complex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_hb(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_ha(Uu),Uub),Uuc))) ).

% ATP.lambda_761
tff(fact_10336_ATP_Olambda__762,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: int] : aa_int_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_gx(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_gw(Uu),Uub),Uuc))) ).

% ATP.lambda_762
tff(fact_10337_ATP_Olambda__763,axiom,
    ! [Uu: set_nat,Uua: fun_re4238438383274954274t_real,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_real(aa_fun6320755270648597735l_real(aa_fun6240130701872165846l_real(aTP_Lamm_he(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(Uua,Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_gs(Uu),Uub),Uuc))) ).

% ATP.lambda_763
tff(fact_10338_ATP_Olambda__764,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_gt(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_gs(Uu),Uub),Uuc))) ).

% ATP.lambda_764
tff(fact_10339_ATP_Olambda__765,axiom,
    ! [Uu: set_complex,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: int] : aa_int_complex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_gp(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_int3545933452204062147omplex(Uua,Uuc),collect_complex(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_go(Uu),Uub),Uuc))) ).

% ATP.lambda_765
tff(fact_10340_ATP_Olambda__766,axiom,
    ! [Uu: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: nat] : aa_nat_complex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_gl(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_gk(Uu),Uub),Uuc))) ).

% ATP.lambda_766
tff(fact_10341_ATP_Olambda__767,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: real] : aa_real_complex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_gh(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_gg(Uu),Uub),Uuc))) ).

% ATP.lambda_767
tff(fact_10342_ATP_Olambda__768,axiom,
    ! [Uu: set_int,Uua: fun_co333780937270390912nt_int,Uub: fun_co3431063785546946928_int_o,Uuc: complex] : aa_complex_int(aa_fun5055871773053899019ex_int(aa_fun7361901416012365340ex_int(aTP_Lamm_gc(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_complex_fun_int_o(aa_fun5644928480641789244_int_o(aTP_Lamm_gb(Uu),Uub),Uuc))) ).

% ATP.lambda_768
tff(fact_10343_ATP_Olambda__769,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_o,Uuc: nat] : aa_nat_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_fy(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_nat_fun_int_int(Uua,Uuc),collect_int(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_fx(Uu),Uub),Uuc))) ).

% ATP.lambda_769
tff(fact_10344_ATP_Olambda__770,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_int,Uub: fun_real_fun_int_o,Uuc: real] : aa_real_int(aa_fun7555737461897725707al_int(aa_fun1679075639499870878al_int(aTP_Lamm_fu(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_real_fun_int_int(Uua,Uuc),collect_int(aa_real_fun_int_o(aa_fun2967211649596192572_int_o(aTP_Lamm_ft(Uu),Uub),Uuc))) ).

% ATP.lambda_770
tff(fact_10345_ATP_Olambda__771,axiom,
    ! [Uu: $o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_kt((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_771
tff(fact_10346_ATP_Olambda__772,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_fn(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_772
tff(fact_10347_ATP_Olambda__773,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_fj(Uu),Uua),Uub),Uuc)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_773
tff(fact_10348_ATP_Olambda__774,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun7461304428735181397nt_int(aTP_Lamm_fk(Uu),Uua),Uub),Uuc) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_774
tff(fact_10349_ATP_Olambda__775,axiom,
    ! [Uu: fun_o_8988054265482332260nt_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_fl(Uu),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_775
tff(fact_10350_ATP_Olambda__776,axiom,
    ! [Uu: fun_o_o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_ff(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_776
tff(fact_10351_ATP_Olambda__777,axiom,
    ! [Uu: fun_o_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun7323606162024495668nt_int(aTP_Lamm_fg(Uu),Uua),Uub),Uuc) = aa_o_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_777
tff(fact_10352_ATP_Olambda__778,axiom,
    ! [Uu: fun_in4211310926729683942nt_int,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun1034830265760468567nt_int(aTP_Lamm_fm(Uu),Uua),Uub),Uuc) = aa_int7462565939943104719nt_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_778
tff(fact_10353_ATP_Olambda__779,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun82482788810740534_int_o(aTP_Lamm_fh(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_779
tff(fact_10354_ATP_Olambda__780,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun6119123525526402106nt_int(aTP_Lamm_fi(Uu),Uua),Uub),Uuc) = aa_int_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_780
tff(fact_10355_ATP_Olambda__781,axiom,
    ! [Uu: fun_fu3275390434952296369_nat_o,Uua: fun_na5914473681949881652_nat_o,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_fo(Uu),Uua),Uub),Uuc) = aa_fun359341089595923442_nat_o(Uu,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_781
tff(fact_10356_ATP_Olambda__782,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_jj(Uu),Uua),Uub),Uuc) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_782
tff(fact_10357_ATP_Olambda__783,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ji(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_783
tff(fact_10358_ATP_Olambda__784,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_tx(Uu),Uua),Uub),Uuc) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_784
tff(fact_10359_ATP_Olambda__785,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aad(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_785
tff(fact_10360_ATP_Olambda__786,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aaf(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_786
tff(fact_10361_ATP_Olambda__787,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aaj(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_787
tff(fact_10362_ATP_Olambda__788,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aah(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_788
tff(fact_10363_ATP_Olambda__789,axiom,
    ! [Uu: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_acs(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_acr(Uua),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_789
tff(fact_10364_ATP_Olambda__790,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_zm(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uu)),Uua)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_790
tff(fact_10365_ATP_Olambda__791,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_zo(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Uu)),Uua)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_791
tff(fact_10366_ATP_Olambda__792,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_zk(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Uu)),Uua)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_792
tff(fact_10367_ATP_Olambda__793,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_zn(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,Uua)),Uud)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Uub)),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_793
tff(fact_10368_ATP_Olambda__794,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_zp(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,Uua)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Uub)),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_794
tff(fact_10369_ATP_Olambda__795,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_zl(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,Uua)),Uud)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Uub)),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_795
tff(fact_10370_ATP_Olambda__796,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_aef(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uue),Uuf)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uud))
           => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(Uua,I3)),X_13)
            <=> aa_nat_o(vEBT_V8194947554948674370ptions(Uub),I3) ) )
        & $ite(
            Uue = Uuf,
            ! [X4: vEBT_VEBT] :
              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Uua))
             => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X4)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X4)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_796
tff(fact_10371_ATP_Olambda__797,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ol(Uu),Uua) = Uu ).

% ATP.lambda_797
tff(fact_10372_ATP_Olambda__798,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_om,Uu),Uua) = Uu ).

% ATP.lambda_798
tff(fact_10373_ATP_Olambda__799,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_aff(Uu),Uua) = Uu ).

% ATP.lambda_799
tff(fact_10374_ATP_Olambda__800,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_lh,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_800
tff(fact_10375_ATP_Olambda__801,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_jk,Uu) = Uu ).

% ATP.lambda_801
tff(fact_10376_ATP_Olambda__802,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_kr,Uu) = Uu ).

% ATP.lambda_802
tff(fact_10377_ATP_Olambda__803,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_eg,Uu) = Uu ).

% ATP.lambda_803
tff(fact_10378_ATP_Olambda__804,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_bu,Uu) = zero_zero_complex ).

% ATP.lambda_804
tff(fact_10379_ATP_Olambda__805,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_bv,Uu) = zero_zero_real ).

% ATP.lambda_805
tff(fact_10380_ATP_Olambda__806,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_bw,Uu) = zero_zero_rat ).

% ATP.lambda_806
tff(fact_10381_ATP_Olambda__807,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_nj,Uu) = zero_zero_complex ).

% ATP.lambda_807
tff(fact_10382_ATP_Olambda__808,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ek,Uu) = zero_zero_real ).

% ATP.lambda_808
tff(fact_10383_ATP_Olambda__809,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_bx,Uu) = zero_zero_nat ).

% ATP.lambda_809
tff(fact_10384_ATP_Olambda__810,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_nk,Uu) = zero_zero_int ).

% ATP.lambda_810
tff(fact_10385_ATP_Olambda__811,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_by,Uu) = zero_zero_int ).

% ATP.lambda_811
tff(fact_10386_ATP_Olambda__812,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_zt,Uu)
    <=> $false ) ).

% ATP.lambda_812
tff(fact_10387_ATP_Olambda__813,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_aaw,Uu)
    <=> $false ) ).

% ATP.lambda_813
tff(fact_10388_ATP_Olambda__814,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_aav,Uu)
    <=> $true ) ).

% ATP.lambda_814

% Helper facts (1)
tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P2: fun_real_o] :
      ( aa_real_o(P2,fChoice_real(P2))
      = ( ? [X7: real] : aa_real_o(P2,X7) ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary),ya)),xa) ).

%------------------------------------------------------------------------------