%------------------------------------------------------------------------------
% File     : ITP210_3 : TPTP v9.0.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Assertions 00353_010205
% 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    : 0024_Assertions_00353_010205 [Des22]

% Status   : Theorem
% Rating   : 0.67 v8.1.0
% Syntax   : Number of formulae    : 14592 (4825 unt;3875 typ;   0 def)
%            Number of atoms       : 23838 (9530 equ)
%            Maximal formula atoms :   38 (   2 avg)
%            Number of connectives : 14522 (2153   ~; 344   |;1027   &)
%                                         (2424 <=>;8574  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (   5 avg)
%            Maximal term depth    :   36 (   2 avg)
%            Number of FOOLs       : 1845 ( 742 fml;1103 var)
%            Number of X terms     :  713 (   0  []; 678 ite;  35 let)
%            Number of types       :  791 ( 790 usr)
%            Number of type conns  : 3706 (2388   >;1318   *;   0   +;   0  <<)
%            Number of predicates  :  224 ( 221 usr;   4 prp; 0-5 aty)
%            Number of functors    : 2874 (2874 usr; 705 con; 0-7 aty)
%            Number of variables   : 25609 (25360   !; 249   ?;25609   :)
% 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 14:36:02.688
%------------------------------------------------------------------------------
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tff(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
    filter_nat: $tType ).

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

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

tff(ty_n_t__fun_It__String__Ochar_M_Eo_J,type,
    fun_char_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__String__Ochar_J,type,
    fun_o_char: $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__Set__Oset_It__Rat__Orat_J,type,
    set_rat: $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__Onatural,type,
    code_natural: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(ty_n_t__String__Ochar,type,
    char: $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 (3085)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: fun_Pr6460944699146269935_nat_o ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: fun_Pr6460944699146269935_nat_o ).

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: fun_int_Code_integer > fun_se6579715966475752165nteger ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: fun_int_rat > fun_se937733209267607904nt_rat ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: fun_int_nat > fun_se7317719546745285464nt_nat ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: fun_nat_Code_integer > fun_se5401815353304058413nteger ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: fun_nat_rat > fun_se4022739043013652648at_rat ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: fun_nat_int > fun_se7912642235559603452at_int ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: fun_se7586849273513198688nteger > fun_se8946511271713717145nteger ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: fun_set_nat_rat > fun_se4050456683449537044at_rat ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: fun_int_Code_integer > fun_int_o ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: fun_nat_Code_integer > fun_nat_o ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: fun_nat_rat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__aaq____,type,
    aTP_Lamm_aaq: fun_nat_int > fun_nat_o ).

tff(sy_c_ATP_058Lamm__aar____,type,
    aTP_Lamm_aar: fun_nat_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: fun_int_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: fun_se7586849273513198688nteger > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: fun_set_nat_rat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: fun_int_Code_integer > fun_se6579715966475752165nteger ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: fun_int_assn > fun_se6298406198028402864t_assn ).

tff(sy_c_ATP_058Lamm__aax____,type,
    aTP_Lamm_aax: fun_int_rat > fun_se937733209267607904nt_rat ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: fun_int_nat > fun_se7317719546745285464nt_nat ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: fun_nat_Code_integer > fun_se5401815353304058413nteger ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: assn > fun_as3747249873083764768_nat_o ).

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: fun_nat_assn > fun_se5107463250893228024t_assn ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: fun_nat_rat > fun_se4022739043013652648at_rat ).

tff(sy_c_ATP_058Lamm__abc____,type,
    aTP_Lamm_abc: fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: fun_nat_int > fun_se7912642235559603452at_int ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: fun_int_Code_integer > fun_int_o ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: fun_int_assn > fun_int_o ).

tff(sy_c_ATP_058Lamm__abh____,type,
    aTP_Lamm_abh: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: fun_nat_Code_integer > fun_nat_o ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: fun_nat_assn > fun_nat_o ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: fun_nat_rat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: fun_nat_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: fun_nat_int > fun_nat_o ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: fun_int_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__abs____,type,
    aTP_Lamm_abs: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abt____,type,
    aTP_Lamm_abt: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abu____,type,
    aTP_Lamm_abu: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__abv____,type,
    aTP_Lamm_abv: fun_int_o > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__abw____,type,
    aTP_Lamm_abw: fun_int_o > fun_fu973771059875708133nt_nat ).

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: fun_nat_o > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: fun_nat_o > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: fun_nat_o > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: assn > fun_as3747249873083764768_nat_o ).

tff(sy_c_ATP_058Lamm__aca____,type,
    aTP_Lamm_aca: fun_int_o > fun_fu5221552698205305849nt_int ).

tff(sy_c_ATP_058Lamm__acb____,type,
    aTP_Lamm_acb: fun_set_nat_o > fun_fu299390977814632459at_rat ).

tff(sy_c_ATP_058Lamm__acc____,type,
    aTP_Lamm_acc: fun_set_nat_o > fun_fu3253069483457113587at_nat ).

tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: fun_set_nat_o > fun_fu1762771019471887623at_int ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: fun_Pr4796943543331071514_nat_o > fun_fu5508233830269119298at_rat ).

tff(sy_c_ATP_058Lamm__acf____,type,
    aTP_Lamm_acf: fun_int_o > fun_fu3890538715107687308nteger ).

tff(sy_c_ATP_058Lamm__acg____,type,
    aTP_Lamm_acg: fun_nat_o > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__ach____,type,
    aTP_Lamm_ach: fun_int_o > fun_fu5832215286117076717t_assn ).

tff(sy_c_ATP_058Lamm__aci____,type,
    aTP_Lamm_aci: fun_nat_o > fun_fu3959123160307459545t_assn ).

tff(sy_c_ATP_058Lamm__acj____,type,
    aTP_Lamm_acj: $o > fun_fu9039538216874201369_o_rat ).

tff(sy_c_ATP_058Lamm__ack____,type,
    aTP_Lamm_ack: $o > fun_fu3581538110357139729_o_nat ).

tff(sy_c_ATP_058Lamm__acl____,type,
    aTP_Lamm_acl: $o > fun_fu9026017786105829997_o_int ).

tff(sy_c_ATP_058Lamm__acm____,type,
    aTP_Lamm_acm: int > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__acn____,type,
    aTP_Lamm_acn: int > fun_fu973771059875708133nt_nat ).

tff(sy_c_ATP_058Lamm__aco____,type,
    aTP_Lamm_aco: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__acp____,type,
    aTP_Lamm_acp: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__acq____,type,
    aTP_Lamm_acq: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__acr____,type,
    aTP_Lamm_acr: int > fun_fu5221552698205305849nt_int ).

tff(sy_c_ATP_058Lamm__acs____,type,
    aTP_Lamm_acs: set_nat > fun_fu299390977814632459at_rat ).

tff(sy_c_ATP_058Lamm__act____,type,
    aTP_Lamm_act: fun_int_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__acu____,type,
    aTP_Lamm_acu: fun_int_nat > fun_nat_fun_int_o ).

tff(sy_c_ATP_058Lamm__acv____,type,
    aTP_Lamm_acv: fun_int_nat > fun_nat_fun_int_o ).

tff(sy_c_ATP_058Lamm__acw____,type,
    aTP_Lamm_acw: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__acx____,type,
    aTP_Lamm_acx: fun_nat_int > fun_int_fun_nat_o ).

tff(sy_c_ATP_058Lamm__acy____,type,
    aTP_Lamm_acy: fun_nat_int > fun_int_fun_nat_o ).

tff(sy_c_ATP_058Lamm__acz____,type,
    aTP_Lamm_acz: fun_int_Code_integer > fun_Co3011819262099091808nteger ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: fun_Pr6460944699146269935_nat_o ).

tff(sy_c_ATP_058Lamm__ada____,type,
    aTP_Lamm_ada: fun_int_Code_integer > fun_Co8353275306710069249_int_o ).

tff(sy_c_ATP_058Lamm__adb____,type,
    aTP_Lamm_adb: fun_int_Code_integer > fun_Co8353275306710069249_int_o ).

tff(sy_c_ATP_058Lamm__adc____,type,
    aTP_Lamm_adc: fun_nat_Code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__add____,type,
    aTP_Lamm_add: fun_nat_Code_integer > fun_Co1375048237525739357_nat_o ).

tff(sy_c_ATP_058Lamm__ade____,type,
    aTP_Lamm_ade: fun_nat_Code_integer > fun_Co1375048237525739357_nat_o ).

tff(sy_c_ATP_058Lamm__adf____,type,
    aTP_Lamm_adf: fun_int_Code_natural > fun_Co6749200540738471926atural ).

tff(sy_c_ATP_058Lamm__adg____,type,
    aTP_Lamm_adg: fun_int_Code_natural > fun_Co7174943067365291894_int_o ).

tff(sy_c_ATP_058Lamm__adh____,type,
    aTP_Lamm_adh: fun_int_Code_natural > fun_Co7174943067365291894_int_o ).

tff(sy_c_ATP_058Lamm__adi____,type,
    aTP_Lamm_adi: fun_nat_Code_natural > fun_Co740390889742897050atural ).

tff(sy_c_ATP_058Lamm__adj____,type,
    aTP_Lamm_adj: fun_nat_Code_natural > fun_Co196715998180962002_nat_o ).

tff(sy_c_ATP_058Lamm__adk____,type,
    aTP_Lamm_adk: fun_nat_Code_natural > fun_Co196715998180962002_nat_o ).

tff(sy_c_ATP_058Lamm__adl____,type,
    aTP_Lamm_adl: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__adm____,type,
    aTP_Lamm_adm: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__adn____,type,
    aTP_Lamm_adn: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ado____,type,
    aTP_Lamm_ado: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__adp____,type,
    aTP_Lamm_adp: fun_int_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__adq____,type,
    aTP_Lamm_adq: fun_int_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__adr____,type,
    aTP_Lamm_adr: fun_set_nat_nat > fun_na106483476154090400at_nat ).

tff(sy_c_ATP_058Lamm__ads____,type,
    aTP_Lamm_ads: fun_set_nat_nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__adt____,type,
    aTP_Lamm_adt: fun_set_nat_nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__adu____,type,
    aTP_Lamm_adu: fun_set_nat_int > fun_in6555721293707029592at_int ).

tff(sy_c_ATP_058Lamm__adv____,type,
    aTP_Lamm_adv: fun_set_nat_int > fun_in7136481858371106712_nat_o ).

tff(sy_c_ATP_058Lamm__adw____,type,
    aTP_Lamm_adw: fun_set_nat_int > fun_in7136481858371106712_nat_o ).

tff(sy_c_ATP_058Lamm__adx____,type,
    aTP_Lamm_adx: $o > fun_fu5442211033179627998nteger ).

tff(sy_c_ATP_058Lamm__ady____,type,
    aTP_Lamm_ady: int > fun_fu3890538715107687308nteger ).

tff(sy_c_ATP_058Lamm__adz____,type,
    aTP_Lamm_adz: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: fun_Pr6460944699146269935_nat_o > fun_fu8419763957888848259_nat_o ).

tff(sy_c_ATP_058Lamm__aea____,type,
    aTP_Lamm_aea: $o > fun_fu556125837173103209o_assn ).

tff(sy_c_ATP_058Lamm__aeb____,type,
    aTP_Lamm_aeb: int > fun_fu5832215286117076717t_assn ).

tff(sy_c_ATP_058Lamm__aec____,type,
    aTP_Lamm_aec: nat > fun_fu3959123160307459545t_assn ).

tff(sy_c_ATP_058Lamm__aed____,type,
    aTP_Lamm_aed: code_integer > fun_na8856381646659805649nteger ).

tff(sy_c_ATP_058Lamm__aee____,type,
    aTP_Lamm_aee: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__aef____,type,
    aTP_Lamm_aef: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__aeg____,type,
    aTP_Lamm_aeg: code_integer > fun_na8856381646659805649nteger ).

tff(sy_c_ATP_058Lamm__aeh____,type,
    aTP_Lamm_aeh: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__aei____,type,
    aTP_Lamm_aei: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__aej____,type,
    aTP_Lamm_aej: code_integer > fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__aek____,type,
    aTP_Lamm_aek: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ael____,type,
    aTP_Lamm_ael: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__aem____,type,
    aTP_Lamm_aem: fun_fu3275390434952296369_nat_o ).

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

tff(sy_c_ATP_058Lamm__aeo____,type,
    aTP_Lamm_aeo: fun_o_o > fun_o_o ).

tff(sy_c_ATP_058Lamm__aep____,type,
    aTP_Lamm_aep: fun_rat_o > fun_rat_o ).

tff(sy_c_ATP_058Lamm__aeq____,type,
    aTP_Lamm_aeq: fun_Pr4796943543331071514_nat_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__aer____,type,
    aTP_Lamm_aer: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aes____,type,
    aTP_Lamm_aes: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aet____,type,
    aTP_Lamm_aet: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aeu____,type,
    aTP_Lamm_aeu: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aev____,type,
    aTP_Lamm_aev: set_nat > fun_fu7641778686208158983_nat_o ).

tff(sy_c_ATP_058Lamm__aew____,type,
    aTP_Lamm_aew: set_int > fun_fu3968995010428139015_int_o ).

tff(sy_c_ATP_058Lamm__aex____,type,
    aTP_Lamm_aex: set_Pr1261947904930325089at_nat > fun_fu3617688454659378439_nat_o ).

tff(sy_c_ATP_058Lamm__aey____,type,
    aTP_Lamm_aey: set_Pr1261947904930325089at_nat > fun_fu2341038929883010567_nat_o ).

tff(sy_c_ATP_058Lamm__aez____,type,
    aTP_Lamm_aez: fun_se4101287982975402108nt_nat > fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_Pr6460944699146269935_nat_o > fun_fu8419763957888848259_nat_o ).

tff(sy_c_ATP_058Lamm__afa____,type,
    aTP_Lamm_afa: fun_se4101287982975402108nt_nat > fun_fu6634461749290962705_int_o ).

tff(sy_c_ATP_058Lamm__afb____,type,
    aTP_Lamm_afb: fun_set_nat_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__afc____,type,
    aTP_Lamm_afc: fun_set_nat_nat > fun_fu1237770272497958833_nat_o ).

tff(sy_c_ATP_058Lamm__afd____,type,
    aTP_Lamm_afd: fun_fu3504661237974713170_o_nat > fun_fu5135969500543783852at_o_o ).

tff(sy_c_ATP_058Lamm__afe____,type,
    aTP_Lamm_afe: fun_fu3504661237974713170_o_nat > fun_fu8326177388093693585at_o_o ).

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

tff(sy_c_ATP_058Lamm__afg____,type,
    aTP_Lamm_afg: fun_nat_nat > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__afh____,type,
    aTP_Lamm_afh: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__afi____,type,
    aTP_Lamm_afi: fun_int_nat > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__afj____,type,
    aTP_Lamm_afj: fun_Pr2602117396471556196at_nat > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__afk____,type,
    aTP_Lamm_afk: fun_Pr2602117396471556196at_nat > fun_fu7237799034916548881_nat_o ).

tff(sy_c_ATP_058Lamm__afl____,type,
    aTP_Lamm_afl: fun_se4101287982975402108nt_nat > fun_se2092758488442489134_int_o ).

tff(sy_c_ATP_058Lamm__afm____,type,
    aTP_Lamm_afm: fun_set_nat_nat > fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__afn____,type,
    aTP_Lamm_afn: fun_fu3504661237974713170_o_nat > fun_fu4164567611050918574at_o_o ).

tff(sy_c_ATP_058Lamm__afo____,type,
    aTP_Lamm_afo: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__afp____,type,
    aTP_Lamm_afp: fun_int_nat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__afq____,type,
    aTP_Lamm_afq: fun_Pr2602117396471556196at_nat > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__afr____,type,
    aTP_Lamm_afr: fun_se4101287982975402108nt_nat > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__afs____,type,
    aTP_Lamm_afs: fun_set_nat_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__aft____,type,
    aTP_Lamm_aft: fun_fu3504661237974713170_o_nat > fun_fu6295675835253126101at_o_o ).

tff(sy_c_ATP_058Lamm__afu____,type,
    aTP_Lamm_afu: fun_nat_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__afv____,type,
    aTP_Lamm_afv: fun_int_nat > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__afw____,type,
    aTP_Lamm_afw: fun_Pr2602117396471556196at_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__afx____,type,
    aTP_Lamm_afx: set_o > fun_fu3819205551299168222un_o_o ).

tff(sy_c_ATP_058Lamm__afy____,type,
    aTP_Lamm_afy: set_o > fun_fu6337961365400742745un_o_o ).

tff(sy_c_ATP_058Lamm__afz____,type,
    aTP_Lamm_afz: set_o > fun_fu7756609662939579309un_o_o ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: set_se6260736226359567993nt_int > fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__aga____,type,
    aTP_Lamm_aga: set_o > fun_fu6781074197733401937un_o_o ).

tff(sy_c_ATP_058Lamm__agb____,type,
    aTP_Lamm_agb: set_int > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__agc____,type,
    aTP_Lamm_agc: set_o > fun_fu7756609662939579309un_o_o ).

tff(sy_c_ATP_058Lamm__agd____,type,
    aTP_Lamm_agd: set_nat > fun_fu8207122332086543297_nat_o ).

tff(sy_c_ATP_058Lamm__age____,type,
    aTP_Lamm_age: set_o > fun_fu3819205551299168222un_o_o ).

tff(sy_c_ATP_058Lamm__agf____,type,
    aTP_Lamm_agf: set_nat > fun_fu5838931486547983824_nat_o ).

tff(sy_c_ATP_058Lamm__agg____,type,
    aTP_Lamm_agg: set_int > fun_fu5271299750659287320_int_o ).

tff(sy_c_ATP_058Lamm__agh____,type,
    aTP_Lamm_agh: set_o > fun_fu7722316861827675049un_o_o ).

tff(sy_c_ATP_058Lamm__agi____,type,
    aTP_Lamm_agi: set_nat > fun_fu2798942646354680517_nat_o ).

tff(sy_c_ATP_058Lamm__agj____,type,
    aTP_Lamm_agj: set_int > fun_fu2231310910465984013_int_o ).

tff(sy_c_ATP_058Lamm__agk____,type,
    aTP_Lamm_agk: fun_in6555721293707029592at_int ).

tff(sy_c_ATP_058Lamm__agl____,type,
    aTP_Lamm_agl: fun_na106483476154090400at_nat ).

tff(sy_c_ATP_058Lamm__agm____,type,
    aTP_Lamm_agm: code_integer > fun_o_Code_integer ).

tff(sy_c_ATP_058Lamm__agn____,type,
    aTP_Lamm_agn: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__ago____,type,
    aTP_Lamm_ago: code_integer > fun_int_Code_integer ).

tff(sy_c_ATP_058Lamm__agp____,type,
    aTP_Lamm_agp: rat > fun_o_rat ).

tff(sy_c_ATP_058Lamm__agq____,type,
    aTP_Lamm_agq: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__agr____,type,
    aTP_Lamm_agr: rat > fun_int_rat ).

tff(sy_c_ATP_058Lamm__ags____,type,
    aTP_Lamm_ags: set_o > fun_fu6781074197733401937un_o_o ).

tff(sy_c_ATP_058Lamm__agt____,type,
    aTP_Lamm_agt: fun_o_o > fun_o_rat ).

tff(sy_c_ATP_058Lamm__agu____,type,
    aTP_Lamm_agu: fun_int_o > fun_int_rat ).

tff(sy_c_ATP_058Lamm__agv____,type,
    aTP_Lamm_agv: fun_nat_o > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__agw____,type,
    aTP_Lamm_agw: fun_o_o > fun_o_int ).

tff(sy_c_ATP_058Lamm__agx____,type,
    aTP_Lamm_agx: fun_nat_o > fun_nat_int ).

tff(sy_c_ATP_058Lamm__agy____,type,
    aTP_Lamm_agy: fun_o_o > fun_o_nat ).

tff(sy_c_ATP_058Lamm__agz____,type,
    aTP_Lamm_agz: fun_int_o > fun_int_nat ).

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

tff(sy_c_ATP_058Lamm__aha____,type,
    aTP_Lamm_aha: fun_nat_o > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ahb____,type,
    aTP_Lamm_ahb: fun_int_o > fun_int_int ).

tff(sy_c_ATP_058Lamm__ahc____,type,
    aTP_Lamm_ahc: fun_list_nat_o > fun_list_nat_rat ).

tff(sy_c_ATP_058Lamm__ahd____,type,
    aTP_Lamm_ahd: set_Pr958786334691620121nt_int > fun_na6120508140265747663_int_o ).

tff(sy_c_ATP_058Lamm__ahe____,type,
    aTP_Lamm_ahe: set_list_nat > fun_na1108963418173300452_nat_o ).

tff(sy_c_ATP_058Lamm__ahf____,type,
    aTP_Lamm_ahf: set_o > fun_nat_fun_set_o_o ).

tff(sy_c_ATP_058Lamm__ahg____,type,
    aTP_Lamm_ahg: set_set_nat > fun_na3711450577076729278_nat_o ).

tff(sy_c_ATP_058Lamm__ahh____,type,
    aTP_Lamm_ahh: set_int > fun_na8635581178960477080_int_o ).

tff(sy_c_ATP_058Lamm__ahi____,type,
    aTP_Lamm_ahi: set_Pr1261947904930325089at_nat > fun_na4518284863240432007_nat_o ).

tff(sy_c_ATP_058Lamm__ahj____,type,
    aTP_Lamm_ahj: set_nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__ahk____,type,
    aTP_Lamm_ahk: fun_nat_Code_integer > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ahl____,type,
    aTP_Lamm_ahl: fun_int_Code_integer > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__ahm____,type,
    aTP_Lamm_ahm: fun_nat_rat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ahn____,type,
    aTP_Lamm_ahn: fun_int_rat > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__aho____,type,
    aTP_Lamm_aho: fun_nat_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ahp____,type,
    aTP_Lamm_ahp: fun_int_nat > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__ahq____,type,
    aTP_Lamm_ahq: fun_nat_int > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ahr____,type,
    aTP_Lamm_ahr: fun_int_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__ahs____,type,
    aTP_Lamm_ahs: fun_se7586849273513198688nteger > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__aht____,type,
    aTP_Lamm_aht: fun_set_nat_rat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__ahu____,type,
    aTP_Lamm_ahu: set_o > fun_fu7066070072329265113un_o_o ).

tff(sy_c_ATP_058Lamm__ahv____,type,
    aTP_Lamm_ahv: set_nat > fun_fu9085782028513977109_nat_o ).

tff(sy_c_ATP_058Lamm__ahw____,type,
    aTP_Lamm_ahw: set_nat > fun_fu6443148758731022984_o_nat ).

tff(sy_c_ATP_058Lamm__ahx____,type,
    aTP_Lamm_ahx: set_nat > fun_fu2757636626010989786nt_nat ).

tff(sy_c_ATP_058Lamm__ahy____,type,
    aTP_Lamm_ahy: set_nat > fun_fu2885402556529144793_nat_o ).

tff(sy_c_ATP_058Lamm__ahz____,type,
    aTP_Lamm_ahz: set_o > fun_fu865690600344432797un_o_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__aia____,type,
    aTP_Lamm_aia: set_o > fun_fu7006494623891191978at_nat ).

tff(sy_c_ATP_058Lamm__aib____,type,
    aTP_Lamm_aib: set_int > fun_fu5586244933174592126at_nat ).

tff(sy_c_ATP_058Lamm__aic____,type,
    aTP_Lamm_aic: set_nat > fun_fu5776950848406856154at_nat ).

tff(sy_c_ATP_058Lamm__aid____,type,
    aTP_Lamm_aid: set_list_nat > fun_fu1834022669976960707_nat_o ).

tff(sy_c_ATP_058Lamm__aie____,type,
    aTP_Lamm_aie: set_nat > fun_fu4568065243339218371_nat_o ).

tff(sy_c_ATP_058Lamm__aif____,type,
    aTP_Lamm_aif: set_nat > fun_fu5576414621873183578at_nat ).

tff(sy_c_ATP_058Lamm__aig____,type,
    aTP_Lamm_aig: set_nat > fun_fu2849355231831568730at_nat ).

tff(sy_c_ATP_058Lamm__aih____,type,
    aTP_Lamm_aih: set_nat > fun_fu225173491512638659_nat_o ).

tff(sy_c_ATP_058Lamm__aii____,type,
    aTP_Lamm_aii: set_list_nat > fun_fu6714502955005156803_nat_o ).

tff(sy_c_ATP_058Lamm__aij____,type,
    aTP_Lamm_aij: set_list_nat > fun_fu6441209753288909002at_nat ).

tff(sy_c_ATP_058Lamm__aik____,type,
    aTP_Lamm_aik: set_set_nat > fun_fu6204623786860714916at_nat ).

tff(sy_c_ATP_058Lamm__ail____,type,
    aTP_Lamm_ail: set_nat > fun_fu7755011012798512067_nat_o ).

tff(sy_c_ATP_058Lamm__aim____,type,
    aTP_Lamm_aim: set_nat > fun_fu6267314603245239642at_nat ).

tff(sy_c_ATP_058Lamm__ain____,type,
    aTP_Lamm_ain: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aio____,type,
    aTP_Lamm_aio: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aip____,type,
    aTP_Lamm_aip: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aiq____,type,
    aTP_Lamm_aiq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__air____,type,
    aTP_Lamm_air: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ais____,type,
    aTP_Lamm_ais: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ait____,type,
    aTP_Lamm_ait: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__aiu____,type,
    aTP_Lamm_aiu: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aiv____,type,
    aTP_Lamm_aiv: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aiw____,type,
    aTP_Lamm_aiw: nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__aix____,type,
    aTP_Lamm_aix: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__aiy____,type,
    aTP_Lamm_aiy: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__aiz____,type,
    aTP_Lamm_aiz: int > fun_in2204242111582549184nt_int ).

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

tff(sy_c_ATP_058Lamm__aja____,type,
    aTP_Lamm_aja: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ajb____,type,
    aTP_Lamm_ajb: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ajc____,type,
    aTP_Lamm_ajc: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ajd____,type,
    aTP_Lamm_ajd: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__aje____,type,
    aTP_Lamm_aje: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ajf____,type,
    aTP_Lamm_ajf: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__ajg____,type,
    aTP_Lamm_ajg: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ajh____,type,
    aTP_Lamm_ajh: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__aji____,type,
    aTP_Lamm_aji: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__ajj____,type,
    aTP_Lamm_ajj: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__ajk____,type,
    aTP_Lamm_ajk: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__ajl____,type,
    aTP_Lamm_ajl: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__ajm____,type,
    aTP_Lamm_ajm: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__ajn____,type,
    aTP_Lamm_ajn: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__ajo____,type,
    aTP_Lamm_ajo: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__ajp____,type,
    aTP_Lamm_ajp: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__ajq____,type,
    aTP_Lamm_ajq: int > fun_in2204242111582549184nt_int ).

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

tff(sy_c_ATP_058Lamm__ajs____,type,
    aTP_Lamm_ajs: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ajt____,type,
    aTP_Lamm_ajt: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aju____,type,
    aTP_Lamm_aju: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ajv____,type,
    aTP_Lamm_ajv: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ajw____,type,
    aTP_Lamm_ajw: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ajx____,type,
    aTP_Lamm_ajx: $o > fun_o_5488095450495718970o_char ).

tff(sy_c_ATP_058Lamm__ajy____,type,
    aTP_Lamm_ajy: $o > fun_o_6360995121869435791o_char ).

tff(sy_c_ATP_058Lamm__ajz____,type,
    aTP_Lamm_ajz: $o > fun_o_2683029531876921444o_char ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: set_fu3555266933746557007_nat_o > fun_fu5135969500543783852at_o_o ).

tff(sy_c_ATP_058Lamm__aka____,type,
    aTP_Lamm_aka: $o > fun_o_3037882179896959673o_char ).

tff(sy_c_ATP_058Lamm__akb____,type,
    aTP_Lamm_akb: $o > fun_o_335877185394996366o_char ).

tff(sy_c_ATP_058Lamm__akc____,type,
    aTP_Lamm_akc: $o > fun_o_2016307092959984611o_char ).

tff(sy_c_ATP_058Lamm__akd____,type,
    aTP_Lamm_akd: $o > fun_Co681828205884031672o_char ).

tff(sy_c_ATP_058Lamm__ake____,type,
    aTP_Lamm_ake: fun_Co681828205884031672o_char ).

tff(sy_c_ATP_058Lamm__akf____,type,
    aTP_Lamm_akf: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__akg____,type,
    aTP_Lamm_akg: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__akh____,type,
    aTP_Lamm_akh: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__aki____,type,
    aTP_Lamm_aki: fun_int_nat ).

tff(sy_c_ATP_058Lamm__akj____,type,
    aTP_Lamm_akj: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__akk____,type,
    aTP_Lamm_akk: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__akl____,type,
    aTP_Lamm_akl: fun_num_option_num ).

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

tff(sy_c_ATP_058Lamm__akn____,type,
    aTP_Lamm_akn: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ako____,type,
    aTP_Lamm_ako: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__akp____,type,
    aTP_Lamm_akp: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__akq____,type,
    aTP_Lamm_akq: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__akr____,type,
    aTP_Lamm_akr: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__aks____,type,
    aTP_Lamm_aks: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__akt____,type,
    aTP_Lamm_akt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aku____,type,
    aTP_Lamm_aku: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__akv____,type,
    aTP_Lamm_akv: fun_int_int ).

tff(sy_c_ATP_058Lamm__akw____,type,
    aTP_Lamm_akw: fun_num_fun_num_int ).

tff(sy_c_ATP_058Lamm__akx____,type,
    aTP_Lamm_akx: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aky____,type,
    aTP_Lamm_aky: fun_nat_o > fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: $o > fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__ala____,type,
    aTP_Lamm_ala: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__alb____,type,
    aTP_Lamm_alb: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__alc____,type,
    aTP_Lamm_alc: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ald____,type,
    aTP_Lamm_ald: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__ale____,type,
    aTP_Lamm_ale: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__alf____,type,
    aTP_Lamm_alf: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__alg____,type,
    aTP_Lamm_alg: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__alh____,type,
    aTP_Lamm_alh: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__ali____,type,
    aTP_Lamm_ali: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__alj____,type,
    aTP_Lamm_alj: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__alk____,type,
    aTP_Lamm_alk: rat > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__all____,type,
    aTP_Lamm_all: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__alm____,type,
    aTP_Lamm_alm: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aln____,type,
    aTP_Lamm_aln: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__alo____,type,
    aTP_Lamm_alo: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__alp____,type,
    aTP_Lamm_alp: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__alq____,type,
    aTP_Lamm_alq: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__alr____,type,
    aTP_Lamm_alr: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__als____,type,
    aTP_Lamm_als: fun_na819858167508000762st_nat ).

tff(sy_c_ATP_058Lamm__alt____,type,
    aTP_Lamm_alt: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__alu____,type,
    aTP_Lamm_alu: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__alv____,type,
    aTP_Lamm_alv: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__alw____,type,
    aTP_Lamm_alw: fun_Pr2118491707810114563at_nat ).

tff(sy_c_ATP_058Lamm__alx____,type,
    aTP_Lamm_alx: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__aly____,type,
    aTP_Lamm_aly: fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__alz____,type,
    aTP_Lamm_alz: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: $o > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__ama____,type,
    aTP_Lamm_ama: fun_char_fun_char_o ).

tff(sy_c_ATP_058Lamm__amb____,type,
    aTP_Lamm_amb: fun_li3177621261580724543t_char ).

tff(sy_c_ATP_058Lamm__amc____,type,
    aTP_Lamm_amc: fun_list_char_o ).

tff(sy_c_ATP_058Lamm__amd____,type,
    aTP_Lamm_amd: fun_o_5935606648521830764t_char ).

tff(sy_c_ATP_058Lamm__ame____,type,
    aTP_Lamm_ame: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__amf____,type,
    aTP_Lamm_amf: fun_nat_int ).

tff(sy_c_ATP_058Lamm__amg____,type,
    aTP_Lamm_amg: code_natural > fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__amh____,type,
    aTP_Lamm_amh: fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__ami____,type,
    aTP_Lamm_ami: code_natural > fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__amj____,type,
    aTP_Lamm_amj: code_natural > fun_Co9168753396221475450atural ).

tff(sy_c_ATP_058Lamm__amk____,type,
    aTP_Lamm_amk: produc7822875418678951345atural > fun_Co1890133979025130057atural ).

tff(sy_c_ATP_058Lamm__aml____,type,
    aTP_Lamm_aml: fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__amm____,type,
    aTP_Lamm_amm: fun_pr6291093682560481277atural ).

tff(sy_c_ATP_058Lamm__amn____,type,
    aTP_Lamm_amn: $o > fun_Product_unit_o ).

tff(sy_c_ATP_058Lamm__amo____,type,
    aTP_Lamm_amo: fun_se4782206882984618170_nat_o ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: $o > fun_fu5135969500543783852at_o_o ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: $o > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: $o > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: $o > fun_rat_o ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_o_fun_nat_o ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_o_fun_int_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: set_se6260736226359567993nt_int > fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: set_fu3555266933746557007_nat_o > fun_fu5135969500543783852at_o_o ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_se7249947736802449410_int_o > fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_set_nat_o > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_int_o > fun_int_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_fu5135969500543783852at_o_o > fun_fu5135969500543783852at_o_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_fu5135969500543783852at_o_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_o_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_nat_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_int_o ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: set_se6260736226359567993nt_int > fun_se3135557758678015630_int_o ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: set_fu3555266933746557007_nat_o > fun_se7198376886676107790at_o_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: set_Pr1261947904930325089at_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: set_Pr958786334691620121nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: set_list_char > fun_list_char_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: set_Pr1261947904930325089at_nat > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_se7249947736802449410_int_o > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_fu5135969500543783852at_o_o > fun_fu6295675835253126101at_o_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_Pr4796943543331071514_nat_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: set_se6260736226359567993nt_int > fun_se3135557758678015630_int_o ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: set_fu3555266933746557007_nat_o > fun_se7198376886676107790at_o_o ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: set_Pr8551490117392284871at_nat > fun_se3981709065971081742_nat_o ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: set_Pr4329608150637261639at_nat > fun_se5282722391638144526_nat_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: set_Pr8551490117392284871at_nat > fun_Pr1099658227000185396_nat_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: set_Pr4329608150637261639at_nat > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_se7249947736802449410_int_o > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_fu5135969500543783852at_o_o > fun_fu6295675835253126101at_o_o ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_Pr1099658227000185396_nat_o > fun_fu714798943701216869_nat_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_Pr3059359313727237556_nat_o > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_se7249947736802449410_int_o > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_fu5135969500543783852at_o_o > fun_fu6295675835253126101at_o_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_Pr1099658227000185396_nat_o > fun_fu714798943701216869_nat_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_Pr3059359313727237556_nat_o > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_assn_assn ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_int_int ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_fu5135969500543783852at_o_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_o_o ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_rat_o ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_nat_o ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_int_o ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: set_Pr3486312109254157132nteger > fun_fu7695273699059480439eger_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: set_Pr1312406793078187350et_nat > fun_fu8419763957888848259_nat_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: set_Pr4040394263813967220et_nat > fun_fu2611741329380808165_nat_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: set_Pr6700895561065273913nt_int > fun_fu6865642474282033866_int_o ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: set_Pr4166680992137330533nt_int > fun_fu2978539155105838238_int_o ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: set_Pr8551490117392284871at_nat > fun_mu7838234450475261998_nat_o ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: set_Pr4329608150637261639at_nat > fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: set_o > fun_na4064227270360537741_o_nat ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: set_o > fun_in6043545733737050673_o_int ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: set_rat > fun_na9140958974696438463at_nat ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: set_rat > fun_in7417575120973741523at_int ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_nat > fun_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: set_int > fun_na6463439003714840339nt_nat ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: set_int > fun_in4740055149992143399nt_int ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: set_set_nat > fun_na7542191908355352173at_nat ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: set_set_nat > fun_in8529793397280000385at_int ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_nat_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_int_nat > fun_fu1756172458508103847_int_o ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: fun_nat_int > fun_fu6128684156180612191_nat_o ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: fun_int_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: fun_nat_set_nat > fun_fu2483414175516609209_nat_o ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: fun_int_set_nat > fun_fu238269207846163293_int_o ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: fun_set_nat_nat > fun_fu69611085475581389_nat_o ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: fun_set_nat_int > fun_fu1277728218403961385_nat_o ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: fun_set_nat_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: fun_na5449828587677346466nt_int > fun_fu8767952198571339512_nat_o ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: fun_fu8419763957888848259_nat_o ).

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

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

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: nat > fun_rat_nat ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: int > fun_rat_int ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: nat > fun_list_nat_nat ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: int > fun_list_nat_int ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_nat_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_nat_int > fun_se3112267675864610456_nat_o ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_int_nat > fun_se7360890175450123928_int_o ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_int_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_set_nat_nat > fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_set_nat_int > fun_se1979679817752774754_nat_o ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_nat_set_nat > fun_se4326259930056832958_nat_o ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_int_set_nat > fun_se2081114962386387042_int_o ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_set_nat_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_se4101287982975402108nt_nat > fun_se7946864661676846469_int_o ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_as3747249873083764768_nat_o ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_int_int ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: set_se6260736226359567993nt_int > fun_se3135557758678015630_int_o ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: set_fu3555266933746557007_nat_o > fun_se7198376886676107790at_o_o ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_se5750412846314922030_nat_o ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_se5750412846314922030_nat_o ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_se5672428230568430382_nat_o ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_se5672428230568430382_nat_o ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_se5038631536132265326et_o_o ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_se5038631536132265326et_o_o ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_fi6805077330906600714_nat_o ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_fi6805077330906600714_nat_o ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: set_se6260736226359567993nt_int > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: set_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: set_fu3555266933746557007_nat_o > fun_fu6295675835253126101at_o_o ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_o_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_o_int > fun_fu6453395667394816515nt_int ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_int_o > fun_fu1481857817119020493_int_o ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_int_int > fun_fu6092417073193843705nt_int ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_Pr982841511892225912nt_int > fun_fu717549361983437348nt_int ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_in4211310926729683942nt_int > fun_fu918106826131887214nt_int ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_Pr2478283820680522243nt_int > fun_fu3898734532474705027nt_int ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_fu3275390434952296369_nat_o > fun_fu6384561554008368101_nat_o ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_na1668100129766719005at_nat > fun_Pr557098546508347797at_nat ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_na5914473681949881652_nat_o > fun_Pr3455331679158536364_nat_o ).

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: $o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: $o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_se7249947736802449410_int_o > fun_se2092758488442489134_int_o ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_set_nat_o > fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_fu5135969500543783852at_o_o > fun_fu4164567611050918574at_o_o ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: set_Pr8057934254988874055nt_int > fun_se8208212259304335032_int_o ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: set_Pr5488025237498180813et_nat > fun_se8555926724756219227_nat_o ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: set_Pr1261947904930325089at_nat > fun_se4095769911529315581_nat_o ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: set_Pr2890355626087285447_nat_o > fun_se9082258560318022882at_o_o ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: set_Pr958786334691620121nt_int > fun_se4642411815161519337_int_o ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: $o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_na4138050610385705875at_nat ).

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

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_Pr5947183769845927726_nat_o ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_se2092758488442489134_int_o ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_fu4164567611050918574at_o_o ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: set_Pr958786334691620121nt_int > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_Pr6460944699146269935_nat_o > fun_fu6295675835253126101at_o_o ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: produc3843707927480180839at_nat > fun_se5282722391638144526_nat_o ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: product_prod_nat_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: $o > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: set_Pr958786334691620121nt_int > fun_se3135557758678015630_int_o ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_Pr6460944699146269935_nat_o > fun_se7198376886676107790at_o_o ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: set_Pr958786334691620121nt_int > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_Pr6460944699146269935_nat_o > fun_fu6295675835253126101at_o_o ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: set_Pr958786334691620121nt_int > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_Pr6460944699146269935_nat_o > fun_fu6295675835253126101at_o_o ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: produc4166570645942440679at_nat > fun_Pr1099658227000185396_nat_o ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: code_integer > fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: int > fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_se7586849273513198688nteger > fun_fu5510336631606277565nteger ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_set_nat_rat > fun_fu3811435807551234611at_rat ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_set_nat_int > fun_fu3492966305048685787at_int ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_se1399197888071115401nteger > fun_fu1922436279269475843nteger ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

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

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

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

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_se4101287982975402108nt_nat > fun_fu8621421401018972867nt_nat ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_fu3504661237974713170_o_nat > fun_fu7083963787269851971_o_nat ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_nat_Code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_na2338080690716861136at_nat > fun_na1945018436707187408at_nat ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_nat_int > fun_nat_fun_int_int ).

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_Code_integer > fun_nat_Code_integer ).

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_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: code_natural > fun_Co740390889742897050atural ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_int_nat > fun_int_int ).

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

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_nat_nat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_int_int ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: code_integer > fun_na7078387940644931716nteger ).

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

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_o_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_o_nat > fun_fu318872218122388049at_nat ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_o_option_num > fun_fu2797749360310127211on_num ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_nat_o > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_nat_option_num > fun_fu5130958774822480081on_num ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_option_num_o > fun_fu2897328509465618819_nat_o ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_option_num_nat > fun_fu1502755112258691281at_nat ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_op8989173174992673105on_num > fun_fu1574286537503582443on_num ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_se2092758488442489134_int_o ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_fu4164567611050918574at_o_o ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: nat > fun_nat_rat ).

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

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

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: nat > fun_fu695563712076452518at_nat ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: nat > fun_fu695563712076452518at_nat ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_nat_fun_nat_int ).

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

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

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: rat > fun_nat_fun_rat_rat ).

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

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_nat_int ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_int_int ).

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

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

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_nat_fun_nat_int > fun_se7912642235559603452at_int ).

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

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_nat_fun_nat_int > fun_se7912642235559603452at_int ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_nat_fun_int_int > fun_se2640714827120536920at_int ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_nat_fun_int_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_nat_fun_int_int > fun_se876191773397849432nt_int ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_int_fun_nat_int > fun_se876191773397849432nt_int ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_int_fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_int_fun_nat_int > fun_se2640714827120536920at_int ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_int_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: nat > fun_nat_fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: code_natural > fun_nat_Code_natural ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_nat_Code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: fun_nat_assn > fun_nat_assn ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_nat_int > fun_na7654559027617327379at_int ).

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

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

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_nat_Code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_nat_assn > fun_na2454419004871237530n_assn ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: nat > fun_na3244735409550470327at_nat ).

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

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_nat_Code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_nat_assn > fun_nat_assn ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: int > fun_in6793278964266839919at_int ).

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

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: nat > fun_fu3959123160307459545t_assn ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: nat > fun_fu3959123160307459545t_assn ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: set_nat > fun_fu5510336631606277565nteger ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: set_nat > fun_fu3811435807551234611at_rat ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: set_nat > fun_fu5510336631606277565nteger ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: set_nat > fun_fu3811435807551234611at_rat ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: nat > fun_fu2995346897264222311t_assn ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: int > fun_fu6568868615275847455t_assn ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: nat > fun_fu2995346897264222311t_assn ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: int > fun_fu6568868615275847455t_assn ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_int_Code_integer > fun_fu6361890901329191134nteger ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: fun_nat_Code_integer > fun_fu7898940055406642654nteger ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_nat_int > fun_fu7052161110045437101at_int ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_int_nat > fun_fu1078644964449000785nt_nat ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_nat_nat > fun_fu318872218122388049at_nat ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_int_int > fun_fu7811933856372049837nt_int ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: fun_se7586849273513198688nteger > fun_fu3412595684604441950nteger ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_set_nat_rat > fun_fu1639022730973683673at_rat ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_int_o > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_nat_o > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_int_o > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_nat_o > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_nat_o > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_int_o > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_nat_o > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_int_o > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_set_nat_o > fun_fu5510336631606277565nteger ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_set_nat_o > fun_fu3811435807551234611at_rat ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: set_Pr958786334691620121nt_int > fun_se2092758488442489134_int_o ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: set_int > fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: set_Pr1261947904930325089at_nat > fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: set_nat > fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: set_nat > fun_fu927436805355353155_nat_o ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: set_int > fun_fu7493154764834273859_int_o ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: set_int > fun_fu5646853623633916995_int_o ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: set_int > fun_fu4584101141160326545at_int ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: set_nat > fun_fu8304507701009772099_nat_o ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: set_nat > fun_fu7802804735324696557nt_int ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: set_int > fun_fu9148247276982919825at_nat ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_nat_fun_int_nat > fun_int_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: set_nat > fun_fu3143578834292514029nt_nat ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: set_nat > fun_fu4073184632390761289at_nat ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: set_nat > fun_fu4073184632390761289at_nat ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: set_nat > fun_fu1834763485345162221nt_int ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: set_int > fun_fu7839431928035568017at_int ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: set_int > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: set_int > fun_fu6909826129937320757nt_int ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: set_int > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: set_int > fun_fu6909826129937320757nt_int ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: set_nat > fun_fu6212260319156258755_nat_o ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: set_nat > fun_fu6829939267127632275at_nat ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_se1179353343636554400at_nat > fun_na106483476154090400at_nat ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: set_set_nat > fun_fu1657029809911482051_nat_o ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: set_set_nat > fun_fu3378010530705604573at_nat ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: set_int > fun_fu7692897720060892355_int_o ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: set_int > fun_fu2189424204742085595at_int ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_se876191773397849432nt_int > fun_in6555721293707029592at_int ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: set_set_nat > fun_fu2875010803360890307_nat_o ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: set_set_nat > fun_fu3776778612278285441nt_int ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: set_set_nat > fun_fu3568225526390654915_nat_o ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: set_set_nat > fun_fu9114370601171615197at_nat ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: fun_na106483476154090400at_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: set_nat > fun_fu8123456035635431619_nat_o ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: set_nat > fun_fu8651063156380458899at_nat ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: set_set_nat > fun_fu1780879341396661443_nat_o ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: set_set_nat > fun_fu1817862469383530113nt_int ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: fun_in6555721293707029592at_int > fun_se876191773397849432nt_int ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: set_int > fun_fu6598766258096663491_int_o ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: set_int > fun_fu6187811966698790363at_int ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: set_int > fun_fu9148247276982919825at_nat ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: set_nat > fun_fu3143578834292514029nt_nat ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: set_nat > fun_fu4073184632390761289at_nat ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: set_nat > fun_fu4073184632390761289at_nat ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: set_nat > fun_fu8732410533422943817at_int ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: set_nat > fun_fu8732410533422943817at_int ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: set_int > fun_fu4584101141160326545at_int ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: set_nat > fun_fu7802804735324696557nt_int ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: set_nat > fun_fu1834763485345162221nt_int ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: set_int > fun_fu7839431928035568017at_int ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: set_int > fun_fu6909826129937320757nt_int ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: set_int > fun_fu6909826129937320757nt_int ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: set_nat > fun_fu6829939267127632275at_nat ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: set_set_nat > fun_fu3378010530705604573at_nat ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: set_nat > fun_fu2855959522417824915at_int ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_se7912642235559603452at_int > fun_na6839772368077139452at_int ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: set_set_nat > fun_fu7137239131779924701at_int ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: set_int > fun_fu2189424204742085595at_int ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: set_set_nat > fun_fu3776778612278285441nt_int ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: fun_se2092758488442489134_int_o ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_fu4164567611050918574at_o_o ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_li4177005666941401710eral_o ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: set_nat > fun_fu5838931486547983824_nat_o ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: set_nat > fun_fu8532066432991188095_nat_o ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: set_int > fun_fu5271299750659287320_int_o ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: set_int > fun_fu63825462011493995_int_o ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: set_int > fun_fu3473604023002724957_int_o ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: set_int > fun_fu7714913403879666293_int_o ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: set_nat > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: set_nat > fun_fu6075382464776002713_nat_o ).

tff(sy_c_ATP_058Lamm__xy____,type,
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tff(sy_c_ATP_058Lamm__xz____,type,
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tff(sy_c_ATP_058Lamm__ya____,type,
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tff(sy_c_ATP_058Lamm__yb____,type,
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tff(sy_c_ATP_058Lamm__yc____,type,
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tff(sy_c_ATP_058Lamm__yd____,type,
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tff(sy_c_ATP_058Lamm__ye____,type,
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tff(sy_c_ATP_058Lamm__yf____,type,
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tff(sy_c_ATP_058Lamm__yg____,type,
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tff(sy_c_ATP_058Lamm__yh____,type,
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tff(sy_c_ATP_058Lamm__yi____,type,
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tff(sy_c_ATP_058Lamm__yj____,type,
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tff(sy_c_ATP_058Lamm__yk____,type,
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tff(sy_c_ATP_058Lamm__yl____,type,
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tff(sy_c_ATP_058Lamm__ym____,type,
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tff(sy_c_ATP_058Lamm__yn____,type,
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tff(sy_c_ATP_058Lamm__yo____,type,
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tff(sy_c_ATP_058Lamm__yp____,type,
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tff(sy_c_ATP_058Lamm__yq____,type,
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tff(sy_c_ATP_058Lamm__yr____,type,
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tff(sy_c_ATP_058Lamm__ys____,type,
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tff(sy_c_ATP_058Lamm__yt____,type,
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tff(sy_c_ATP_058Lamm__yu____,type,
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tff(sy_c_ATP_058Lamm__yv____,type,
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tff(sy_c_ATP_058Lamm__yw____,type,
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tff(sy_c_ATP_058Lamm__yx____,type,
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tff(sy_c_ATP_058Lamm__yy____,type,
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tff(sy_c_ATP_058Lamm__yz____,type,
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tff(sy_c_ATP_058Lamm__za____,type,
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tff(sy_c_ATP_058Lamm__zb____,type,
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tff(sy_c_ATP_058Lamm__zc____,type,
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tff(sy_c_ATP_058Lamm__zd____,type,
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tff(sy_c_ATP_058Lamm__ze____,type,
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tff(sy_c_ATP_058Lamm__zf____,type,
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tff(sy_c_ATP_058Lamm__zg____,type,
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tff(sy_c_ATP_058Lamm__zh____,type,
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tff(sy_c_ATP_058Lamm__zi____,type,
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tff(sy_c_ATP_058Lamm__zj____,type,
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tff(sy_c_ATP_058Lamm__zk____,type,
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tff(sy_c_ATP_058Lamm__zl____,type,
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tff(sy_c_ATP_058Lamm__zm____,type,
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tff(sy_c_ATP_058Lamm__zn____,type,
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tff(sy_c_ATP_058Lamm__zo____,type,
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tff(sy_c_ATP_058Lamm__zp____,type,
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tff(sy_c_ATP_058Lamm__zq____,type,
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tff(sy_c_ATP_058Lamm__zr____,type,
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tff(sy_c_ATP_058Lamm__zs____,type,
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tff(sy_c_ATP_058Lamm__zt____,type,
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tff(sy_c_ATP_058Lamm__zu____,type,
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tff(sy_c_ATP_058Lamm__zv____,type,
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tff(sy_c_ATP_058Lamm__zw____,type,
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tff(sy_c_ATP_058Lamm__zx____,type,
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tff(sy_c_ATP_058Lamm__zy____,type,
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tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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tff(sy_c_Assertions_Oassn_OAbs__assn,type,
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tff(sy_c_Assertions_Oassn_ORep__assn,type,
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tff(sy_c_Assertions_Oin__range,type,
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tff(sy_c_Assertions_Oone__assn__raw,type,
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tff(sy_c_Assertions_Oproper,type,
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tff(sy_c_Assertions_OrelH,type,
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tff(sy_c_Assertions_Otimes__assn__raw,type,
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tff(sy_c_Assertions_Otimes__assn__raw__rel,type,
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tff(sy_c_Assertions_Owand__assn,type,
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tff(sy_c_Assertions_Owand__raw,type,
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tff(sy_c_Assertions_Owand__raw__rel,type,
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tff(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
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tff(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
    bNF_Cardinal_ctwo: set_Product_prod_o_o ).

tff(sy_c_BNF__Cardinal__Arithmetic_Oczero_001_Eo,type,
    bNF_Cardinal_czero_o: set_Product_prod_o_o ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001_Eo,type,
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tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Nat__Onat,type,
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tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Product____Type__Ounit,type,
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tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001_Eo,type,
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tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001t__Nat__Onat,type,
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tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
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tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001_Eo,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__Int__Oint,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__Nat__Onat,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__Rat__Orat,type,
    bNF_id_bnf_rat: fun_rat_rat ).

tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_I_Eo_M_Eo_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_I_Eo_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_I_Eo_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__Int__Oint_M_Eo_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__Int__Oint_Mt__fun_I_Eo_M_Eo_J_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__Nat__Onat_Mt__fun_I_Eo_M_Eo_J_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_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,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_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_M_Eo_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_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_Mt__fun_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_M_Eo_J_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_I_Eo_M_Eo_J_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J_J,type,
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tff(sy_c_BNF__Composition_Oid__bnf_001t__fun_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_Eo_J,type,
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tff(sy_c_BNF__Def_Oeq__onp_001t__List__Olist_It__String__Ochar_J,type,
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tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J,type,
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tff(sy_c_Misc_Opairself_001t__Int__Oint_001t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J,type,
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tff(sy_c_Misc_Orel__restrict_001_Eo,type,
    rel_restrict_o: ( set_Product_prod_o_o * set_o ) > set_Product_prod_o_o ).

tff(sy_c_Misc_Orel__restrict_001t__Int__Oint,type,
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tff(sy_c_Misc_Orel__restrict_001t__Nat__Onat,type,
    rel_restrict_nat: ( set_Pr1261947904930325089at_nat * set_nat ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_Misc_Orel__restrict_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    rel_re2197483123211154633at_nat: ( set_Pr4759807026041935047at_nat * set_Pr8551490117392284871at_nat ) > set_Pr4759807026041935047at_nat ).

tff(sy_c_Misc_Orel__restrict_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Misc_Orel__restrict_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Misc_Orel__restrict_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(sy_c_Misc_Ouncurry_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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tff(sy_c_Misc_Ouncurry_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Misc_Ouncurry_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
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tff(sy_c_Misc_Ouncurry_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,
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tff(sy_c_Multiset_Oadd__mset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Multiset_Oms__strict,type,
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tff(sy_c_Multiset_Oms__weak,type,
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tff(sy_c_Multiset_Opw__leq,type,
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tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

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

tff(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat_Onat_Opred,type,
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tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__String__Ochar_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__String__Oliteral,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__decode,type,
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tff(sy_c_Nat__Bijection_Olist__decode__rel,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,
    nat_list_encode_rel: fun_li1025265955262268170_nat_o ).

tff(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: fun_na1476630412504255114at_nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > fun_na1476630412504255114at_nat ).

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,
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tff(sy_c_Nat__Bijection_Osum__decode,type,
    nat_sum_decode: fun_na3635074868517587788at_nat ).

tff(sy_c_Nat__Bijection_Osum__encode,type,
    nat_sum_encode: fun_Su884838341992399436at_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
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tff(sy_c_Num_OBitM,type,
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tff(sy_c_Num_Oinc,type,
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tff(sy_c_Num_Onat__of__num,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__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_Osub_001t__Int__Oint,type,
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tff(sy_c_Num_Oneg__numeral__class_Osub_001t__Rat__Orat,type,
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tff(sy_c_Num_Onum_OBit0,type,
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tff(sy_c_Num_Onum_OBit1,type,
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tff(sy_c_Num_Onum_OOne,type,
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tff(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,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__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_Opred__numeral,type,
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tff(sy_c_Num_Oring__1__class_Oiszero_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Oring__1__class_Oiszero_001t__Int__Oint,type,
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tff(sy_c_Num_Oring__1__class_Oiszero_001t__Rat__Orat,type,
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tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
    case_option_int_num: ( int * fun_num_int * option_num ) > int ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
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tff(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
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tff(sy_c_Order__Relation_Owell__order__on_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Assertions__Oassn,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Filter__Ofilter_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    member9174161442226468181et_nat: ( produc3112576180687729086et_nat * set_Pr4040394263813967220et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    member3012145478770660624_nat_o: ( produc6671579610312156647_nat_o * set_Pr2890355626087285447_nat_o ) > $o ).

tff(sy_c_member_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,
    member753276645541486746nt_int: ( produc3043004577601930755nt_int * set_Pr6700895561065273913nt_int ) > $o ).

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

tff(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > fun_set_set_nat_o ).

tff(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member2340774599025711042nt_int: set_Pr958786334691620121nt_int > fun_se9106460625052465826_int_o ).

tff(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: ( set_Pr1261947904930325089at_nat * set_se7855581050983116737at_nat ) > $o ).

tff(sy_c_member_001t__String__Ochar,type,
    member_char: ( char * set_char ) > $o ).

tff(sy_c_member_001t__String__Oliteral,type,
    member_literal: literal > fun_set_literal_o ).

tff(sy_c_member_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    member7544390268018397336_nat_o: fun_Pr6460944699146269935_nat_o > fun_se7459991829846211660at_o_o ).

tff(sy_v_x,type,
    x: assn ).

tff(sy_v_y,type,
    y: assn ).

% Relevant facts (10692)
tff(fact_0_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_ass4243947775541419017_nat_o(rep_assn,X) = aa_ass4243947775541419017_nat_o(rep_assn,Y) )
    <=> ( X = Y ) ) ).

% Rep_assn_inject
tff(fact_1_Rep__assn__inverse,axiom,
    ! [X: assn] : ( aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(rep_assn,X)) = X ) ).

% Rep_assn_inverse
tff(fact_2_Abs__assn__eqI_I2_J,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Pr: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(P,H)
        <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pr),H) )
     => ( Pr = aa_fun5526454687433275355o_assn(abs_assn,P) ) ) ).

% Abs_assn_eqI(2)
tff(fact_3_Abs__assn__eqI_I1_J,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Pr: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(P,H)
        <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pr),H) )
     => ( aa_fun5526454687433275355o_assn(abs_assn,P) = Pr ) ) ).

% Abs_assn_eqI(1)
tff(fact_4_wand__assn__def,axiom,
    ! [P: assn,Q: assn] : ( wand_assn(P,Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_fun2638059832795814268_nat_o(wand_raw(aa_ass4243947775541419017_nat_o(rep_assn,P)),aa_ass4243947775541419017_nat_o(rep_assn,Q))) ) ).

% wand_assn_def
tff(fact_5_Abs__assn__inverse,axiom,
    ! [Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),collec6407654713642994138_nat_o(proper))
     => ( aa_ass4243947775541419017_nat_o(rep_assn,aa_fun5526454687433275355o_assn(abs_assn,Y)) = Y ) ) ).

% Abs_assn_inverse
tff(fact_6_inf__assn__def,axiom,
    ! [P: assn,Q: assn] : ( inf_inf_assn(P,Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(aTP_Lamm_ab(P),Q)) ) ).

% inf_assn_def
tff(fact_7_sup__assn__def,axiom,
    ! [P: assn,Q: assn] : ( sup_sup_assn(P,Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(aTP_Lamm_ac(P),Q)) ) ).

% sup_assn_def
tff(fact_8_bot__assn__def,axiom,
    bot_bot_assn = aa_fun5526454687433275355o_assn(abs_assn,aTP_Lamm_ad) ).

% bot_assn_def
tff(fact_9_Abs__assn__cases,axiom,
    ! [X: assn] :
      ~ ! [Y2: fun_Pr6460944699146269935_nat_o] :
          ( ( X = aa_fun5526454687433275355o_assn(abs_assn,Y2) )
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),collec6407654713642994138_nat_o(proper)) ) ).

% Abs_assn_cases
tff(fact_10_Abs__assn__induct,axiom,
    ! [P: fun_assn_o,X: assn] :
      ( ! [Y2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),collec6407654713642994138_nat_o(proper))
         => aa_assn_o(P,aa_fun5526454687433275355o_assn(abs_assn,Y2)) )
     => aa_assn_o(P,X) ) ).

% Abs_assn_induct
tff(fact_11_Abs__assn__inject,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),collec6407654713642994138_nat_o(proper))
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),collec6407654713642994138_nat_o(proper))
       => ( ( aa_fun5526454687433275355o_assn(abs_assn,X) = aa_fun5526454687433275355o_assn(abs_assn,Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_assn_inject
tff(fact_12_Rep__assn,axiom,
    ! [X: assn] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,X)),collec6407654713642994138_nat_o(proper)) ).

% Rep_assn
tff(fact_13_Rep__assn__cases,axiom,
    ! [Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),collec6407654713642994138_nat_o(proper))
     => ~ ! [X2: assn] : ( Y != aa_ass4243947775541419017_nat_o(rep_assn,X2) ) ) ).

% Rep_assn_cases
tff(fact_14_Rep__assn__induct,axiom,
    ! [Y: fun_Pr6460944699146269935_nat_o,P: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),collec6407654713642994138_nat_o(proper))
     => ( ! [X2: assn] : aa_fun1675921005893230629at_o_o(P,aa_ass4243947775541419017_nat_o(rep_assn,X2))
       => aa_fun1675921005893230629at_o_o(P,Y) ) ) ).

% Rep_assn_induct
tff(fact_15_times__assn__def,axiom,
    ! [P: assn,Q: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_fun2638059832795814268_nat_o(times_assn_raw(aa_ass4243947775541419017_nat_o(rep_assn,P)),aa_ass4243947775541419017_nat_o(rep_assn,Q))) ) ).

% times_assn_def
tff(fact_16_bool__assn__proper_I2_J,axiom,
    aa_fun1675921005893230629at_o_o(proper,aTP_Lamm_ad) ).

% bool_assn_proper(2)
tff(fact_17_bool__assn__proper_I3_J,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
     => ( aa_fun1675921005893230629at_o_o(proper,Q)
       => aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aTP_Lamm_ae(P),Q)) ) ) ).

% bool_assn_proper(3)
tff(fact_18_bool__assn__proper_I4_J,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
     => ( aa_fun1675921005893230629at_o_o(proper,Q)
       => aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aTP_Lamm_af(P),Q)) ) ) ).

% bool_assn_proper(4)
tff(fact_19_wand__proper,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] : aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(wand_raw(P),Q)) ).

% wand_proper
tff(fact_20_times__assn__proper,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
     => ( aa_fun1675921005893230629at_o_o(proper,Q)
       => aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(times_assn_raw(P),Q)) ) ) ).

% times_assn_proper
tff(fact_21_assn__times__comm,axiom,
    ! [P: assn,Q: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),P) ) ).

% assn_times_comm
tff(fact_22_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),R) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),R)) ) ).

% assn_times_assoc
tff(fact_23_mod__starE,axiom,
    ! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),H2)
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,A),X_1)
         => ! [H_2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,B),H_2) ) ) ).

% mod_starE
tff(fact_24_mod__starD,axiom,
    ! [A2: assn,B2: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),H2)
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,A2),H1)
          & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,B2),H22) ) ) ).

% mod_starD
tff(fact_25_inf__sup__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)) = X ) ).

% inf_sup_absorb
tff(fact_26_inf__sup__absorb,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)) = X ) ).

% inf_sup_absorb
tff(fact_27_inf__sup__absorb,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),Y)) = X ) ).

% inf_sup_absorb
tff(fact_28_inf__sup__absorb,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) = X ) ).

% inf_sup_absorb
tff(fact_29_inf__sup__absorb,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = X ) ).

% inf_sup_absorb
tff(fact_30_inf__sup__absorb,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = X ) ).

% inf_sup_absorb
tff(fact_31_inf__sup__absorb,axiom,
    ! [X: nat,Y: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) = X ) ).

% inf_sup_absorb
tff(fact_32_inf__sup__absorb,axiom,
    ! [X: int,Y: int] : ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) = X ) ).

% inf_sup_absorb
tff(fact_33_sup__inf__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = X ) ).

% sup_inf_absorb
tff(fact_34_sup__inf__absorb,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)) = X ) ).

% sup_inf_absorb
tff(fact_35_sup__inf__absorb,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)) = X ) ).

% sup_inf_absorb
tff(fact_36_sup__inf__absorb,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y)) = X ) ).

% sup_inf_absorb
tff(fact_37_sup__inf__absorb,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)) = X ) ).

% sup_inf_absorb
tff(fact_38_sup__inf__absorb,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = X ) ).

% sup_inf_absorb
tff(fact_39_sup__inf__absorb,axiom,
    ! [X: nat,Y: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)) = X ) ).

% sup_inf_absorb
tff(fact_40_sup__inf__absorb,axiom,
    ! [X: int,Y: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)) = X ) ).

% sup_inf_absorb
tff(fact_41_sup__bot__left,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,bot_bo8422036546324065075at_nat),X) = X ) ).

% sup_bot_left
tff(fact_42_sup__bot__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),X) = X ) ).

% sup_bot_left
tff(fact_43_sup__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),X) = X ) ).

% sup_bot_left
tff(fact_44_sup__bot__left,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,bot_bot_set_o),X) = X ) ).

% sup_bot_left
tff(fact_45_sup__bot__left,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),X) = X ) ).

% sup_bot_left
tff(fact_46_sup__bot__left,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),X) = X ) ).

% sup_bot_left
tff(fact_47_sup__bot__right,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),bot_bo8422036546324065075at_nat) = X ) ).

% sup_bot_right
tff(fact_48_sup__bot__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),bot_bo228742789529271731at_nat) = X ) ).

% sup_bot_right
tff(fact_49_sup__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),bot_bo2099793752762293965at_nat) = X ) ).

% sup_bot_right
tff(fact_50_sup__bot__right,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),bot_bot_set_o) = X ) ).

% sup_bot_right
tff(fact_51_sup__bot__right,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),bot_bot_set_nat) = X ) ).

% sup_bot_right
tff(fact_52_sup__bot__right,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),bot_bot_set_int) = X ) ).

% sup_bot_right
tff(fact_53_bot__eq__sup__iff,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( bot_bo8422036546324065075at_nat = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) )
    <=> ( ( X = bot_bo8422036546324065075at_nat )
        & ( Y = bot_bo8422036546324065075at_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_54_bot__eq__sup__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) )
    <=> ( ( X = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_55_bot__eq__sup__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y) )
    <=> ( ( X = bot_bo2099793752762293965at_nat )
        & ( Y = bot_bo2099793752762293965at_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_56_bot__eq__sup__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),Y) )
    <=> ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% bot_eq_sup_iff
tff(fact_57_bot__eq__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) )
    <=> ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_58_bot__eq__sup__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) )
    <=> ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
tff(fact_59_sup__eq__bot__iff,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = bot_bo8422036546324065075at_nat )
    <=> ( ( X = bot_bo8422036546324065075at_nat )
        & ( Y = bot_bo8422036546324065075at_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_60_sup__eq__bot__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = bot_bo228742789529271731at_nat )
    <=> ( ( X = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_61_sup__eq__bot__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y) = bot_bo2099793752762293965at_nat )
    <=> ( ( X = bot_bo2099793752762293965at_nat )
        & ( Y = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_62_sup__eq__bot__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),Y) = bot_bot_set_o )
    <=> ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% sup_eq_bot_iff
tff(fact_63_sup__eq__bot__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = bot_bot_set_nat )
    <=> ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_64_sup__eq__bot__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = bot_bot_set_int )
    <=> ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
tff(fact_65_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = bot_bo8422036546324065075at_nat )
    <=> ( ( A = bot_bo8422036546324065075at_nat )
        & ( B = bot_bo8422036546324065075at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_66_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = bot_bo228742789529271731at_nat )
    <=> ( ( A = bot_bo228742789529271731at_nat )
        & ( B = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_67_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B) = bot_bo2099793752762293965at_nat )
    <=> ( ( A = bot_bo2099793752762293965at_nat )
        & ( B = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_68_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B) = bot_bot_set_o )
    <=> ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_69_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = bot_bot_set_nat )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_70_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = bot_bot_set_int )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_71_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,bot_bo8422036546324065075at_nat),A) = A ) ).

% sup_bot.left_neutral
tff(fact_72_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),A) = A ) ).

% sup_bot.left_neutral
tff(fact_73_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),A) = A ) ).

% sup_bot.left_neutral
tff(fact_74_sup__bot_Oleft__neutral,axiom,
    ! [A: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,bot_bot_set_o),A) = A ) ).

% sup_bot.left_neutral
tff(fact_75_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),A) = A ) ).

% sup_bot.left_neutral
tff(fact_76_sup__bot_Oleft__neutral,axiom,
    ! [A: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),A) = A ) ).

% sup_bot.left_neutral
tff(fact_77_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( bot_bo8422036546324065075at_nat = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) )
    <=> ( ( A = bot_bo8422036546324065075at_nat )
        & ( B = bot_bo8422036546324065075at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_78_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) )
    <=> ( ( A = bot_bo228742789529271731at_nat )
        & ( B = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_79_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B) )
    <=> ( ( A = bot_bo2099793752762293965at_nat )
        & ( B = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_80_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B) )
    <=> ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_81_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_82_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_83_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),bot_bo8422036546324065075at_nat) = A ) ).

% sup_bot.right_neutral
tff(fact_84_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),bot_bo228742789529271731at_nat) = A ) ).

% sup_bot.right_neutral
tff(fact_85_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),bot_bo2099793752762293965at_nat) = A ) ).

% sup_bot.right_neutral
tff(fact_86_sup__bot_Oright__neutral,axiom,
    ! [A: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),bot_bot_set_o) = A ) ).

% sup_bot.right_neutral
tff(fact_87_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),bot_bot_set_nat) = A ) ).

% sup_bot.right_neutral
tff(fact_88_sup__bot_Oright__neutral,axiom,
    ! [A: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),bot_bot_set_int) = A ) ).

% sup_bot.right_neutral
tff(fact_89_inf__bot__left,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,bot_bo3516217578726220278_int_o),X) = bot_bo3516217578726220278_int_o ) ).

% inf_bot_left
tff(fact_90_inf__bot__left,axiom,
    ! [X: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,bot_bo8860746597864711612char_o),X) = bot_bo8860746597864711612char_o ) ).

% inf_bot_left
tff(fact_91_inf__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),X) = bot_bo2099793752762293965at_nat ) ).

% inf_bot_left
tff(fact_92_inf__bot__left,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,bot_bot_set_o),X) = bot_bot_set_o ) ).

% inf_bot_left
tff(fact_93_inf__bot__left,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ) ).

% inf_bot_left
tff(fact_94_inf__bot__left,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),X) = bot_bot_set_int ) ).

% inf_bot_left
tff(fact_95_inf__bot__right,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),bot_bo3516217578726220278_int_o) = bot_bo3516217578726220278_int_o ) ).

% inf_bot_right
tff(fact_96_inf__bot__right,axiom,
    ! [X: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),bot_bo8860746597864711612char_o) = bot_bo8860746597864711612char_o ) ).

% inf_bot_right
tff(fact_97_inf__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ) ).

% inf_bot_right
tff(fact_98_inf__bot__right,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),bot_bot_set_o) = bot_bot_set_o ) ).

% inf_bot_right
tff(fact_99_inf__bot__right,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ) ).

% inf_bot_right
tff(fact_100_inf__bot__right,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),bot_bot_set_int) = bot_bot_set_int ) ).

% inf_bot_right
tff(fact_101_inf__apply,axiom,
    ! [F: fun_Pr3238374608701176162_int_o,G: fun_Pr3238374608701176162_int_o,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,F),G),X)
    <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_Pro4592068669816069595_int_o(F,X)),aa_Pro4592068669816069595_int_o(G,X)) ) ).

% inf_apply
tff(fact_102_inf__apply,axiom,
    ! [F: fun_list_char_o,G: fun_list_char_o,X: list_char] :
      ( aa_list_char_o(aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,F),G),X)
    <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_list_char_o(F,X)),aa_list_char_o(G,X)) ) ).

% inf_apply
tff(fact_103_inf__right__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) ) ).

% inf_right_idem
tff(fact_104_inf__right__idem,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ) ).

% inf_right_idem
tff(fact_105_inf__right__idem,axiom,
    ! [X: nat,Y: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) ) ).

% inf_right_idem
tff(fact_106_inf__right__idem,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Y) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) ) ).

% inf_right_idem
tff(fact_107_inf__right__idem,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)),Y) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y) ) ).

% inf_right_idem
tff(fact_108_inf_Oright__idem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ).

% inf.right_idem
tff(fact_109_inf_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ).

% inf.right_idem
tff(fact_110_inf_Oright__idem,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ).

% inf.right_idem
tff(fact_111_inf_Oright__idem,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),B) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) ) ).

% inf.right_idem
tff(fact_112_inf_Oright__idem,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)),B) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) ) ).

% inf.right_idem
tff(fact_113_inf__left__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) ) ).

% inf_left_idem
tff(fact_114_inf__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ) ).

% inf_left_idem
tff(fact_115_inf__left__idem,axiom,
    ! [X: nat,Y: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) ) ).

% inf_left_idem
tff(fact_116_inf__left__idem,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) ) ).

% inf_left_idem
tff(fact_117_inf__left__idem,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y) ) ).

% inf_left_idem
tff(fact_118_inf_Oleft__idem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ).

% inf.left_idem
tff(fact_119_inf_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ).

% inf.left_idem
tff(fact_120_inf_Oleft__idem,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ).

% inf.left_idem
tff(fact_121_inf_Oleft__idem,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) ) ).

% inf.left_idem
tff(fact_122_inf_Oleft__idem,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) ) ).

% inf.left_idem
tff(fact_123_inf__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),X) = X ) ).

% inf_idem
tff(fact_124_inf__idem,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),X) = X ) ).

% inf_idem
tff(fact_125_inf__idem,axiom,
    ! [X: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),X) = X ) ).

% inf_idem
tff(fact_126_inf__idem,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),X) = X ) ).

% inf_idem
tff(fact_127_inf__idem,axiom,
    ! [X: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),X) = X ) ).

% inf_idem
tff(fact_128_inf_Oidem,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),A) = A ) ).

% inf.idem
tff(fact_129_inf_Oidem,axiom,
    ! [A: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),A) = A ) ).

% inf.idem
tff(fact_130_inf_Oidem,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),A) = A ) ).

% inf.idem
tff(fact_131_inf_Oidem,axiom,
    ! [A: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),A) = A ) ).

% inf.idem
tff(fact_132_inf_Oidem,axiom,
    ! [A: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),A) = A ) ).

% inf.idem
tff(fact_133_mem__Collect__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int,P: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),collec5210948495886036740nt_int(P))
    <=> aa_set6170482117830564987_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_134_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(A),collect_set_nat(P))
    <=> aa_set_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_135_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_o] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),collect_nat(P))
    <=> aa_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_136_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_o] :
      ( aa_set_int_o(member_int(A),collect_int(P))
    <=> aa_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_137_mem__Collect__eq,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,P: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),collec6407654713642994138_nat_o(P))
    <=> aa_fun1675921005893230629at_o_o(P,A) ) ).

% mem_Collect_eq
tff(fact_138_Collect__mem__eq,axiom,
    ! [A2: set_se6260736226359567993nt_int] : ( collec5210948495886036740nt_int(aTP_Lamm_ag(A2)) = A2 ) ).

% Collect_mem_eq
tff(fact_139_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : ( collect_set_nat(aTP_Lamm_ah(A2)) = A2 ) ).

% Collect_mem_eq
tff(fact_140_Collect__mem__eq,axiom,
    ! [A2: set_nat] : ( collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ai,A2)) = A2 ) ).

% Collect_mem_eq
tff(fact_141_Collect__mem__eq,axiom,
    ! [A2: set_int] : ( collect_int(aTP_Lamm_aj(A2)) = A2 ) ).

% Collect_mem_eq
tff(fact_142_Collect__mem__eq,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] : ( collec6407654713642994138_nat_o(aTP_Lamm_ak(A2)) = A2 ) ).

% Collect_mem_eq
tff(fact_143_Collect__cong,axiom,
    ! [P: fun_se7249947736802449410_int_o,Q: fun_se7249947736802449410_int_o] :
      ( ! [X2: set_Pr958786334691620121nt_int] :
          ( aa_set6170482117830564987_int_o(P,X2)
        <=> aa_set6170482117830564987_int_o(Q,X2) )
     => ( collec5210948495886036740nt_int(P) = collec5210948495886036740nt_int(Q) ) ) ).

% Collect_cong
tff(fact_144_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X2: set_nat] :
          ( aa_set_nat_o(P,X2)
        <=> aa_set_nat_o(Q,X2) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_145_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(P,X2)
        <=> aa_nat_o(Q,X2) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_146_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( aa_int_o(P,X2)
        <=> aa_int_o(Q,X2) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_147_Collect__cong,axiom,
    ! [P: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_fun1675921005893230629at_o_o(P,X2)
        <=> aa_fun1675921005893230629at_o_o(Q,X2) )
     => ( collec6407654713642994138_nat_o(P) = collec6407654713642994138_nat_o(Q) ) ) ).

% Collect_cong
tff(fact_148_sup_Oright__idem,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)),B) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) ) ).

% sup.right_idem
tff(fact_149_sup_Oright__idem,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),B) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ) ).

% sup.right_idem
tff(fact_150_sup_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ).

% sup.right_idem
tff(fact_151_sup_Oright__idem,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ).

% sup.right_idem
tff(fact_152_sup_Oright__idem,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ).

% sup.right_idem
tff(fact_153_sup__left__idem,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) ) ).

% sup_left_idem
tff(fact_154_sup__left__idem,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) ) ).

% sup_left_idem
tff(fact_155_sup__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) ) ).

% sup_left_idem
tff(fact_156_sup__left__idem,axiom,
    ! [X: nat,Y: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) ) ).

% sup_left_idem
tff(fact_157_sup__left__idem,axiom,
    ! [X: int,Y: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) ) ).

% sup_left_idem
tff(fact_158_sup_Oleft__idem,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) ) ).

% sup.left_idem
tff(fact_159_sup_Oleft__idem,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ) ).

% sup.left_idem
tff(fact_160_sup_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ).

% sup.left_idem
tff(fact_161_sup_Oleft__idem,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ).

% sup.left_idem
tff(fact_162_sup_Oleft__idem,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ).

% sup.left_idem
tff(fact_163_sup__idem,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),X) = X ) ).

% sup_idem
tff(fact_164_sup__idem,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),X) = X ) ).

% sup_idem
tff(fact_165_sup__idem,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),X) = X ) ).

% sup_idem
tff(fact_166_sup__idem,axiom,
    ! [X: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),X) = X ) ).

% sup_idem
tff(fact_167_sup__idem,axiom,
    ! [X: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),X) = X ) ).

% sup_idem
tff(fact_168_sup_Oidem,axiom,
    ! [A: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),A) = A ) ).

% sup.idem
tff(fact_169_sup_Oidem,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),A) = A ) ).

% sup.idem
tff(fact_170_sup_Oidem,axiom,
    ! [A: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),A) = A ) ).

% sup.idem
tff(fact_171_sup_Oidem,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),A) = A ) ).

% sup.idem
tff(fact_172_sup_Oidem,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),A) = A ) ).

% sup.idem
tff(fact_173_inf__fun__def,axiom,
    ! [F: fun_Pr3238374608701176162_int_o,G: fun_Pr3238374608701176162_int_o,X3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,F),G),X3)
    <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_Pro4592068669816069595_int_o(F,X3)),aa_Pro4592068669816069595_int_o(G,X3)) ) ).

% inf_fun_def
tff(fact_174_inf__fun__def,axiom,
    ! [F: fun_list_char_o,G: fun_list_char_o,X3: list_char] :
      ( aa_list_char_o(aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,F),G),X3)
    <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_list_char_o(F,X3)),aa_list_char_o(G,X3)) ) ).

% inf_fun_def
tff(fact_175_inf__left__commute,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ) ).

% inf_left_commute
tff(fact_176_inf__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ) ).

% inf_left_commute
tff(fact_177_inf__left__commute,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z)) ) ).

% inf_left_commute
tff(fact_178_inf__left__commute,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z)) ) ).

% inf_left_commute
tff(fact_179_inf__left__commute,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),Z)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Z)) ) ).

% inf_left_commute
tff(fact_180_inf_Oleft__commute,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),C)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) ) ).

% inf.left_commute
tff(fact_181_inf_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ) ).

% inf.left_commute
tff(fact_182_inf_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C)) ) ).

% inf.left_commute
tff(fact_183_inf_Oleft__commute,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),C)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C)) ) ).

% inf.left_commute
tff(fact_184_inf_Oleft__commute,axiom,
    ! [B: fun_list_char_o,A: fun_list_char_o,C: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,B),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),C)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,B),C)) ) ).

% inf.left_commute
tff(fact_185_inf__commute,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X) ) ).

% inf_commute
tff(fact_186_inf__commute,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X) ) ).

% inf_commute
tff(fact_187_inf__commute,axiom,
    ! [X: nat,Y: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),X) ) ).

% inf_commute
tff(fact_188_inf__commute,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),X) ) ).

% inf_commute
tff(fact_189_inf__commute,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),X) ) ).

% inf_commute
tff(fact_190_inf_Ocommute,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),A) ) ).

% inf.commute
tff(fact_191_inf_Ocommute,axiom,
    ! [A: set_nat,B: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),A) ) ).

% inf.commute
tff(fact_192_inf_Ocommute,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),A) ) ).

% inf.commute
tff(fact_193_inf_Ocommute,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),A) ) ).

% inf.commute
tff(fact_194_inf_Ocommute,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,B),A) ) ).

% inf.commute
tff(fact_195_inf__assoc,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Z) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) ) ).

% inf_assoc
tff(fact_196_inf__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ) ).

% inf_assoc
tff(fact_197_inf__assoc,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) ) ).

% inf_assoc
tff(fact_198_inf__assoc,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Z) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) ) ).

% inf_assoc
tff(fact_199_inf__assoc,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)),Z) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),Z)) ) ).

% inf_assoc
tff(fact_200_inf_Oassoc,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) ) ).

% inf.assoc
tff(fact_201_inf_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ) ).

% inf.assoc
tff(fact_202_inf_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C)) ) ).

% inf.assoc
tff(fact_203_inf_Oassoc,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),C) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C)) ) ).

% inf.assoc
tff(fact_204_inf_Oassoc,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o,C: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)),C) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,B),C)) ) ).

% inf.assoc
tff(fact_205_inf__sup__aci_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X) ) ).

% inf_sup_aci(1)
tff(fact_206_inf__sup__aci_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X) ) ).

% inf_sup_aci(1)
tff(fact_207_inf__sup__aci_I1_J,axiom,
    ! [X: nat,Y: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),X) ) ).

% inf_sup_aci(1)
tff(fact_208_inf__sup__aci_I1_J,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),X) ) ).

% inf_sup_aci(1)
tff(fact_209_inf__sup__aci_I1_J,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),X) ) ).

% inf_sup_aci(1)
tff(fact_210_inf__sup__aci_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Z) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) ) ).

% inf_sup_aci(2)
tff(fact_211_inf__sup__aci_I2_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ) ).

% inf_sup_aci(2)
tff(fact_212_inf__sup__aci_I2_J,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) ) ).

% inf_sup_aci(2)
tff(fact_213_inf__sup__aci_I2_J,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Z) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) ) ).

% inf_sup_aci(2)
tff(fact_214_inf__sup__aci_I2_J,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)),Z) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),Z)) ) ).

% inf_sup_aci(2)
tff(fact_215_inf__sup__aci_I3_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ) ).

% inf_sup_aci(3)
tff(fact_216_inf__sup__aci_I3_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ) ).

% inf_sup_aci(3)
tff(fact_217_inf__sup__aci_I3_J,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z)) ) ).

% inf_sup_aci(3)
tff(fact_218_inf__sup__aci_I3_J,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z)) ) ).

% inf_sup_aci(3)
tff(fact_219_inf__sup__aci_I3_J,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),Z)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Z)) ) ).

% inf_sup_aci(3)
tff(fact_220_inf__sup__aci_I4_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) ) ).

% inf_sup_aci(4)
tff(fact_221_inf__sup__aci_I4_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ) ).

% inf_sup_aci(4)
tff(fact_222_inf__sup__aci_I4_J,axiom,
    ! [X: nat,Y: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) ) ).

% inf_sup_aci(4)
tff(fact_223_inf__sup__aci_I4_J,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) ) ).

% inf_sup_aci(4)
tff(fact_224_inf__sup__aci_I4_J,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y) ) ).

% inf_sup_aci(4)
tff(fact_225_sup__left__commute,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Z)) ) ).

% sup_left_commute
tff(fact_226_sup__left__commute,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z)) ) ).

% sup_left_commute
tff(fact_227_sup__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ) ).

% sup_left_commute
tff(fact_228_sup__left__commute,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z)) ) ).

% sup_left_commute
tff(fact_229_sup__left__commute,axiom,
    ! [X: int,Y: int,Z: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z)) ) ).

% sup_left_commute
tff(fact_230_sup_Oleft__commute,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),C)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),C)) ) ).

% sup.left_commute
tff(fact_231_sup_Oleft__commute,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),C)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)) ) ).

% sup.left_commute
tff(fact_232_sup_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) ) ).

% sup.left_commute
tff(fact_233_sup_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)) ) ).

% sup.left_commute
tff(fact_234_sup_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,B),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),C)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)) ) ).

% sup.left_commute
tff(fact_235_sup__commute,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),X) ) ).

% sup_commute
tff(fact_236_sup__commute,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X) ) ).

% sup_commute
tff(fact_237_sup__commute,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X) ) ).

% sup_commute
tff(fact_238_sup__commute,axiom,
    ! [X: nat,Y: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),X) ) ).

% sup_commute
tff(fact_239_sup__commute,axiom,
    ! [X: int,Y: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),X) ) ).

% sup_commute
tff(fact_240_sup_Ocommute,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),A) ) ).

% sup.commute
tff(fact_241_sup_Ocommute,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),A) ) ).

% sup.commute
tff(fact_242_sup_Ocommute,axiom,
    ! [A: set_nat,B: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),A) ) ).

% sup.commute
tff(fact_243_sup_Ocommute,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),A) ) ).

% sup.commute
tff(fact_244_sup_Ocommute,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = aa_int_int(aa_int_fun_int_int(sup_sup_int,B),A) ) ).

% sup.commute
tff(fact_245_sup__assoc,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)),Z) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z)) ) ).

% sup_assoc
tff(fact_246_sup__assoc,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),Z) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) ) ).

% sup_assoc
tff(fact_247_sup__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ) ).

% sup_assoc
tff(fact_248_sup__assoc,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) ) ).

% sup_assoc
tff(fact_249_sup__assoc,axiom,
    ! [X: int,Y: int,Z: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)) ) ).

% sup_assoc
tff(fact_250_sup_Oassoc,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)),C) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),C)) ) ).

% sup.assoc
tff(fact_251_sup_Oassoc,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),C) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)) ) ).

% sup.assoc
tff(fact_252_sup_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) ) ).

% sup.assoc
tff(fact_253_sup_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)) ) ).

% sup.assoc
tff(fact_254_sup_Oassoc,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)) ) ).

% sup.assoc
tff(fact_255_inf__sup__aci_I5_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),X) ) ).

% inf_sup_aci(5)
tff(fact_256_inf__sup__aci_I5_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X) ) ).

% inf_sup_aci(5)
tff(fact_257_inf__sup__aci_I5_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X) ) ).

% inf_sup_aci(5)
tff(fact_258_inf__sup__aci_I5_J,axiom,
    ! [X: nat,Y: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),X) ) ).

% inf_sup_aci(5)
tff(fact_259_inf__sup__aci_I5_J,axiom,
    ! [X: int,Y: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),X) ) ).

% inf_sup_aci(5)
tff(fact_260_inf__sup__aci_I6_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)),Z) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z)) ) ).

% inf_sup_aci(6)
tff(fact_261_inf__sup__aci_I6_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),Z) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) ) ).

% inf_sup_aci(6)
tff(fact_262_inf__sup__aci_I6_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ) ).

% inf_sup_aci(6)
tff(fact_263_inf__sup__aci_I6_J,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) ) ).

% inf_sup_aci(6)
tff(fact_264_inf__sup__aci_I6_J,axiom,
    ! [X: int,Y: int,Z: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)) ) ).

% inf_sup_aci(6)
tff(fact_265_inf__sup__aci_I7_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Z)) ) ).

% inf_sup_aci(7)
tff(fact_266_inf__sup__aci_I7_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z)) ) ).

% inf_sup_aci(7)
tff(fact_267_inf__sup__aci_I7_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ) ).

% inf_sup_aci(7)
tff(fact_268_inf__sup__aci_I7_J,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z)) ) ).

% inf_sup_aci(7)
tff(fact_269_inf__sup__aci_I7_J,axiom,
    ! [X: int,Y: int,Z: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z)) ) ).

% inf_sup_aci(7)
tff(fact_270_inf__sup__aci_I8_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) ) ).

% inf_sup_aci(8)
tff(fact_271_inf__sup__aci_I8_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) ) ).

% inf_sup_aci(8)
tff(fact_272_inf__sup__aci_I8_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) ) ).

% inf_sup_aci(8)
tff(fact_273_inf__sup__aci_I8_J,axiom,
    ! [X: nat,Y: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) ) ).

% inf_sup_aci(8)
tff(fact_274_inf__sup__aci_I8_J,axiom,
    ! [X: int,Y: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) ) ).

% inf_sup_aci(8)
tff(fact_275_sup__inf__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Z),X)) ) ).

% sup_inf_distrib2
tff(fact_276_sup__inf__distrib2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)),X) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),X)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Z),X)) ) ).

% sup_inf_distrib2
tff(fact_277_sup__inf__distrib2,axiom,
    ! [Y: fun_list_char_o,Z: fun_list_char_o,X: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),Z)),X) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,Y),X)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,Z),X)) ) ).

% sup_inf_distrib2
tff(fact_278_sup__inf__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),Z)),X) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),X)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Z),X)) ) ).

% sup_inf_distrib2
tff(fact_279_sup__inf__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)),X) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Z),X)) ) ).

% sup_inf_distrib2
tff(fact_280_sup__inf__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z),X)) ) ).

% sup_inf_distrib2
tff(fact_281_sup__inf__distrib2,axiom,
    ! [Y: nat,Z: nat,X: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)),X) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),X)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Z),X)) ) ).

% sup_inf_distrib2
tff(fact_282_sup__inf__distrib2,axiom,
    ! [Y: int,Z: int,X: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)),X) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),X)),aa_int_int(aa_int_fun_int_int(sup_sup_int,Z),X)) ) ).

% sup_inf_distrib2
tff(fact_283_sup__inf__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z)) ) ).

% sup_inf_distrib1
tff(fact_284_sup__inf__distrib1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Z)) ) ).

% sup_inf_distrib1
tff(fact_285_sup__inf__distrib1,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),Z)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),Y)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),Z)) ) ).

% sup_inf_distrib1
tff(fact_286_sup__inf__distrib1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),Z)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Z)) ) ).

% sup_inf_distrib1
tff(fact_287_sup__inf__distrib1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z)) ) ).

% sup_inf_distrib1
tff(fact_288_sup__inf__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ) ).

% sup_inf_distrib1
tff(fact_289_sup__inf__distrib1,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z)) ) ).

% sup_inf_distrib1
tff(fact_290_sup__inf__distrib1,axiom,
    ! [X: int,Y: int,Z: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z)) ) ).

% sup_inf_distrib1
tff(fact_291_inf__sup__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Z),X)) ) ).

% inf_sup_distrib2
tff(fact_292_inf__sup__distrib2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z)),X) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),X)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Z),X)) ) ).

% inf_sup_distrib2
tff(fact_293_inf__sup__distrib2,axiom,
    ! [Y: fun_list_char_o,Z: fun_list_char_o,X: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,Y),Z)),X) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),X)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Z),X)) ) ).

% inf_sup_distrib2
tff(fact_294_inf__sup__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z)),X) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),X)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Z),X)) ) ).

% inf_sup_distrib2
tff(fact_295_inf__sup__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)),X) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Z),X)) ) ).

% inf_sup_distrib2
tff(fact_296_inf__sup__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z),X)) ) ).

% inf_sup_distrib2
tff(fact_297_inf__sup__distrib2,axiom,
    ! [Y: nat,Z: nat,X: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)),X) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),X)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Z),X)) ) ).

% inf_sup_distrib2
tff(fact_298_inf__sup__distrib2,axiom,
    ! [Y: int,Z: int,X: int] : ( aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)),X) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),X)),aa_int_int(aa_int_fun_int_int(inf_inf_int,Z),X)) ) ).

% inf_sup_distrib2
tff(fact_299_inf__sup__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ) ).

% inf_sup_distrib1
tff(fact_300_inf__sup__distrib1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z)) ) ).

% inf_sup_distrib1
tff(fact_301_inf__sup__distrib1,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,Y),Z)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Z)) ) ).

% inf_sup_distrib1
tff(fact_302_inf__sup__distrib1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Z)) ) ).

% inf_sup_distrib1
tff(fact_303_inf__sup__distrib1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z)) ) ).

% inf_sup_distrib1
tff(fact_304_inf__sup__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ) ).

% inf_sup_distrib1
tff(fact_305_inf__sup__distrib1,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z)) ) ).

% inf_sup_distrib1
tff(fact_306_inf__sup__distrib1,axiom,
    ! [X: int,Y: int,Z: int] : ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Z)) ) ).

% inf_sup_distrib1
tff(fact_307_distrib__imp2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y2),Z2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),Y2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),Z2)) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_308_distrib__imp2,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: fun_Pr3238374608701176162_int_o,Y2: fun_Pr3238374608701176162_int_o,Z2: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X2),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y2),Z2)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X2),Y2)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X2),Z2)) )
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z)) ) ) ).

% distrib_imp2
tff(fact_309_distrib__imp2,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] :
      ( ! [X2: fun_list_char_o,Y2: fun_list_char_o,Z2: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X2),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y2),Z2)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X2),Y2)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X2),Z2)) )
     => ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,Y),Z)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Z)) ) ) ).

% distrib_imp2
tff(fact_310_distrib__imp2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ! [X2: set_Pr8551490117392284871at_nat,Y2: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y2),Z2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X2),Y2)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X2),Z2)) )
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_311_distrib__imp2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y2),Z2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X2),Y2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X2),Z2)) )
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_312_distrib__imp2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y2),Z2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Y2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Z2)) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_313_distrib__imp2,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ! [X2: nat,Y2: nat,Z2: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y2),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),Z2)) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_314_distrib__imp2,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ! [X2: int,Y2: int,Z2: int] : ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X2),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y2),Z2)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X2),Y2)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X2),Z2)) )
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Z)) ) ) ).

% distrib_imp2
tff(fact_315_distrib__imp1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y2),Z2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),Y2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),Z2)) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_316_distrib__imp1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: fun_Pr3238374608701176162_int_o,Y2: fun_Pr3238374608701176162_int_o,Z2: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X2),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y2),Z2)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X2),Y2)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X2),Z2)) )
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Z)) ) ) ).

% distrib_imp1
tff(fact_317_distrib__imp1,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] :
      ( ! [X2: fun_list_char_o,Y2: fun_list_char_o,Z2: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X2),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,Y2),Z2)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X2),Y2)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X2),Z2)) )
     => ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),Z)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),Y)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),Z)) ) ) ).

% distrib_imp1
tff(fact_318_distrib__imp1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ! [X2: set_Pr8551490117392284871at_nat,Y2: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y2),Z2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X2),Y2)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X2),Z2)) )
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),Z)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_319_distrib__imp1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y2),Z2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X2),Y2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X2),Z2)) )
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_320_distrib__imp1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y2),Z2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Y2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Z2)) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_321_distrib__imp1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ! [X2: nat,Y2: nat,Z2: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y2),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),Z2)) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_322_distrib__imp1,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ! [X2: int,Y2: int,Z2: int] : ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X2),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y2),Z2)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X2),Y2)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X2),Z2)) )
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z)) ) ) ).

% distrib_imp1
tff(fact_323_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),bot_bo3516217578726220278_int_o) = bot_bo3516217578726220278_int_o ) ).

% boolean_algebra.conj_zero_right
tff(fact_324_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),bot_bo8860746597864711612char_o) = bot_bo8860746597864711612char_o ) ).

% boolean_algebra.conj_zero_right
tff(fact_325_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_right
tff(fact_326_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),bot_bot_set_o) = bot_bot_set_o ) ).

% boolean_algebra.conj_zero_right
tff(fact_327_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_right
tff(fact_328_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),bot_bot_set_int) = bot_bot_set_int ) ).

% boolean_algebra.conj_zero_right
tff(fact_329_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,bot_bo3516217578726220278_int_o),X) = bot_bo3516217578726220278_int_o ) ).

% boolean_algebra.conj_zero_left
tff(fact_330_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,bot_bo8860746597864711612char_o),X) = bot_bo8860746597864711612char_o ) ).

% boolean_algebra.conj_zero_left
tff(fact_331_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),X) = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_left
tff(fact_332_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,bot_bot_set_o),X) = bot_bot_set_o ) ).

% boolean_algebra.conj_zero_left
tff(fact_333_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_left
tff(fact_334_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),X) = bot_bot_set_int ) ).

% boolean_algebra.conj_zero_left
tff(fact_335_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Z),X)) ) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_336_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)),X) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),X)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Z),X)) ) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_337_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: fun_list_char_o,Z: fun_list_char_o,X: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),Z)),X) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,Y),X)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,Z),X)) ) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_338_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),Z)),X) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),X)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Z),X)) ) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_339_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)),X) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Z),X)) ) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_340_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z),X)) ) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_341_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Z),X)) ) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_342_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z)),X) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),X)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Z),X)) ) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_343_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: fun_list_char_o,Z: fun_list_char_o,X: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,Y),Z)),X) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),X)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Z),X)) ) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_344_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z)),X) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),X)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Z),X)) ) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_345_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)),X) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Z),X)) ) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_346_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z),X)) ) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_347_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z)) ) ).

% boolean_algebra.disj_conj_distrib
tff(fact_348_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Z)) ) ).

% boolean_algebra.disj_conj_distrib
tff(fact_349_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),Z)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),Y)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),Z)) ) ).

% boolean_algebra.disj_conj_distrib
tff(fact_350_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),Z)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Z)) ) ).

% boolean_algebra.disj_conj_distrib
tff(fact_351_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z)) ) ).

% boolean_algebra.disj_conj_distrib
tff(fact_352_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ) ).

% boolean_algebra.disj_conj_distrib
tff(fact_353_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ) ).

% boolean_algebra.conj_disj_distrib
tff(fact_354_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z)) ) ).

% boolean_algebra.conj_disj_distrib
tff(fact_355_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,Y),Z)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Z)) ) ).

% boolean_algebra.conj_disj_distrib
tff(fact_356_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Z)) ) ).

% boolean_algebra.conj_disj_distrib
tff(fact_357_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z)) ) ).

% boolean_algebra.conj_disj_distrib
tff(fact_358_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ) ).

% boolean_algebra.conj_disj_distrib
tff(fact_359_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),bot_bo8422036546324065075at_nat) = X ) ).

% boolean_algebra.disj_zero_right
tff(fact_360_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),bot_bo228742789529271731at_nat) = X ) ).

% boolean_algebra.disj_zero_right
tff(fact_361_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),bot_bo2099793752762293965at_nat) = X ) ).

% boolean_algebra.disj_zero_right
tff(fact_362_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),bot_bot_set_o) = X ) ).

% boolean_algebra.disj_zero_right
tff(fact_363_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),bot_bot_set_nat) = X ) ).

% boolean_algebra.disj_zero_right
tff(fact_364_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),bot_bot_set_int) = X ) ).

% boolean_algebra.disj_zero_right
tff(fact_365_one__assn__proper,axiom,
    aa_fun1675921005893230629at_o_o(proper,one_assn_raw) ).

% one_assn_proper
tff(fact_366_type__definition__assn,axiom,
    aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(rep_assn,abs_assn),collec6407654713642994138_nat_o(proper)) ).

% type_definition_assn
tff(fact_367_boolean__algebra__cancel_Osup2,axiom,
    ! [B2: set_Pr8551490117392284871at_nat,K: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( B2 = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,K),B) )
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,K),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_368_boolean__algebra__cancel_Osup2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,K: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( B2 = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,K),B) )
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,K),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_369_boolean__algebra__cancel_Osup2,axiom,
    ! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
      ( ( B2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_370_boolean__algebra__cancel_Osup2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_371_boolean__algebra__cancel_Osup2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(sup_sup_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2) = aa_int_int(aa_int_fun_int_int(sup_sup_int,K),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_372_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,K: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( A2 = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,K),A) )
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,K),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_373_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,K: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( A2 = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,K),A) )
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,K),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_374_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
      ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),A) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_375_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_376_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(sup_sup_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B) = aa_int_int(aa_int_fun_int_int(sup_sup_int,K),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_377_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),A) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_378_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
      ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),A) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_379_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_380_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: fun_Pr3238374608701176162_int_o,K: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( ( A2 = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,K),A) )
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A2),B) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,K),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_381_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: fun_list_char_o,K: fun_list_char_o,A: fun_list_char_o,B: fun_list_char_o] :
      ( ( A2 = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,K),A) )
     => ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A2),B) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,K),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_382_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( B2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),B) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_383_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
      ( ( B2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_384_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_385_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: fun_Pr3238374608701176162_int_o,K: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( ( B2 = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,K),B) )
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B2) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,K),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_386_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: fun_list_char_o,K: fun_list_char_o,B: fun_list_char_o,A: fun_list_char_o] :
      ( ( B2 = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,K),B) )
     => ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B2) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,K),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_387_type__definition_ORep,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A2: set_o,X: product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A2)
     => aa_set_o_o(member_o(aa_Product_unit_o(Rep,X)),A2) ) ).

% type_definition.Rep
tff(fact_388_type__definition_ORep,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A2: set_fu3555266933746557007_nat_o,X: assn] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A2)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_ass4243947775541419017_nat_o(Rep,X)),A2) ) ).

% type_definition.Rep
tff(fact_389_type__definition_ORep,axiom,
    ! [Rep: fun_li7473841323753414984t_char,Abs: fun_li5857714827575272844iteral,A2: set_list_char,X: literal] :
      ( type_d4752411451802217481t_char(Rep,Abs,A2)
     => member_list_char(aa_literal_list_char(Rep,X),A2) ) ).

% type_definition.Rep
tff(fact_390_type__definition_ORep,axiom,
    ! [Rep: fun_ra7314870381713115802nt_int,Abs: fun_se593059424582005380nt_rat,A2: set_se6260736226359567993nt_int,X: rat] :
      ( aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(Rep,Abs),A2)
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(aa_rat8624586542854104963nt_int(Rep,X)),A2) ) ).

% type_definition.Rep
tff(fact_391_type__definition_ORep,axiom,
    ! [Rep: fun_in8804710076054006414at_nat,Abs: fun_se9142667792015812128at_int,A2: set_se7855581050983116737at_nat,X: int] :
      ( type_d8752208705193531015at_nat(Rep,Abs,A2)
     => member2643936169264416010at_nat(aa_int8822742982810151799at_nat(Rep,X),A2) ) ).

% type_definition.Rep
tff(fact_392_type__definition_Ointro,axiom,
    ! [Rep: fun_Product_unit_o,A2: set_o,Abs: fun_o_Product_unit] :
      ( ! [X2: product_unit] : aa_set_o_o(member_o(aa_Product_unit_o(Rep,X2)),A2)
     => ( ! [X2: product_unit] : ( aa_o_Product_unit(Abs,aa_Product_unit_o(Rep,X2)) = X2 )
       => ( ! [Y2: $o] :
              ( aa_set_o_o(member_o((Y2)),A2)
             => ( aa_Product_unit_o(Rep,aa_o_Product_unit(Abs,(Y2)))
              <=> (Y2) ) )
         => aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A2) ) ) ) ).

% type_definition.intro
tff(fact_393_type__definition_Ointro,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,A2: set_fu3555266933746557007_nat_o,Abs: fun_fu3790787410453625514o_assn] :
      ( ! [X2: assn] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_ass4243947775541419017_nat_o(Rep,X2)),A2)
     => ( ! [X2: assn] : ( aa_fun5526454687433275355o_assn(Abs,aa_ass4243947775541419017_nat_o(Rep,X2)) = X2 )
       => ( ! [Y2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A2)
             => ( aa_ass4243947775541419017_nat_o(Rep,aa_fun5526454687433275355o_assn(Abs,Y2)) = Y2 ) )
         => aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A2) ) ) ) ).

% type_definition.intro
tff(fact_394_type__definition_Ointro,axiom,
    ! [Rep: fun_li7473841323753414984t_char,A2: set_list_char,Abs: fun_li5857714827575272844iteral] :
      ( ! [X2: literal] : member_list_char(aa_literal_list_char(Rep,X2),A2)
     => ( ! [X2: literal] : ( aa_list_char_literal(Abs,aa_literal_list_char(Rep,X2)) = X2 )
       => ( ! [Y2: list_char] :
              ( member_list_char(Y2,A2)
             => ( aa_literal_list_char(Rep,aa_list_char_literal(Abs,Y2)) = Y2 ) )
         => type_d4752411451802217481t_char(Rep,Abs,A2) ) ) ) ).

% type_definition.intro
tff(fact_395_type__definition_Ointro,axiom,
    ! [Rep: fun_ra7314870381713115802nt_int,A2: set_se6260736226359567993nt_int,Abs: fun_se593059424582005380nt_rat] :
      ( ! [X2: rat] : aa_set6820988702716878619_int_o(member2340774599025711042nt_int(aa_rat8624586542854104963nt_int(Rep,X2)),A2)
     => ( ! [X2: rat] : ( aa_set4118085860027418549nt_rat(Abs,aa_rat8624586542854104963nt_int(Rep,X2)) = X2 )
       => ( ! [Y2: set_Pr958786334691620121nt_int] :
              ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y2),A2)
             => ( aa_rat8624586542854104963nt_int(Rep,aa_set4118085860027418549nt_rat(Abs,Y2)) = Y2 ) )
         => aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(Rep,Abs),A2) ) ) ) ).

% type_definition.intro
tff(fact_396_type__definition_Ointro,axiom,
    ! [Rep: fun_in8804710076054006414at_nat,A2: set_se7855581050983116737at_nat,Abs: fun_se9142667792015812128at_int] :
      ( ! [X2: int] : member2643936169264416010at_nat(aa_int8822742982810151799at_nat(Rep,X2),A2)
     => ( ! [X2: int] : ( aa_set4746584241299694929at_int(Abs,aa_int8822742982810151799at_nat(Rep,X2)) = X2 )
       => ( ! [Y2: set_Pr1261947904930325089at_nat] :
              ( member2643936169264416010at_nat(Y2,A2)
             => ( aa_int8822742982810151799at_nat(Rep,aa_set4746584241299694929at_int(Abs,Y2)) = Y2 ) )
         => type_d8752208705193531015at_nat(Rep,Abs,A2) ) ) ) ).

% type_definition.intro
tff(fact_397_type__definition_OAbs__cases,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A2: set_o,X: product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A2)
     => ~ ! [Y2: $o] :
            ( ( X = aa_o_Product_unit(Abs,(Y2)) )
           => ~ aa_set_o_o(member_o((Y2)),A2) ) ) ).

% type_definition.Abs_cases
tff(fact_398_type__definition_OAbs__cases,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A2: set_fu3555266933746557007_nat_o,X: assn] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A2)
     => ~ ! [Y2: fun_Pr6460944699146269935_nat_o] :
            ( ( X = aa_fun5526454687433275355o_assn(Abs,Y2) )
           => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A2) ) ) ).

% type_definition.Abs_cases
tff(fact_399_type__definition_OAbs__cases,axiom,
    ! [Rep: fun_li7473841323753414984t_char,Abs: fun_li5857714827575272844iteral,A2: set_list_char,X: literal] :
      ( type_d4752411451802217481t_char(Rep,Abs,A2)
     => ~ ! [Y2: list_char] :
            ( ( X = aa_list_char_literal(Abs,Y2) )
           => ~ member_list_char(Y2,A2) ) ) ).

% type_definition.Abs_cases
tff(fact_400_type__definition_OAbs__cases,axiom,
    ! [Rep: fun_ra7314870381713115802nt_int,Abs: fun_se593059424582005380nt_rat,A2: set_se6260736226359567993nt_int,X: rat] :
      ( aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(Rep,Abs),A2)
     => ~ ! [Y2: set_Pr958786334691620121nt_int] :
            ( ( X = aa_set4118085860027418549nt_rat(Abs,Y2) )
           => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y2),A2) ) ) ).

% type_definition.Abs_cases
tff(fact_401_type__definition_OAbs__cases,axiom,
    ! [Rep: fun_in8804710076054006414at_nat,Abs: fun_se9142667792015812128at_int,A2: set_se7855581050983116737at_nat,X: int] :
      ( type_d8752208705193531015at_nat(Rep,Abs,A2)
     => ~ ! [Y2: set_Pr1261947904930325089at_nat] :
            ( ( X = aa_set4746584241299694929at_int(Abs,Y2) )
           => ~ member2643936169264416010at_nat(Y2,A2) ) ) ).

% type_definition.Abs_cases
tff(fact_402_type__definition_ORep__cases,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A2: set_o,Y: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A2)
     => ( aa_set_o_o(member_o((Y)),A2)
       => ~ ! [X2: product_unit] :
              ( (Y)
            <=> ~ aa_Product_unit_o(Rep,X2) ) ) ) ).

% type_definition.Rep_cases
tff(fact_403_type__definition_ORep__cases,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A2: set_fu3555266933746557007_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),A2)
       => ~ ! [X2: assn] : ( Y != aa_ass4243947775541419017_nat_o(Rep,X2) ) ) ) ).

% type_definition.Rep_cases
tff(fact_404_type__definition_ORep__cases,axiom,
    ! [Rep: fun_li7473841323753414984t_char,Abs: fun_li5857714827575272844iteral,A2: set_list_char,Y: list_char] :
      ( type_d4752411451802217481t_char(Rep,Abs,A2)
     => ( member_list_char(Y,A2)
       => ~ ! [X2: literal] : ( Y != aa_literal_list_char(Rep,X2) ) ) ) ).

% type_definition.Rep_cases
tff(fact_405_type__definition_ORep__cases,axiom,
    ! [Rep: fun_ra7314870381713115802nt_int,Abs: fun_se593059424582005380nt_rat,A2: set_se6260736226359567993nt_int,Y: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(Rep,Abs),A2)
     => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y),A2)
       => ~ ! [X2: rat] : ( Y != aa_rat8624586542854104963nt_int(Rep,X2) ) ) ) ).

% type_definition.Rep_cases
tff(fact_406_type__definition_ORep__cases,axiom,
    ! [Rep: fun_in8804710076054006414at_nat,Abs: fun_se9142667792015812128at_int,A2: set_se7855581050983116737at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( type_d8752208705193531015at_nat(Rep,Abs,A2)
     => ( member2643936169264416010at_nat(Y,A2)
       => ~ ! [X2: int] : ( Y != aa_int8822742982810151799at_nat(Rep,X2) ) ) ) ).

% type_definition.Rep_cases
tff(fact_407_type__definition_OAbs__induct,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A2: set_o,P: fun_Product_unit_o,X: product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A2)
     => ( ! [Y2: $o] :
            ( aa_set_o_o(member_o((Y2)),A2)
           => aa_Product_unit_o(P,aa_o_Product_unit(Abs,(Y2))) )
       => aa_Product_unit_o(P,X) ) ) ).

% type_definition.Abs_induct
tff(fact_408_type__definition_OAbs__induct,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A2: set_fu3555266933746557007_nat_o,P: fun_assn_o,X: assn] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A2)
     => ( ! [Y2: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A2)
           => aa_assn_o(P,aa_fun5526454687433275355o_assn(Abs,Y2)) )
       => aa_assn_o(P,X) ) ) ).

% type_definition.Abs_induct
tff(fact_409_type__definition_OAbs__induct,axiom,
    ! [Rep: fun_li7473841323753414984t_char,Abs: fun_li5857714827575272844iteral,A2: set_list_char,P: fun_literal_o,X: literal] :
      ( type_d4752411451802217481t_char(Rep,Abs,A2)
     => ( ! [Y2: list_char] :
            ( member_list_char(Y2,A2)
           => aa_literal_o(P,aa_list_char_literal(Abs,Y2)) )
       => aa_literal_o(P,X) ) ) ).

% type_definition.Abs_induct
tff(fact_410_type__definition_OAbs__induct,axiom,
    ! [Rep: fun_ra7314870381713115802nt_int,Abs: fun_se593059424582005380nt_rat,A2: set_se6260736226359567993nt_int,P: fun_rat_o,X: rat] :
      ( aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(Rep,Abs),A2)
     => ( ! [Y2: set_Pr958786334691620121nt_int] :
            ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y2),A2)
           => aa_rat_o(P,aa_set4118085860027418549nt_rat(Abs,Y2)) )
       => aa_rat_o(P,X) ) ) ).

% type_definition.Abs_induct
tff(fact_411_type__definition_OAbs__induct,axiom,
    ! [Rep: fun_in8804710076054006414at_nat,Abs: fun_se9142667792015812128at_int,A2: set_se7855581050983116737at_nat,P: fun_int_o,X: int] :
      ( type_d8752208705193531015at_nat(Rep,Abs,A2)
     => ( ! [Y2: set_Pr1261947904930325089at_nat] :
            ( member2643936169264416010at_nat(Y2,A2)
           => aa_int_o(P,aa_set4746584241299694929at_int(Abs,Y2)) )
       => aa_int_o(P,X) ) ) ).

% type_definition.Abs_induct
tff(fact_412_type__definition_OAbs__inject,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A2: set_o,X: $o,Y: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( aa_set_o_o(member_o((Y)),A2)
         => ( ( aa_o_Product_unit(Abs,(X)) = aa_o_Product_unit(Abs,(Y)) )
          <=> ( (X)
            <=> (Y) ) ) ) ) ) ).

% type_definition.Abs_inject
tff(fact_413_type__definition_OAbs__inject,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),A2)
         => ( ( aa_fun5526454687433275355o_assn(Abs,X) = aa_fun5526454687433275355o_assn(Abs,Y) )
          <=> ( X = Y ) ) ) ) ) ).

% type_definition.Abs_inject
tff(fact_414_type__definition_OAbs__inject,axiom,
    ! [Rep: fun_li7473841323753414984t_char,Abs: fun_li5857714827575272844iteral,A2: set_list_char,X: list_char,Y: list_char] :
      ( type_d4752411451802217481t_char(Rep,Abs,A2)
     => ( member_list_char(X,A2)
       => ( member_list_char(Y,A2)
         => ( ( aa_list_char_literal(Abs,X) = aa_list_char_literal(Abs,Y) )
          <=> ( X = Y ) ) ) ) ) ).

% type_definition.Abs_inject
tff(fact_415_type__definition_OAbs__inject,axiom,
    ! [Rep: fun_ra7314870381713115802nt_int,Abs: fun_se593059424582005380nt_rat,A2: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(Rep,Abs),A2)
     => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A2)
       => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y),A2)
         => ( ( aa_set4118085860027418549nt_rat(Abs,X) = aa_set4118085860027418549nt_rat(Abs,Y) )
          <=> ( X = Y ) ) ) ) ) ).

% type_definition.Abs_inject
tff(fact_416_type__definition_OAbs__inject,axiom,
    ! [Rep: fun_in8804710076054006414at_nat,Abs: fun_se9142667792015812128at_int,A2: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( type_d8752208705193531015at_nat(Rep,Abs,A2)
     => ( member2643936169264416010at_nat(X,A2)
       => ( member2643936169264416010at_nat(Y,A2)
         => ( ( aa_set4746584241299694929at_int(Abs,X) = aa_set4746584241299694929at_int(Abs,Y) )
          <=> ( X = Y ) ) ) ) ) ).

% type_definition.Abs_inject
tff(fact_417_type__definition_ORep__induct,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A2: set_o,Y: $o,P: fun_o_o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A2)
     => ( aa_set_o_o(member_o((Y)),A2)
       => ( ! [X2: product_unit] : aa_o_o(P,aa_Product_unit_o(Rep,X2))
         => aa_o_o(P,(Y)) ) ) ) ).

% type_definition.Rep_induct
tff(fact_418_type__definition_ORep__induct,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A2: set_fu3555266933746557007_nat_o,Y: fun_Pr6460944699146269935_nat_o,P: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),A2)
       => ( ! [X2: assn] : aa_fun1675921005893230629at_o_o(P,aa_ass4243947775541419017_nat_o(Rep,X2))
         => aa_fun1675921005893230629at_o_o(P,Y) ) ) ) ).

% type_definition.Rep_induct
tff(fact_419_type__definition_ORep__induct,axiom,
    ! [Rep: fun_li7473841323753414984t_char,Abs: fun_li5857714827575272844iteral,A2: set_list_char,Y: list_char,P: fun_list_char_o] :
      ( type_d4752411451802217481t_char(Rep,Abs,A2)
     => ( member_list_char(Y,A2)
       => ( ! [X2: literal] : aa_list_char_o(P,aa_literal_list_char(Rep,X2))
         => aa_list_char_o(P,Y) ) ) ) ).

% type_definition.Rep_induct
tff(fact_420_type__definition_ORep__induct,axiom,
    ! [Rep: fun_ra7314870381713115802nt_int,Abs: fun_se593059424582005380nt_rat,A2: set_se6260736226359567993nt_int,Y: set_Pr958786334691620121nt_int,P: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(Rep,Abs),A2)
     => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y),A2)
       => ( ! [X2: rat] : aa_set6170482117830564987_int_o(P,aa_rat8624586542854104963nt_int(Rep,X2))
         => aa_set6170482117830564987_int_o(P,Y) ) ) ) ).

% type_definition.Rep_induct
tff(fact_421_type__definition_ORep__induct,axiom,
    ! [Rep: fun_in8804710076054006414at_nat,Abs: fun_se9142667792015812128at_int,A2: set_se7855581050983116737at_nat,Y: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( type_d8752208705193531015at_nat(Rep,Abs,A2)
     => ( member2643936169264416010at_nat(Y,A2)
       => ( ! [X2: int] : aa_set3702741264012733747_nat_o(P,aa_int8822742982810151799at_nat(Rep,X2))
         => aa_set3702741264012733747_nat_o(P,Y) ) ) ) ).

% type_definition.Rep_induct
tff(fact_422_type__definition_ORep__inject,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A2: set_o,X: product_unit,Y: product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A2)
     => ( ( aa_Product_unit_o(Rep,X)
        <=> aa_Product_unit_o(Rep,Y) )
      <=> ( X = Y ) ) ) ).

% type_definition.Rep_inject
tff(fact_423_type__definition_ORep__inject,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A2: set_fu3555266933746557007_nat_o,X: assn,Y: assn] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A2)
     => ( ( aa_ass4243947775541419017_nat_o(Rep,X) = aa_ass4243947775541419017_nat_o(Rep,Y) )
      <=> ( X = Y ) ) ) ).

% type_definition.Rep_inject
tff(fact_424_type__definition_ORep__inject,axiom,
    ! [Rep: fun_li7473841323753414984t_char,Abs: fun_li5857714827575272844iteral,A2: set_list_char,X: literal,Y: literal] :
      ( type_d4752411451802217481t_char(Rep,Abs,A2)
     => ( ( aa_literal_list_char(Rep,X) = aa_literal_list_char(Rep,Y) )
      <=> ( X = Y ) ) ) ).

% type_definition.Rep_inject
tff(fact_425_type__definition_ORep__inject,axiom,
    ! [Rep: fun_ra7314870381713115802nt_int,Abs: fun_se593059424582005380nt_rat,A2: set_se6260736226359567993nt_int,X: rat,Y: rat] :
      ( aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(Rep,Abs),A2)
     => ( ( aa_rat8624586542854104963nt_int(Rep,X) = aa_rat8624586542854104963nt_int(Rep,Y) )
      <=> ( X = Y ) ) ) ).

% type_definition.Rep_inject
tff(fact_426_type__definition_ORep__inject,axiom,
    ! [Rep: fun_in8804710076054006414at_nat,Abs: fun_se9142667792015812128at_int,A2: set_se7855581050983116737at_nat,X: int,Y: int] :
      ( type_d8752208705193531015at_nat(Rep,Abs,A2)
     => ( ( aa_int8822742982810151799at_nat(Rep,X) = aa_int8822742982810151799at_nat(Rep,Y) )
      <=> ( X = Y ) ) ) ).

% type_definition.Rep_inject
tff(fact_427_type__definition_OAbs__inverse,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A2: set_o,Y: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A2)
     => ( aa_set_o_o(member_o((Y)),A2)
       => ( aa_Product_unit_o(Rep,aa_o_Product_unit(Abs,(Y)))
        <=> (Y) ) ) ) ).

% type_definition.Abs_inverse
tff(fact_428_type__definition_OAbs__inverse,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A2: set_fu3555266933746557007_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),A2)
       => ( aa_ass4243947775541419017_nat_o(Rep,aa_fun5526454687433275355o_assn(Abs,Y)) = Y ) ) ) ).

% type_definition.Abs_inverse
tff(fact_429_type__definition_OAbs__inverse,axiom,
    ! [Rep: fun_li7473841323753414984t_char,Abs: fun_li5857714827575272844iteral,A2: set_list_char,Y: list_char] :
      ( type_d4752411451802217481t_char(Rep,Abs,A2)
     => ( member_list_char(Y,A2)
       => ( aa_literal_list_char(Rep,aa_list_char_literal(Abs,Y)) = Y ) ) ) ).

% type_definition.Abs_inverse
tff(fact_430_type__definition_OAbs__inverse,axiom,
    ! [Rep: fun_ra7314870381713115802nt_int,Abs: fun_se593059424582005380nt_rat,A2: set_se6260736226359567993nt_int,Y: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(Rep,Abs),A2)
     => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y),A2)
       => ( aa_rat8624586542854104963nt_int(Rep,aa_set4118085860027418549nt_rat(Abs,Y)) = Y ) ) ) ).

% type_definition.Abs_inverse
tff(fact_431_type__definition_OAbs__inverse,axiom,
    ! [Rep: fun_in8804710076054006414at_nat,Abs: fun_se9142667792015812128at_int,A2: set_se7855581050983116737at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( type_d8752208705193531015at_nat(Rep,Abs,A2)
     => ( member2643936169264416010at_nat(Y,A2)
       => ( aa_int8822742982810151799at_nat(Rep,aa_set4746584241299694929at_int(Abs,Y)) = Y ) ) ) ).

% type_definition.Abs_inverse
tff(fact_432_type__definition_ORep__inverse,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A2: set_o,X: product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A2)
     => ( aa_o_Product_unit(Abs,aa_Product_unit_o(Rep,X)) = X ) ) ).

% type_definition.Rep_inverse
tff(fact_433_type__definition_ORep__inverse,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A2: set_fu3555266933746557007_nat_o,X: assn] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A2)
     => ( aa_fun5526454687433275355o_assn(Abs,aa_ass4243947775541419017_nat_o(Rep,X)) = X ) ) ).

% type_definition.Rep_inverse
tff(fact_434_type__definition_ORep__inverse,axiom,
    ! [Rep: fun_li7473841323753414984t_char,Abs: fun_li5857714827575272844iteral,A2: set_list_char,X: literal] :
      ( type_d4752411451802217481t_char(Rep,Abs,A2)
     => ( aa_list_char_literal(Abs,aa_literal_list_char(Rep,X)) = X ) ) ).

% type_definition.Rep_inverse
tff(fact_435_type__definition_ORep__inverse,axiom,
    ! [Rep: fun_ra7314870381713115802nt_int,Abs: fun_se593059424582005380nt_rat,A2: set_se6260736226359567993nt_int,X: rat] :
      ( aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(Rep,Abs),A2)
     => ( aa_set4118085860027418549nt_rat(Abs,aa_rat8624586542854104963nt_int(Rep,X)) = X ) ) ).

% type_definition.Rep_inverse
tff(fact_436_type__definition_ORep__inverse,axiom,
    ! [Rep: fun_in8804710076054006414at_nat,Abs: fun_se9142667792015812128at_int,A2: set_se7855581050983116737at_nat,X: int] :
      ( type_d8752208705193531015at_nat(Rep,Abs,A2)
     => ( aa_set4746584241299694929at_int(Abs,aa_int8822742982810151799at_nat(Rep,X)) = X ) ) ).

% type_definition.Rep_inverse
tff(fact_437_type__definition__def,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A2: set_o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A2)
    <=> ( ! [X4: product_unit] : aa_set_o_o(member_o(aa_Product_unit_o(Rep,X4)),A2)
        & ! [X4: product_unit] : ( aa_o_Product_unit(Abs,aa_Product_unit_o(Rep,X4)) = X4 )
        & ! [Y3: $o] :
            ( aa_set_o_o(member_o((Y3)),A2)
           => ( aa_Product_unit_o(Rep,aa_o_Product_unit(Abs,(Y3)))
            <=> (Y3) ) ) ) ) ).

% type_definition_def
tff(fact_438_type__definition__def,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A2)
    <=> ( ! [X4: assn] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_ass4243947775541419017_nat_o(Rep,X4)),A2)
        & ! [X4: assn] : ( aa_fun5526454687433275355o_assn(Abs,aa_ass4243947775541419017_nat_o(Rep,X4)) = X4 )
        & ! [Y3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y3),A2)
           => ( aa_ass4243947775541419017_nat_o(Rep,aa_fun5526454687433275355o_assn(Abs,Y3)) = Y3 ) ) ) ) ).

% type_definition_def
tff(fact_439_type__definition__def,axiom,
    ! [Rep: fun_li7473841323753414984t_char,Abs: fun_li5857714827575272844iteral,A2: set_list_char] :
      ( type_d4752411451802217481t_char(Rep,Abs,A2)
    <=> ( ! [X4: literal] : member_list_char(aa_literal_list_char(Rep,X4),A2)
        & ! [X4: literal] : ( aa_list_char_literal(Abs,aa_literal_list_char(Rep,X4)) = X4 )
        & ! [Y3: list_char] :
            ( member_list_char(Y3,A2)
           => ( aa_literal_list_char(Rep,aa_list_char_literal(Abs,Y3)) = Y3 ) ) ) ) ).

% type_definition_def
tff(fact_440_type__definition__def,axiom,
    ! [Rep: fun_ra7314870381713115802nt_int,Abs: fun_se593059424582005380nt_rat,A2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(Rep,Abs),A2)
    <=> ( ! [X4: rat] : aa_set6820988702716878619_int_o(member2340774599025711042nt_int(aa_rat8624586542854104963nt_int(Rep,X4)),A2)
        & ! [X4: rat] : ( aa_set4118085860027418549nt_rat(Abs,aa_rat8624586542854104963nt_int(Rep,X4)) = X4 )
        & ! [Y3: set_Pr958786334691620121nt_int] :
            ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y3),A2)
           => ( aa_rat8624586542854104963nt_int(Rep,aa_set4118085860027418549nt_rat(Abs,Y3)) = Y3 ) ) ) ) ).

% type_definition_def
tff(fact_441_type__definition__def,axiom,
    ! [Rep: fun_in8804710076054006414at_nat,Abs: fun_se9142667792015812128at_int,A2: set_se7855581050983116737at_nat] :
      ( type_d8752208705193531015at_nat(Rep,Abs,A2)
    <=> ( ! [X4: int] : member2643936169264416010at_nat(aa_int8822742982810151799at_nat(Rep,X4),A2)
        & ! [X4: int] : ( aa_set4746584241299694929at_int(Abs,aa_int8822742982810151799at_nat(Rep,X4)) = X4 )
        & ! [Y3: set_Pr1261947904930325089at_nat] :
            ( member2643936169264416010at_nat(Y3,A2)
           => ( aa_int8822742982810151799at_nat(Rep,aa_set4746584241299694929at_int(Abs,Y3)) = Y3 ) ) ) ) ).

% type_definition_def
tff(fact_442_mult_Oright__assoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% mult.right_assoc
tff(fact_443_mult_Oright__assoc,axiom,
    ! [A: assn,B: assn,C: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ) ).

% mult.right_assoc
tff(fact_444_mult_Oright__assoc,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% mult.right_assoc
tff(fact_445_mult_Oright__assoc,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% mult.right_assoc
tff(fact_446_mult_Oright__assoc,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% mult.right_assoc
tff(fact_447_mult_Oright__commute,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ).

% mult.right_commute
tff(fact_448_mult_Oright__commute,axiom,
    ! [A: assn,B: assn,C: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)),B) ) ).

% mult.right_commute
tff(fact_449_mult_Oright__commute,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ).

% mult.right_commute
tff(fact_450_mult_Oright__commute,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ).

% mult.right_commute
tff(fact_451_mult_Oright__commute,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ).

% mult.right_commute
tff(fact_452_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_453_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: assn,A: assn,C: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_454_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_455_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_456_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_457_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_458_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: assn,B: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_459_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_460_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_461_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_462_mult_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% mult.assoc
tff(fact_463_mult_Oassoc,axiom,
    ! [A: assn,B: assn,C: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ) ).

% mult.assoc
tff(fact_464_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% mult.assoc
tff(fact_465_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% mult.assoc
tff(fact_466_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% mult.assoc
tff(fact_467_mult_Osafe__commute,axiom,
    ! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(syntax380677951125133566nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) ) ) ).

% mult.safe_commute
tff(fact_468_mult_Osafe__commute,axiom,
    ! [X: assn,Y: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(syntax7398250324933576852n_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y)),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ) ) ).

% mult.safe_commute
tff(fact_469_mult_Osafe__commute,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(syntax3730441303064801268at_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A) ) ) ).

% mult.safe_commute
tff(fact_470_mult_Osafe__commute,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(syntax4682126007086162916at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)),A)
     => ( 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.safe_commute
tff(fact_471_mult_Osafe__commute,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(syntax5678989248478167196nt_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),A)
     => ( 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.safe_commute
tff(fact_472_sup__bot_Osemilattice__neutr__axioms,axiom,
    semila6683195626996185257at_nat(sup_su3035147773818789531at_nat,bot_bo8422036546324065075at_nat) ).

% sup_bot.semilattice_neutr_axioms
tff(fact_473_sup__bot_Osemilattice__neutr__axioms,axiom,
    aa_set5844364881195198285_nat_o(semila3675781407871293737at_nat(sup_su5525570899277871387at_nat),bot_bo228742789529271731at_nat) ).

% sup_bot.semilattice_neutr_axioms
tff(fact_474_sup__bot_Osemilattice__neutr__axioms,axiom,
    aa_set3702741264012733747_nat_o(semila858173770281612099at_nat(sup_su6327502436637775413at_nat),bot_bo2099793752762293965at_nat) ).

% sup_bot.semilattice_neutr_axioms
tff(fact_475_sup__bot_Osemilattice__neutr__axioms,axiom,
    aa_set_o_o(semila1065760912024005978_set_o(sup_sup_set_o),bot_bot_set_o) ).

% sup_bot.semilattice_neutr_axioms
tff(fact_476_sup__bot_Osemilattice__neutr__axioms,axiom,
    aa_set_nat_o(semila1241773964035338532et_nat(sup_sup_set_nat),bot_bot_set_nat) ).

% sup_bot.semilattice_neutr_axioms
tff(fact_477_sup__bot_Osemilattice__neutr__axioms,axiom,
    aa_set_int_o(semila6287294981380917632et_int(sup_sup_set_int),bot_bot_set_int) ).

% sup_bot.semilattice_neutr_axioms
tff(fact_478_sup__bot_Omonoid__axioms,axiom,
    monoid3323479476634721864at_nat(sup_su3035147773818789531at_nat,bot_bo8422036546324065075at_nat) ).

% sup_bot.monoid_axioms
tff(fact_479_sup__bot_Omonoid__axioms,axiom,
    aa_set5844364881195198285_nat_o(monoid534952638934301128at_nat(sup_su5525570899277871387at_nat),bot_bo228742789529271731at_nat) ).

% sup_bot.monoid_axioms
tff(fact_480_sup__bot_Omonoid__axioms,axiom,
    aa_set3702741264012733747_nat_o(monoid1519094961268053602at_nat(sup_su6327502436637775413at_nat),bot_bo2099793752762293965at_nat) ).

% sup_bot.monoid_axioms
tff(fact_481_sup__bot_Omonoid__axioms,axiom,
    aa_set_o_o(monoid_set_o(sup_sup_set_o),bot_bot_set_o) ).

% sup_bot.monoid_axioms
tff(fact_482_sup__bot_Omonoid__axioms,axiom,
    aa_set_nat_o(monoid_set_nat(sup_sup_set_nat),bot_bot_set_nat) ).

% sup_bot.monoid_axioms
tff(fact_483_sup__bot_Omonoid__axioms,axiom,
    aa_set_int_o(monoid_set_int(sup_sup_set_int),bot_bot_set_int) ).

% sup_bot.monoid_axioms
tff(fact_484_one__assn__def,axiom,
    one_one_assn = aa_fun5526454687433275355o_assn(abs_assn,one_assn_raw) ).

% one_assn_def
tff(fact_485_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),X) = bot_bo3516217578726220278_int_o )
     => ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,A),X) = top_to8487252020945707026_int_o )
       => ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),Y) = bot_bo3516217578726220278_int_o )
         => ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,A),Y) = top_to8487252020945707026_int_o )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_486_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: fun_list_char_o,X: fun_list_char_o,Y: fun_list_char_o] :
      ( ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),X) = bot_bo8860746597864711612char_o )
     => ( ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,A),X) = top_to7404587909000090784char_o )
       => ( ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),Y) = bot_bo8860746597864711612char_o )
         => ( ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,A),Y) = top_to7404587909000090784char_o )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_487_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A),X) = bot_bo8422036546324065075at_nat )
     => ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),X) = top_to8764741339697478167at_nat )
       => ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A),Y) = bot_bo8422036546324065075at_nat )
         => ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),Y) = top_to8764741339697478167at_nat )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_488_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),X) = bot_bo228742789529271731at_nat )
     => ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),X) = top_to6833984726390702231at_nat )
       => ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),Y) = bot_bo228742789529271731at_nat )
         => ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),Y) = top_to6833984726390702231at_nat )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_489_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),X) = bot_bo2099793752762293965at_nat )
     => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),X) = top_to4669805908274784177at_nat )
       => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),Y) = bot_bo2099793752762293965at_nat )
         => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),Y) = top_to4669805908274784177at_nat )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_490_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_list_nat,X: set_list_nat,Y: set_list_nat] :
      ( ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,A),X) = bot_bot_set_list_nat )
     => ( ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,A),X) = top_top_set_list_nat )
       => ( ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,A),Y) = bot_bot_set_list_nat )
         => ( ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,A),Y) = top_top_set_list_nat )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_491_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_o,X: set_o,Y: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A),X) = bot_bot_set_o )
     => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),X) = top_top_set_o )
       => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A),Y) = bot_bot_set_o )
         => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),Y) = top_top_set_o )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_492_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_rat,X: set_rat,Y: set_rat] :
      ( ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,A),X) = bot_bot_set_rat )
     => ( ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,A),X) = top_top_set_rat )
       => ( ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,A),Y) = bot_bot_set_rat )
         => ( ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,A),Y) = top_top_set_rat )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_493_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_nat,X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),X) = bot_bot_set_nat )
     => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),X) = top_top_set_nat )
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),Y) = bot_bot_set_nat )
         => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),Y) = top_top_set_nat )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_494_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_int,X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),X) = bot_bot_set_int )
     => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),X) = top_top_set_int )
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),Y) = bot_bot_set_int )
         => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),Y) = top_top_set_int )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_495_mod__star__conv,axiom,
    ! [A2: assn,B2: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),H2)
    <=> ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As2: set_nat] :
          ( ( H2 = produc7507926704131184380et_nat(Hr,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As2)) )
          & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As2) = bot_bot_set_nat )
          & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,A2),produc7507926704131184380et_nat(Hr,As1))
          & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,B2),produc7507926704131184380et_nat(Hr,As2)) ) ) ).

% mod_star_conv
tff(fact_496_star__assnI,axiom,
    ! [P: assn,H2: heap_e7401611519738050253t_unit,As: set_nat,Q: assn,As3: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),produc7507926704131184380et_nat(H2,As))
     => ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),produc7507926704131184380et_nat(H2,As3))
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As3) = bot_bot_set_nat )
         => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),produc7507926704131184380et_nat(H2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As3))) ) ) ) ).

% star_assnI
tff(fact_497_sup__bot_Ocomm__monoid__axioms,axiom,
    comm_m4707310049023135612at_nat(sup_su3035147773818789531at_nat,bot_bo8422036546324065075at_nat) ).

% sup_bot.comm_monoid_axioms
tff(fact_498_sup__bot_Ocomm__monoid__axioms,axiom,
    aa_set5844364881195198285_nat_o(comm_m2457627138559323900at_nat(sup_su5525570899277871387at_nat),bot_bo228742789529271731at_nat) ).

% sup_bot.comm_monoid_axioms
tff(fact_499_sup__bot_Ocomm__monoid__axioms,axiom,
    aa_set3702741264012733747_nat_o(comm_m5709050211238513046at_nat(sup_su6327502436637775413at_nat),bot_bo2099793752762293965at_nat) ).

% sup_bot.comm_monoid_axioms
tff(fact_500_sup__bot_Ocomm__monoid__axioms,axiom,
    aa_set_o_o(comm_monoid_set_o(sup_sup_set_o),bot_bot_set_o) ).

% sup_bot.comm_monoid_axioms
tff(fact_501_sup__bot_Ocomm__monoid__axioms,axiom,
    aa_set_nat_o(comm_monoid_set_nat(sup_sup_set_nat),bot_bot_set_nat) ).

% sup_bot.comm_monoid_axioms
tff(fact_502_sup__bot_Ocomm__monoid__axioms,axiom,
    aa_set_int_o(comm_monoid_set_int(sup_sup_set_int),bot_bot_set_int) ).

% sup_bot.comm_monoid_axioms
tff(fact_503_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Y)) ) ).

% boolean_algebra.de_Morgan_disj
tff(fact_504_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,Y)) ) ).

% boolean_algebra.de_Morgan_disj
tff(fact_505_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : ( aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),Y)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,X)),aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,Y)) ) ).

% boolean_algebra.de_Morgan_disj
tff(fact_506_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,X)),aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,Y)) ) ).

% boolean_algebra.de_Morgan_disj
tff(fact_507_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,uminus935396558254630718at_nat(X)),uminus935396558254630718at_nat(Y)) ) ).

% boolean_algebra.de_Morgan_disj
tff(fact_508_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ) ).

% boolean_algebra.de_Morgan_disj
tff(fact_509_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: product_unit,Y: product_unit] : ( aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)) ) ).

% boolean_algebra.de_Morgan_disj
tff(fact_510_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_511_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ) ).

% add.inverse_inverse
tff(fact_512_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_513_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_514_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_515_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_516_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X) = aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y) )
    <=> ( X = Y ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
tff(fact_517_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [X: product_unit] : ( aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)) = X ) ).

% boolean_algebra_class.boolean_algebra.double_compl
tff(fact_518_mult_Oright__neutral,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),one_one_Code_integer) = A ) ).

% mult.right_neutral
tff(fact_519_mult_Oright__neutral,axiom,
    ! [A: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ) ).

% mult.right_neutral
tff(fact_520_mult_Oright__neutral,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ) ).

% mult.right_neutral
tff(fact_521_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_522_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_523_mult__1,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A) = A ) ).

% mult_1
tff(fact_524_mult__1,axiom,
    ! [A: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ) ).

% mult_1
tff(fact_525_mult__1,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ) ).

% mult_1
tff(fact_526_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_527_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_528_inf__top__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,top_to4669805908274784177at_nat),X) = X ) ).

% inf_top_left
tff(fact_529_inf__top__left,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,top_to8487252020945707026_int_o),X) = X ) ).

% inf_top_left
tff(fact_530_inf__top__left,axiom,
    ! [X: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,top_to7404587909000090784char_o),X) = X ) ).

% inf_top_left
tff(fact_531_inf__top__left,axiom,
    ! [X: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,top_top_set_list_nat),X) = X ) ).

% inf_top_left
tff(fact_532_inf__top__left,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,top_top_set_o),X) = X ) ).

% inf_top_left
tff(fact_533_inf__top__left,axiom,
    ! [X: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,top_top_set_rat),X) = X ) ).

% inf_top_left
tff(fact_534_inf__top__left,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,top_top_set_nat),X) = X ) ).

% inf_top_left
tff(fact_535_inf__top__left,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,top_top_set_int),X) = X ) ).

% inf_top_left
tff(fact_536_inf__top__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),top_to4669805908274784177at_nat) = X ) ).

% inf_top_right
tff(fact_537_inf__top__right,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),top_to8487252020945707026_int_o) = X ) ).

% inf_top_right
tff(fact_538_inf__top__right,axiom,
    ! [X: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),top_to7404587909000090784char_o) = X ) ).

% inf_top_right
tff(fact_539_inf__top__right,axiom,
    ! [X: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,X),top_top_set_list_nat) = X ) ).

% inf_top_right
tff(fact_540_inf__top__right,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),top_top_set_o) = X ) ).

% inf_top_right
tff(fact_541_inf__top__right,axiom,
    ! [X: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,X),top_top_set_rat) = X ) ).

% inf_top_right
tff(fact_542_inf__top__right,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),top_top_set_nat) = X ) ).

% inf_top_right
tff(fact_543_inf__top__right,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),top_top_set_int) = X ) ).

% inf_top_right
tff(fact_544_inf__eq__top__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = top_to4669805908274784177at_nat )
    <=> ( ( X = top_to4669805908274784177at_nat )
        & ( Y = top_to4669805908274784177at_nat ) ) ) ).

% inf_eq_top_iff
tff(fact_545_inf__eq__top__iff,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = top_to8487252020945707026_int_o )
    <=> ( ( X = top_to8487252020945707026_int_o )
        & ( Y = top_to8487252020945707026_int_o ) ) ) ).

% inf_eq_top_iff
tff(fact_546_inf__eq__top__iff,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] :
      ( ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y) = top_to7404587909000090784char_o )
    <=> ( ( X = top_to7404587909000090784char_o )
        & ( Y = top_to7404587909000090784char_o ) ) ) ).

% inf_eq_top_iff
tff(fact_547_inf__eq__top__iff,axiom,
    ! [X: set_list_nat,Y: set_list_nat] :
      ( ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,X),Y) = top_top_set_list_nat )
    <=> ( ( X = top_top_set_list_nat )
        & ( Y = top_top_set_list_nat ) ) ) ).

% inf_eq_top_iff
tff(fact_548_inf__eq__top__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),Y) = top_top_set_o )
    <=> ( ( X = top_top_set_o )
        & ( Y = top_top_set_o ) ) ) ).

% inf_eq_top_iff
tff(fact_549_inf__eq__top__iff,axiom,
    ! [X: set_rat,Y: set_rat] :
      ( ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,X),Y) = top_top_set_rat )
    <=> ( ( X = top_top_set_rat )
        & ( Y = top_top_set_rat ) ) ) ).

% inf_eq_top_iff
tff(fact_550_inf__eq__top__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = top_top_set_nat )
    <=> ( ( X = top_top_set_nat )
        & ( Y = top_top_set_nat ) ) ) ).

% inf_eq_top_iff
tff(fact_551_inf__eq__top__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = top_top_set_int )
    <=> ( ( X = top_top_set_int )
        & ( Y = top_top_set_int ) ) ) ).

% inf_eq_top_iff
tff(fact_552_top__eq__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( top_to4669805908274784177at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) )
    <=> ( ( X = top_to4669805908274784177at_nat )
        & ( Y = top_to4669805908274784177at_nat ) ) ) ).

% top_eq_inf_iff
tff(fact_553_top__eq__inf__iff,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( ( top_to8487252020945707026_int_o = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) )
    <=> ( ( X = top_to8487252020945707026_int_o )
        & ( Y = top_to8487252020945707026_int_o ) ) ) ).

% top_eq_inf_iff
tff(fact_554_top__eq__inf__iff,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] :
      ( ( top_to7404587909000090784char_o = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y) )
    <=> ( ( X = top_to7404587909000090784char_o )
        & ( Y = top_to7404587909000090784char_o ) ) ) ).

% top_eq_inf_iff
tff(fact_555_top__eq__inf__iff,axiom,
    ! [X: set_list_nat,Y: set_list_nat] :
      ( ( top_top_set_list_nat = aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,X),Y) )
    <=> ( ( X = top_top_set_list_nat )
        & ( Y = top_top_set_list_nat ) ) ) ).

% top_eq_inf_iff
tff(fact_556_top__eq__inf__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( top_top_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),Y) )
    <=> ( ( X = top_top_set_o )
        & ( Y = top_top_set_o ) ) ) ).

% top_eq_inf_iff
tff(fact_557_top__eq__inf__iff,axiom,
    ! [X: set_rat,Y: set_rat] :
      ( ( top_top_set_rat = aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,X),Y) )
    <=> ( ( X = top_top_set_rat )
        & ( Y = top_top_set_rat ) ) ) ).

% top_eq_inf_iff
tff(fact_558_top__eq__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( top_top_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) )
    <=> ( ( X = top_top_set_nat )
        & ( Y = top_top_set_nat ) ) ) ).

% top_eq_inf_iff
tff(fact_559_top__eq__inf__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( top_top_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) )
    <=> ( ( X = top_top_set_int )
        & ( Y = top_top_set_int ) ) ) ).

% top_eq_inf_iff
tff(fact_560_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = top_to4669805908274784177at_nat )
    <=> ( ( A = top_to4669805908274784177at_nat )
        & ( B = top_to4669805908274784177at_nat ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_561_inf__top_Oeq__neutr__iff,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = top_to8487252020945707026_int_o )
    <=> ( ( A = top_to8487252020945707026_int_o )
        & ( B = top_to8487252020945707026_int_o ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_562_inf__top_Oeq__neutr__iff,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o] :
      ( ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) = top_to7404587909000090784char_o )
    <=> ( ( A = top_to7404587909000090784char_o )
        & ( B = top_to7404587909000090784char_o ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_563_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_list_nat,B: set_list_nat] :
      ( ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,A),B) = top_top_set_list_nat )
    <=> ( ( A = top_top_set_list_nat )
        & ( B = top_top_set_list_nat ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_564_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A),B) = top_top_set_o )
    <=> ( ( A = top_top_set_o )
        & ( B = top_top_set_o ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_565_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_rat,B: set_rat] :
      ( ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,A),B) = top_top_set_rat )
    <=> ( ( A = top_top_set_rat )
        & ( B = top_top_set_rat ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_566_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = top_top_set_nat )
    <=> ( ( A = top_top_set_nat )
        & ( B = top_top_set_nat ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_567_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = top_top_set_int )
    <=> ( ( A = top_top_set_int )
        & ( B = top_top_set_int ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_568_inf__top_Oleft__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,top_to4669805908274784177at_nat),A) = A ) ).

% inf_top.left_neutral
tff(fact_569_inf__top_Oleft__neutral,axiom,
    ! [A: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,top_to8487252020945707026_int_o),A) = A ) ).

% inf_top.left_neutral
tff(fact_570_inf__top_Oleft__neutral,axiom,
    ! [A: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,top_to7404587909000090784char_o),A) = A ) ).

% inf_top.left_neutral
tff(fact_571_inf__top_Oleft__neutral,axiom,
    ! [A: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,top_top_set_list_nat),A) = A ) ).

% inf_top.left_neutral
tff(fact_572_inf__top_Oleft__neutral,axiom,
    ! [A: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,top_top_set_o),A) = A ) ).

% inf_top.left_neutral
tff(fact_573_inf__top_Oleft__neutral,axiom,
    ! [A: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,top_top_set_rat),A) = A ) ).

% inf_top.left_neutral
tff(fact_574_inf__top_Oleft__neutral,axiom,
    ! [A: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,top_top_set_nat),A) = A ) ).

% inf_top.left_neutral
tff(fact_575_inf__top_Oleft__neutral,axiom,
    ! [A: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,top_top_set_int),A) = A ) ).

% inf_top.left_neutral
tff(fact_576_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( top_to4669805908274784177at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) )
    <=> ( ( A = top_to4669805908274784177at_nat )
        & ( B = top_to4669805908274784177at_nat ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_577_inf__top_Oneutr__eq__iff,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( ( top_to8487252020945707026_int_o = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) )
    <=> ( ( A = top_to8487252020945707026_int_o )
        & ( B = top_to8487252020945707026_int_o ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_578_inf__top_Oneutr__eq__iff,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o] :
      ( ( top_to7404587909000090784char_o = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) )
    <=> ( ( A = top_to7404587909000090784char_o )
        & ( B = top_to7404587909000090784char_o ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_579_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_list_nat,B: set_list_nat] :
      ( ( top_top_set_list_nat = aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,A),B) )
    <=> ( ( A = top_top_set_list_nat )
        & ( B = top_top_set_list_nat ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_580_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( top_top_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A),B) )
    <=> ( ( A = top_top_set_o )
        & ( B = top_top_set_o ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_581_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_rat,B: set_rat] :
      ( ( top_top_set_rat = aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,A),B) )
    <=> ( ( A = top_top_set_rat )
        & ( B = top_top_set_rat ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_582_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( top_top_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
    <=> ( ( A = top_top_set_nat )
        & ( B = top_top_set_nat ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_583_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( top_top_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) )
    <=> ( ( A = top_top_set_int )
        & ( B = top_top_set_int ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_584_inf__top_Oright__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),top_to4669805908274784177at_nat) = A ) ).

% inf_top.right_neutral
tff(fact_585_inf__top_Oright__neutral,axiom,
    ! [A: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),top_to8487252020945707026_int_o) = A ) ).

% inf_top.right_neutral
tff(fact_586_inf__top_Oright__neutral,axiom,
    ! [A: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),top_to7404587909000090784char_o) = A ) ).

% inf_top.right_neutral
tff(fact_587_inf__top_Oright__neutral,axiom,
    ! [A: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,A),top_top_set_list_nat) = A ) ).

% inf_top.right_neutral
tff(fact_588_inf__top_Oright__neutral,axiom,
    ! [A: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A),top_top_set_o) = A ) ).

% inf_top.right_neutral
tff(fact_589_inf__top_Oright__neutral,axiom,
    ! [A: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,A),top_top_set_rat) = A ) ).

% inf_top.right_neutral
tff(fact_590_inf__top_Oright__neutral,axiom,
    ! [A: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),top_top_set_nat) = A ) ).

% inf_top.right_neutral
tff(fact_591_inf__top_Oright__neutral,axiom,
    ! [A: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),top_top_set_int) = A ) ).

% inf_top.right_neutral
tff(fact_592_boolean__algebra_Odisj__one__left,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,top_to8764741339697478167at_nat),X) = top_to8764741339697478167at_nat ) ).

% boolean_algebra.disj_one_left
tff(fact_593_boolean__algebra_Odisj__one__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,top_to6833984726390702231at_nat),X) = top_to6833984726390702231at_nat ) ).

% boolean_algebra.disj_one_left
tff(fact_594_boolean__algebra_Odisj__one__left,axiom,
    ! [X: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,top_top_set_list_nat),X) = top_top_set_list_nat ) ).

% boolean_algebra.disj_one_left
tff(fact_595_boolean__algebra_Odisj__one__left,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,top_top_set_o),X) = top_top_set_o ) ).

% boolean_algebra.disj_one_left
tff(fact_596_boolean__algebra_Odisj__one__left,axiom,
    ! [X: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,top_top_set_rat),X) = top_top_set_rat ) ).

% boolean_algebra.disj_one_left
tff(fact_597_boolean__algebra_Odisj__one__left,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,top_top_set_nat),X) = top_top_set_nat ) ).

% boolean_algebra.disj_one_left
tff(fact_598_boolean__algebra_Odisj__one__left,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,top_top_set_int),X) = top_top_set_int ) ).

% boolean_algebra.disj_one_left
tff(fact_599_boolean__algebra_Odisj__one__right,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),top_to8764741339697478167at_nat) = top_to8764741339697478167at_nat ) ).

% boolean_algebra.disj_one_right
tff(fact_600_boolean__algebra_Odisj__one__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),top_to6833984726390702231at_nat) = top_to6833984726390702231at_nat ) ).

% boolean_algebra.disj_one_right
tff(fact_601_boolean__algebra_Odisj__one__right,axiom,
    ! [X: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,X),top_top_set_list_nat) = top_top_set_list_nat ) ).

% boolean_algebra.disj_one_right
tff(fact_602_boolean__algebra_Odisj__one__right,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),top_top_set_o) = top_top_set_o ) ).

% boolean_algebra.disj_one_right
tff(fact_603_boolean__algebra_Odisj__one__right,axiom,
    ! [X: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,X),top_top_set_rat) = top_top_set_rat ) ).

% boolean_algebra.disj_one_right
tff(fact_604_boolean__algebra_Odisj__one__right,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),top_top_set_nat) = top_top_set_nat ) ).

% boolean_algebra.disj_one_right
tff(fact_605_boolean__algebra_Odisj__one__right,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),top_top_set_int) = top_top_set_int ) ).

% boolean_algebra.disj_one_right
tff(fact_606_sup__top__left,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,top_to8764741339697478167at_nat),X) = top_to8764741339697478167at_nat ) ).

% sup_top_left
tff(fact_607_sup__top__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,top_to6833984726390702231at_nat),X) = top_to6833984726390702231at_nat ) ).

% sup_top_left
tff(fact_608_sup__top__left,axiom,
    ! [X: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,top_top_set_list_nat),X) = top_top_set_list_nat ) ).

% sup_top_left
tff(fact_609_sup__top__left,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,top_top_set_o),X) = top_top_set_o ) ).

% sup_top_left
tff(fact_610_sup__top__left,axiom,
    ! [X: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,top_top_set_rat),X) = top_top_set_rat ) ).

% sup_top_left
tff(fact_611_sup__top__left,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,top_top_set_nat),X) = top_top_set_nat ) ).

% sup_top_left
tff(fact_612_sup__top__left,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,top_top_set_int),X) = top_top_set_int ) ).

% sup_top_left
tff(fact_613_sup__top__right,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),top_to8764741339697478167at_nat) = top_to8764741339697478167at_nat ) ).

% sup_top_right
tff(fact_614_sup__top__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),top_to6833984726390702231at_nat) = top_to6833984726390702231at_nat ) ).

% sup_top_right
tff(fact_615_sup__top__right,axiom,
    ! [X: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,X),top_top_set_list_nat) = top_top_set_list_nat ) ).

% sup_top_right
tff(fact_616_sup__top__right,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),top_top_set_o) = top_top_set_o ) ).

% sup_top_right
tff(fact_617_sup__top__right,axiom,
    ! [X: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,X),top_top_set_rat) = top_top_set_rat ) ).

% sup_top_right
tff(fact_618_sup__top__right,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),top_top_set_nat) = top_top_set_nat ) ).

% sup_top_right
tff(fact_619_sup__top__right,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),top_top_set_int) = top_top_set_int ) ).

% sup_top_right
tff(fact_620_boolean__algebra_Ocompl__one,axiom,
    aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,top_to4669805908274784177at_nat) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.compl_one
tff(fact_621_boolean__algebra_Ocompl__one,axiom,
    aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,top_top_set_list_nat) = bot_bot_set_list_nat ).

% boolean_algebra.compl_one
tff(fact_622_boolean__algebra_Ocompl__one,axiom,
    aa_set_o_set_o(uminus_uminus_set_o,top_top_set_o) = bot_bot_set_o ).

% boolean_algebra.compl_one
tff(fact_623_boolean__algebra_Ocompl__one,axiom,
    aa_set_rat_set_rat(uminus2201863774496077783et_rat,top_top_set_rat) = bot_bot_set_rat ).

% boolean_algebra.compl_one
tff(fact_624_boolean__algebra_Ocompl__one,axiom,
    aa_set_nat_set_nat(uminus5710092332889474511et_nat,top_top_set_nat) = bot_bot_set_nat ).

% boolean_algebra.compl_one
tff(fact_625_boolean__algebra_Ocompl__one,axiom,
    aa_set_int_set_int(uminus1532241313380277803et_int,top_top_set_int) = bot_bot_set_int ).

% boolean_algebra.compl_one
tff(fact_626_boolean__algebra_Ocompl__one,axiom,
    aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,top_top_Product_unit) = bot_bot_Product_unit ).

% boolean_algebra.compl_one
tff(fact_627_boolean__algebra_Ocompl__zero,axiom,
    aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,bot_bo2099793752762293965at_nat) = top_to4669805908274784177at_nat ).

% boolean_algebra.compl_zero
tff(fact_628_boolean__algebra_Ocompl__zero,axiom,
    aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,bot_bot_set_list_nat) = top_top_set_list_nat ).

% boolean_algebra.compl_zero
tff(fact_629_boolean__algebra_Ocompl__zero,axiom,
    aa_set_o_set_o(uminus_uminus_set_o,bot_bot_set_o) = top_top_set_o ).

% boolean_algebra.compl_zero
tff(fact_630_boolean__algebra_Ocompl__zero,axiom,
    aa_set_rat_set_rat(uminus2201863774496077783et_rat,bot_bot_set_rat) = top_top_set_rat ).

% boolean_algebra.compl_zero
tff(fact_631_boolean__algebra_Ocompl__zero,axiom,
    aa_set_nat_set_nat(uminus5710092332889474511et_nat,bot_bot_set_nat) = top_top_set_nat ).

% boolean_algebra.compl_zero
tff(fact_632_boolean__algebra_Ocompl__zero,axiom,
    aa_set_int_set_int(uminus1532241313380277803et_int,bot_bot_set_int) = top_top_set_int ).

% boolean_algebra.compl_zero
tff(fact_633_boolean__algebra_Ocompl__zero,axiom,
    aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,bot_bot_Product_unit) = top_top_Product_unit ).

% boolean_algebra.compl_zero
tff(fact_634_inf__compl__bot__left1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)) = bot_bo3516217578726220278_int_o ) ).

% inf_compl_bot_left1
tff(fact_635_inf__compl__bot__left1,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,X)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)) = bot_bo8860746597864711612char_o ) ).

% inf_compl_bot_left1
tff(fact_636_inf__compl__bot__left1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_left1
tff(fact_637_inf__compl__bot__left1,axiom,
    ! [X: set_o,Y: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),Y)) = bot_bot_set_o ) ).

% inf_compl_bot_left1
tff(fact_638_inf__compl__bot__left1,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = bot_bot_set_nat ) ).

% inf_compl_bot_left1
tff(fact_639_inf__compl__bot__left1,axiom,
    ! [X: set_int,Y: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)) = bot_bot_set_int ) ).

% inf_compl_bot_left1
tff(fact_640_inf__compl__bot__left1,axiom,
    ! [X: product_unit,Y: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)) = bot_bot_Product_unit ) ).

% inf_compl_bot_left1
tff(fact_641_inf__compl__bot__left2,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)),Y)) = bot_bo3516217578726220278_int_o ) ).

% inf_compl_bot_left2
tff(fact_642_inf__compl__bot__left2,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,X)),Y)) = bot_bo8860746597864711612char_o ) ).

% inf_compl_bot_left2
tff(fact_643_inf__compl__bot__left2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),Y)) = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_left2
tff(fact_644_inf__compl__bot__left2,axiom,
    ! [X: set_o,Y: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),Y)) = bot_bot_set_o ) ).

% inf_compl_bot_left2
tff(fact_645_inf__compl__bot__left2,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),Y)) = bot_bot_set_nat ) ).

% inf_compl_bot_left2
tff(fact_646_inf__compl__bot__left2,axiom,
    ! [X: set_int,Y: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),Y)) = bot_bot_set_int ) ).

% inf_compl_bot_left2
tff(fact_647_inf__compl__bot__left2,axiom,
    ! [X: product_unit,Y: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),Y)) = bot_bot_Product_unit ) ).

% inf_compl_bot_left2
tff(fact_648_inf__compl__bot__right,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X))) = bot_bo3516217578726220278_int_o ) ).

% inf_compl_bot_right
tff(fact_649_inf__compl__bot__right,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,X))) = bot_bo8860746597864711612char_o ) ).

% inf_compl_bot_right
tff(fact_650_inf__compl__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X))) = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_right
tff(fact_651_inf__compl__bot__right,axiom,
    ! [X: set_o,Y: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Y),aa_set_o_set_o(uminus_uminus_set_o,X))) = bot_bot_set_o ) ).

% inf_compl_bot_right
tff(fact_652_inf__compl__bot__right,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(uminus5710092332889474511et_nat,X))) = bot_bot_set_nat ) ).

% inf_compl_bot_right
tff(fact_653_inf__compl__bot__right,axiom,
    ! [X: set_int,Y: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),aa_set_int_set_int(uminus1532241313380277803et_int,X))) = bot_bot_set_int ) ).

% inf_compl_bot_right
tff(fact_654_inf__compl__bot__right,axiom,
    ! [X: product_unit,Y: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X))) = bot_bot_Product_unit ) ).

% inf_compl_bot_right
tff(fact_655_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)),X) = bot_bo3516217578726220278_int_o ) ).

% boolean_algebra.conj_cancel_left
tff(fact_656_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,X)),X) = bot_bo8860746597864711612char_o ) ).

% boolean_algebra.conj_cancel_left
tff(fact_657_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),X) = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_cancel_left
tff(fact_658_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),X) = bot_bot_set_o ) ).

% boolean_algebra.conj_cancel_left
tff(fact_659_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),X) = bot_bot_set_nat ) ).

% boolean_algebra.conj_cancel_left
tff(fact_660_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),X) = bot_bot_set_int ) ).

% boolean_algebra.conj_cancel_left
tff(fact_661_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),X) = bot_bot_Product_unit ) ).

% boolean_algebra.conj_cancel_left
tff(fact_662_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)) = bot_bo3516217578726220278_int_o ) ).

% boolean_algebra.conj_cancel_right
tff(fact_663_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,X)) = bot_bo8860746597864711612char_o ) ).

% boolean_algebra.conj_cancel_right
tff(fact_664_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)) = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_cancel_right
tff(fact_665_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),aa_set_o_set_o(uminus_uminus_set_o,X)) = bot_bot_set_o ) ).

% boolean_algebra.conj_cancel_right
tff(fact_666_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)) = bot_bot_set_nat ) ).

% boolean_algebra.conj_cancel_right
tff(fact_667_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(uminus1532241313380277803et_int,X)) = bot_bot_set_int ) ).

% boolean_algebra.conj_cancel_right
tff(fact_668_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)) = bot_bot_Product_unit ) ).

% boolean_algebra.conj_cancel_right
tff(fact_669_sup__compl__top__left1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,X)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) = top_to8764741339697478167at_nat ) ).

% sup_compl_top_left1
tff(fact_670_sup__compl__top__left1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = top_to6833984726390702231at_nat ) ).

% sup_compl_top_left1
tff(fact_671_sup__compl__top__left1,axiom,
    ! [X: set_list_nat,Y: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,X)),aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,X),Y)) = top_top_set_list_nat ) ).

% sup_compl_top_left1
tff(fact_672_sup__compl__top__left1,axiom,
    ! [X: set_o,Y: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),Y)) = top_top_set_o ) ).

% sup_compl_top_left1
tff(fact_673_sup__compl__top__left1,axiom,
    ! [X: set_rat,Y: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,aa_set_rat_set_rat(uminus2201863774496077783et_rat,X)),aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,X),Y)) = top_top_set_rat ) ).

% sup_compl_top_left1
tff(fact_674_sup__compl__top__left1,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = top_top_set_nat ) ).

% sup_compl_top_left1
tff(fact_675_sup__compl__top__left1,axiom,
    ! [X: set_int,Y: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)) = top_top_set_int ) ).

% sup_compl_top_left1
tff(fact_676_sup__compl__top__left1,axiom,
    ! [X: product_unit,Y: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y)) = top_top_Product_unit ) ).

% sup_compl_top_left1
tff(fact_677_sup__compl__top__left2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,X)),Y)) = top_to8764741339697478167at_nat ) ).

% sup_compl_top_left2
tff(fact_678_sup__compl__top__left2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(X)),Y)) = top_to6833984726390702231at_nat ) ).

% sup_compl_top_left2
tff(fact_679_sup__compl__top__left2,axiom,
    ! [X: set_list_nat,Y: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,X),aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,X)),Y)) = top_top_set_list_nat ) ).

% sup_compl_top_left2
tff(fact_680_sup__compl__top__left2,axiom,
    ! [X: set_o,Y: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),Y)) = top_top_set_o ) ).

% sup_compl_top_left2
tff(fact_681_sup__compl__top__left2,axiom,
    ! [X: set_rat,Y: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,X),aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,aa_set_rat_set_rat(uminus2201863774496077783et_rat,X)),Y)) = top_top_set_rat ) ).

% sup_compl_top_left2
tff(fact_682_sup__compl__top__left2,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),Y)) = top_top_set_nat ) ).

% sup_compl_top_left2
tff(fact_683_sup__compl__top__left2,axiom,
    ! [X: set_int,Y: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),Y)) = top_top_set_int ) ).

% sup_compl_top_left2
tff(fact_684_sup__compl__top__left2,axiom,
    ! [X: product_unit,Y: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),Y)) = top_top_Product_unit ) ).

% sup_compl_top_left2
tff(fact_685_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,X)),X) = top_to8764741339697478167at_nat ) ).

% boolean_algebra.disj_cancel_left
tff(fact_686_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(X)),X) = top_to6833984726390702231at_nat ) ).

% boolean_algebra.disj_cancel_left
tff(fact_687_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,X)),X) = top_top_set_list_nat ) ).

% boolean_algebra.disj_cancel_left
tff(fact_688_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),X) = top_top_set_o ) ).

% boolean_algebra.disj_cancel_left
tff(fact_689_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,aa_set_rat_set_rat(uminus2201863774496077783et_rat,X)),X) = top_top_set_rat ) ).

% boolean_algebra.disj_cancel_left
tff(fact_690_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),X) = top_top_set_nat ) ).

% boolean_algebra.disj_cancel_left
tff(fact_691_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),X) = top_top_set_int ) ).

% boolean_algebra.disj_cancel_left
tff(fact_692_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),X) = top_top_Product_unit ) ).

% boolean_algebra.disj_cancel_left
tff(fact_693_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,X)) = top_to8764741339697478167at_nat ) ).

% boolean_algebra.disj_cancel_right
tff(fact_694_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),uminus935396558254630718at_nat(X)) = top_to6833984726390702231at_nat ) ).

% boolean_algebra.disj_cancel_right
tff(fact_695_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,X),aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,X)) = top_top_set_list_nat ) ).

% boolean_algebra.disj_cancel_right
tff(fact_696_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),aa_set_o_set_o(uminus_uminus_set_o,X)) = top_top_set_o ) ).

% boolean_algebra.disj_cancel_right
tff(fact_697_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,X),aa_set_rat_set_rat(uminus2201863774496077783et_rat,X)) = top_top_set_rat ) ).

% boolean_algebra.disj_cancel_right
tff(fact_698_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)) = top_top_set_nat ) ).

% boolean_algebra.disj_cancel_right
tff(fact_699_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),aa_set_int_set_int(uminus1532241313380277803et_int,X)) = top_top_set_int ) ).

% boolean_algebra.disj_cancel_right
tff(fact_700_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)) = top_top_Product_unit ) ).

% boolean_algebra.disj_cancel_right
tff(fact_701_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Y)) ) ).

% boolean_algebra.de_Morgan_conj
tff(fact_702_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,Y)) ) ).

% boolean_algebra.de_Morgan_conj
tff(fact_703_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : ( aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,X)),aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,Y)) ) ).

% boolean_algebra.de_Morgan_conj
tff(fact_704_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,X)),aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,Y)) ) ).

% boolean_algebra.de_Morgan_conj
tff(fact_705_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(X)),uminus935396558254630718at_nat(Y)) ) ).

% boolean_algebra.de_Morgan_conj
tff(fact_706_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ) ).

% boolean_algebra.de_Morgan_conj
tff(fact_707_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: product_unit,Y: product_unit] : ( aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)) ) ).

% boolean_algebra.de_Morgan_conj
tff(fact_708_mult_Omonoid__axioms,axiom,
    aa_Code_integer_o(monoid_Code_integer(times_3573771949741848930nteger),one_one_Code_integer) ).

% mult.monoid_axioms
tff(fact_709_mult_Omonoid__axioms,axiom,
    aa_assn_o(monoid_assn(times_times_assn),one_one_assn) ).

% mult.monoid_axioms
tff(fact_710_mult_Omonoid__axioms,axiom,
    aa_rat_o(monoid_rat(times_times_rat),one_one_rat) ).

% mult.monoid_axioms
tff(fact_711_mult_Omonoid__axioms,axiom,
    aa_nat_o(monoid_nat(times_times_nat),one_one_nat) ).

% mult.monoid_axioms
tff(fact_712_mult_Omonoid__axioms,axiom,
    aa_int_o(monoid_int(times_times_int),one_one_int) ).

% mult.monoid_axioms
tff(fact_713_mult_Ocomm__monoid__axioms,axiom,
    aa_Code_integer_o(comm_m1654649897431166824nteger(times_3573771949741848930nteger),one_one_Code_integer) ).

% mult.comm_monoid_axioms
tff(fact_714_mult_Ocomm__monoid__axioms,axiom,
    aa_assn_o(comm_monoid_assn(times_times_assn),one_one_assn) ).

% mult.comm_monoid_axioms
tff(fact_715_mult_Ocomm__monoid__axioms,axiom,
    aa_rat_o(comm_monoid_rat(times_times_rat),one_one_rat) ).

% mult.comm_monoid_axioms
tff(fact_716_mult_Ocomm__monoid__axioms,axiom,
    aa_nat_o(comm_monoid_nat(times_times_nat),one_one_nat) ).

% mult.comm_monoid_axioms
tff(fact_717_mult_Ocomm__monoid__axioms,axiom,
    aa_int_o(comm_monoid_int(times_times_int),one_one_int) ).

% mult.comm_monoid_axioms
tff(fact_718_semilattice__neutr_Oaxioms_I2_J,axiom,
    ! [F: fun_nat_fun_nat_nat,Z: nat] :
      ( aa_nat_o(semila9081495762789891438tr_nat(F),Z)
     => aa_nat_o(comm_monoid_nat(F),Z) ) ).

% semilattice_neutr.axioms(2)
tff(fact_719_monoid_Oleft__neutral,axiom,
    ! [F: fun_nat_fun_nat_nat,Z: nat,A: nat] :
      ( aa_nat_o(monoid_nat(F),Z)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(F,Z),A) = A ) ) ).

% monoid.left_neutral
tff(fact_720_monoid_Oright__neutral,axiom,
    ! [F: fun_nat_fun_nat_nat,Z: nat,A: nat] :
      ( aa_nat_o(monoid_nat(F),Z)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(F,A),Z) = A ) ) ).

% monoid.right_neutral
tff(fact_721_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] : ( X != produc7507926704131184380et_nat(H,As4) ) ).

% one_assn_raw.cases
tff(fact_722_comm__monoid_Ocomm__neutral,axiom,
    ! [F: fun_nat_fun_nat_nat,Z: nat,A: nat] :
      ( aa_nat_o(comm_monoid_nat(F),Z)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(F,A),Z) = A ) ) ).

% comm_monoid.comm_neutral
tff(fact_723_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_724_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_725_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_726_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_727_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_728_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_729_one__reorient,axiom,
    ! [X: code_integer] :
      ( ( one_one_Code_integer = X )
    <=> ( X = one_one_Code_integer ) ) ).

% one_reorient
tff(fact_730_one__reorient,axiom,
    ! [X: assn] :
      ( ( one_one_assn = X )
    <=> ( X = one_one_assn ) ) ).

% one_reorient
tff(fact_731_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_732_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_733_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_734_sup__cancel__left1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),A)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,X)),B)) = top_to8764741339697478167at_nat ) ).

% sup_cancel_left1
tff(fact_735_sup__cancel__left1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),A)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(X)),B)) = top_to6833984726390702231at_nat ) ).

% sup_cancel_left1
tff(fact_736_sup__cancel__left1,axiom,
    ! [X: set_list_nat,A: set_list_nat,B: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,X),A)),aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,X)),B)) = top_top_set_list_nat ) ).

% sup_cancel_left1
tff(fact_737_sup__cancel__left1,axiom,
    ! [X: set_o,A: set_o,B: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),A)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),B)) = top_top_set_o ) ).

% sup_cancel_left1
tff(fact_738_sup__cancel__left1,axiom,
    ! [X: set_rat,A: set_rat,B: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,X),A)),aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,aa_set_rat_set_rat(uminus2201863774496077783et_rat,X)),B)) = top_top_set_rat ) ).

% sup_cancel_left1
tff(fact_739_sup__cancel__left1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),B)) = top_top_set_nat ) ).

% sup_cancel_left1
tff(fact_740_sup__cancel__left1,axiom,
    ! [X: set_int,A: set_int,B: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),A)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),B)) = top_top_set_int ) ).

% sup_cancel_left1
tff(fact_741_sup__cancel__left1,axiom,
    ! [X: product_unit,A: product_unit,B: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),A)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),B)) = top_top_Product_unit ) ).

% sup_cancel_left1
tff(fact_742_sup__cancel__left2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,X)),A)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),B)) = top_to8764741339697478167at_nat ) ).

% sup_cancel_left2
tff(fact_743_sup__cancel__left2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(X)),A)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),B)) = top_to6833984726390702231at_nat ) ).

% sup_cancel_left2
tff(fact_744_sup__cancel__left2,axiom,
    ! [X: set_list_nat,A: set_list_nat,B: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,X)),A)),aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,X),B)) = top_top_set_list_nat ) ).

% sup_cancel_left2
tff(fact_745_sup__cancel__left2,axiom,
    ! [X: set_o,A: set_o,B: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),A)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),B)) = top_top_set_o ) ).

% sup_cancel_left2
tff(fact_746_sup__cancel__left2,axiom,
    ! [X: set_rat,A: set_rat,B: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,aa_set_rat_set_rat(uminus2201863774496077783et_rat,X)),A)),aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,X),B)) = top_top_set_rat ) ).

% sup_cancel_left2
tff(fact_747_sup__cancel__left2,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),B)) = top_top_set_nat ) ).

% sup_cancel_left2
tff(fact_748_sup__cancel__left2,axiom,
    ! [X: set_int,A: set_int,B: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),A)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),B)) = top_top_set_int ) ).

% sup_cancel_left2
tff(fact_749_sup__cancel__left2,axiom,
    ! [X: product_unit,A: product_unit,B: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),A)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),B)) = top_top_Product_unit ) ).

% sup_cancel_left2
tff(fact_750_inf__top_Ocomm__monoid__axioms,axiom,
    aa_set3702741264012733747_nat_o(comm_m5709050211238513046at_nat(inf_in2572325071724192079at_nat),top_to4669805908274784177at_nat) ).

% inf_top.comm_monoid_axioms
tff(fact_751_inf__top_Ocomm__monoid__axioms,axiom,
    comm_m285090526141560045_int_o(inf_in1782598092406300404_int_o,top_to8487252020945707026_int_o) ).

% inf_top.comm_monoid_axioms
tff(fact_752_inf__top_Ocomm__monoid__axioms,axiom,
    comm_m6974863962029961733char_o(inf_in94156054473216830char_o,top_to7404587909000090784char_o) ).

% inf_top.comm_monoid_axioms
tff(fact_753_inf__top_Ocomm__monoid__axioms,axiom,
    comm_m4511514676032769697st_nat(inf_inf_set_list_nat,top_top_set_list_nat) ).

% inf_top.comm_monoid_axioms
tff(fact_754_inf__top_Ocomm__monoid__axioms,axiom,
    aa_set_o_o(comm_monoid_set_o(inf_inf_set_o),top_top_set_o) ).

% inf_top.comm_monoid_axioms
tff(fact_755_inf__top_Ocomm__monoid__axioms,axiom,
    aa_set_rat_o(comm_monoid_set_rat(inf_inf_set_rat),top_top_set_rat) ).

% inf_top.comm_monoid_axioms
tff(fact_756_inf__top_Ocomm__monoid__axioms,axiom,
    aa_set_nat_o(comm_monoid_set_nat(inf_inf_set_nat),top_top_set_nat) ).

% inf_top.comm_monoid_axioms
tff(fact_757_inf__top_Ocomm__monoid__axioms,axiom,
    aa_set_int_o(comm_monoid_set_int(inf_inf_set_int),top_top_set_int) ).

% inf_top.comm_monoid_axioms
tff(fact_758_inf__top_Omonoid__axioms,axiom,
    aa_set3702741264012733747_nat_o(monoid1519094961268053602at_nat(inf_in2572325071724192079at_nat),top_to4669805908274784177at_nat) ).

% inf_top.monoid_axioms
tff(fact_759_inf__top_Omonoid__axioms,axiom,
    monoid8055414245428884769_int_o(inf_in1782598092406300404_int_o,top_to8487252020945707026_int_o) ).

% inf_top.monoid_axioms
tff(fact_760_inf__top_Omonoid__axioms,axiom,
    monoid7854550568138350801char_o(inf_in94156054473216830char_o,top_to7404587909000090784char_o) ).

% inf_top.monoid_axioms
tff(fact_761_inf__top_Omonoid__axioms,axiom,
    monoid_set_list_nat(inf_inf_set_list_nat,top_top_set_list_nat) ).

% inf_top.monoid_axioms
tff(fact_762_inf__top_Omonoid__axioms,axiom,
    aa_set_o_o(monoid_set_o(inf_inf_set_o),top_top_set_o) ).

% inf_top.monoid_axioms
tff(fact_763_inf__top_Omonoid__axioms,axiom,
    aa_set_rat_o(monoid_set_rat(inf_inf_set_rat),top_top_set_rat) ).

% inf_top.monoid_axioms
tff(fact_764_inf__top_Omonoid__axioms,axiom,
    aa_set_nat_o(monoid_set_nat(inf_inf_set_nat),top_top_set_nat) ).

% inf_top.monoid_axioms
tff(fact_765_inf__top_Omonoid__axioms,axiom,
    aa_set_int_o(monoid_set_int(inf_inf_set_int),top_top_set_int) ).

% inf_top.monoid_axioms
tff(fact_766_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A) = A ) ).

% comm_monoid_mult_class.mult_1
tff(fact_767_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ) ).

% comm_monoid_mult_class.mult_1
tff(fact_768_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ) ).

% comm_monoid_mult_class.mult_1
tff(fact_769_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_770_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_771_mult_Ocomm__neutral,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),one_one_Code_integer) = A ) ).

% mult.comm_neutral
tff(fact_772_mult_Ocomm__neutral,axiom,
    ! [A: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ) ).

% mult.comm_neutral
tff(fact_773_mult_Ocomm__neutral,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ) ).

% mult.comm_neutral
tff(fact_774_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_775_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_776_inf__top_Osemilattice__neutr__axioms,axiom,
    aa_set3702741264012733747_nat_o(semila858173770281612099at_nat(inf_in2572325071724192079at_nat),top_to4669805908274784177at_nat) ).

% inf_top.semilattice_neutr_axioms
tff(fact_777_inf__top_Osemilattice__neutr__axioms,axiom,
    semila8824639500271816704_int_o(inf_in1782598092406300404_int_o,top_to8487252020945707026_int_o) ).

% inf_top.semilattice_neutr_axioms
tff(fact_778_inf__top_Osemilattice__neutr__axioms,axiom,
    semila1992739488332178226char_o(inf_in94156054473216830char_o,top_to7404587909000090784char_o) ).

% inf_top.semilattice_neutr_axioms
tff(fact_779_inf__top_Osemilattice__neutr__axioms,axiom,
    semila7136768128396663732st_nat(inf_inf_set_list_nat,top_top_set_list_nat) ).

% inf_top.semilattice_neutr_axioms
tff(fact_780_inf__top_Osemilattice__neutr__axioms,axiom,
    aa_set_o_o(semila1065760912024005978_set_o(inf_inf_set_o),top_top_set_o) ).

% inf_top.semilattice_neutr_axioms
tff(fact_781_inf__top_Osemilattice__neutr__axioms,axiom,
    aa_set_rat_o(semila6956917442496717612et_rat(inf_inf_set_rat),top_top_set_rat) ).

% inf_top.semilattice_neutr_axioms
tff(fact_782_inf__top_Osemilattice__neutr__axioms,axiom,
    aa_set_nat_o(semila1241773964035338532et_nat(inf_inf_set_nat),top_top_set_nat) ).

% inf_top.semilattice_neutr_axioms
tff(fact_783_inf__top_Osemilattice__neutr__axioms,axiom,
    aa_set_int_o(semila6287294981380917632et_int(inf_inf_set_int),top_top_set_int) ).

% inf_top.semilattice_neutr_axioms
tff(fact_784_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),top_to4669805908274784177at_nat) = X ) ).

% boolean_algebra.conj_one_right
tff(fact_785_boolean__algebra_Oconj__one__right,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),top_to8487252020945707026_int_o) = X ) ).

% boolean_algebra.conj_one_right
tff(fact_786_boolean__algebra_Oconj__one__right,axiom,
    ! [X: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),top_to7404587909000090784char_o) = X ) ).

% boolean_algebra.conj_one_right
tff(fact_787_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,X),top_top_set_list_nat) = X ) ).

% boolean_algebra.conj_one_right
tff(fact_788_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),top_top_set_o) = X ) ).

% boolean_algebra.conj_one_right
tff(fact_789_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,X),top_top_set_rat) = X ) ).

% boolean_algebra.conj_one_right
tff(fact_790_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),top_top_set_nat) = X ) ).

% boolean_algebra.conj_one_right
tff(fact_791_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),top_top_set_int) = X ) ).

% boolean_algebra.conj_one_right
tff(fact_792_mod__h__bot__indep,axiom,
    ! [P: assn,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
    <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),produc7507926704131184380et_nat(H3,bot_bot_set_nat)) ) ).

% mod_h_bot_indep
tff(fact_793_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = bot_bot_Product_unit )
     => ( ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y) = top_top_Product_unit )
       => ( aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_794_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),Y) = bot_bot_set_o )
     => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),Y) = top_top_set_o )
       => ( aa_set_o_set_o(uminus_uminus_set_o,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_795_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_rat,Y: set_rat] :
      ( ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,X),Y) = bot_bot_set_rat )
     => ( ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,X),Y) = top_top_set_rat )
       => ( aa_set_rat_set_rat(uminus2201863774496077783et_rat,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_796_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = bot_bot_set_nat )
     => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = top_top_set_nat )
       => ( aa_set_nat_set_nat(uminus5710092332889474511et_nat,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_797_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = bot_bot_set_int )
     => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = top_top_set_int )
       => ( aa_set_int_set_int(uminus1532241313380277803et_int,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_798_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_list_nat,Y: set_list_nat] :
      ( ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,X),Y) = bot_bot_set_list_nat )
     => ( ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,X),Y) = top_top_set_list_nat )
       => ( aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_799_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] :
      ( ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y) = bot_bo8860746597864711612char_o )
     => ( ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),Y) = top_to7404587909000090784char_o )
       => ( aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_800_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = bot_bo2099793752762293965at_nat )
     => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y) = top_to4669805908274784177at_nat )
       => ( aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_801_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = bot_bo3516217578726220278_int_o )
     => ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y) = top_to8487252020945707026_int_o )
       => ( aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_802_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y) = bot_bo8422036546324065075at_nat )
     => ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = top_to8764741339697478167at_nat )
       => ( aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_803_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As4) )
           => ( As4 = bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(3)
tff(fact_804_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As4) )
           => ( As4 != bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(2)
tff(fact_805_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As4) )
           => ( (Y)
            <=> ( As4 != bot_bot_set_nat ) ) ) ) ).

% one_assn_raw.elims(1)
tff(fact_806_one__assn__raw_Osimps,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,produc7507926704131184380et_nat(H2,As))
    <=> ( As = bot_bot_set_nat ) ) ).

% one_assn_raw.simps
tff(fact_807_times__assn__raw_Osimps,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(P),Q),produc7507926704131184380et_nat(H2,As))
    <=> ? [As1: set_nat,As2: set_nat] :
          ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As2) )
          & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As2) = bot_bot_set_nat )
          & aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H2,As1))
          & aa_Pro653082453696054616_nat_o(Q,produc7507926704131184380et_nat(H2,As2)) ) ) ).

% times_assn_raw.simps
tff(fact_808_times__assn__raw_Oelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As4) )
           => ( (Y)
            <=> ~ ? [As1: set_nat,As2: set_nat] :
                    ( ( As4 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As2) )
                    & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As2) = bot_bot_set_nat )
                    & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H,As1))
                    & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As2)) ) ) ) ) ).

% times_assn_raw.elims(1)
tff(fact_809_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
     => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As4) )
           => ~ ? [As12: set_nat,As22: set_nat] :
                  ( ( As4 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As12),As22) )
                  & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As12),As22) = bot_bot_set_nat )
                  & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H,As12))
                  & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As22)) ) ) ) ).

% times_assn_raw.elims(2)
tff(fact_810_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
     => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As4) )
           => ? [As13: set_nat,As23: set_nat] :
                ( ( As4 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As13),As23) )
                & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As13),As23) = bot_bot_set_nat )
                & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H,As13))
                & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As23)) ) ) ) ).

% times_assn_raw.elims(3)
tff(fact_811_assn__one__left,axiom,
    ! [P: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),P) = P ) ).

% assn_one_left
tff(fact_812_inf__cancel__left1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),A)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)),B)) = bot_bo3516217578726220278_int_o ) ).

% inf_cancel_left1
tff(fact_813_inf__cancel__left1,axiom,
    ! [X: fun_list_char_o,A: fun_list_char_o,B: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),A)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,X)),B)) = bot_bo8860746597864711612char_o ) ).

% inf_cancel_left1
tff(fact_814_inf__cancel__left1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),A)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),B)) = bot_bo2099793752762293965at_nat ) ).

% inf_cancel_left1
tff(fact_815_inf__cancel__left1,axiom,
    ! [X: set_o,A: set_o,B: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),A)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),B)) = bot_bot_set_o ) ).

% inf_cancel_left1
tff(fact_816_inf__cancel__left1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),B)) = bot_bot_set_nat ) ).

% inf_cancel_left1
tff(fact_817_inf__cancel__left1,axiom,
    ! [X: set_int,A: set_int,B: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),A)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),B)) = bot_bot_set_int ) ).

% inf_cancel_left1
tff(fact_818_inf__cancel__left1,axiom,
    ! [X: product_unit,A: product_unit,B: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),A)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),B)) = bot_bot_Product_unit ) ).

% inf_cancel_left1
tff(fact_819_inf__cancel__left2,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)),A)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),B)) = bot_bo3516217578726220278_int_o ) ).

% inf_cancel_left2
tff(fact_820_inf__cancel__left2,axiom,
    ! [X: fun_list_char_o,A: fun_list_char_o,B: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,X)),A)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),B)) = bot_bo8860746597864711612char_o ) ).

% inf_cancel_left2
tff(fact_821_inf__cancel__left2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),A)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),B)) = bot_bo2099793752762293965at_nat ) ).

% inf_cancel_left2
tff(fact_822_inf__cancel__left2,axiom,
    ! [X: set_o,A: set_o,B: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),A)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),B)) = bot_bot_set_o ) ).

% inf_cancel_left2
tff(fact_823_inf__cancel__left2,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),B)) = bot_bot_set_nat ) ).

% inf_cancel_left2
tff(fact_824_inf__cancel__left2,axiom,
    ! [X: set_int,A: set_int,B: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),A)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),B)) = bot_bot_set_int ) ).

% inf_cancel_left2
tff(fact_825_inf__cancel__left2,axiom,
    ! [X: product_unit,A: product_unit,B: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),A)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),B)) = bot_bot_Product_unit ) ).

% inf_cancel_left2
tff(fact_826_mult__minus1,axiom,
    ! [Z: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,one_one_int)),Z) = aa_int_int(uminus_uminus_int,Z) ) ).

% mult_minus1
tff(fact_827_mult__minus1,axiom,
    ! [Z: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Z) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ) ).

% mult_minus1
tff(fact_828_mult__minus1,axiom,
    ! [Z: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ) ).

% mult_minus1
tff(fact_829_mult__minus1__right,axiom,
    ! [Z: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z) ) ).

% mult_minus1_right
tff(fact_830_mult__minus1__right,axiom,
    ! [Z: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ) ).

% mult_minus1_right
tff(fact_831_mult__minus1__right,axiom,
    ! [Z: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ) ).

% mult_minus1_right
tff(fact_832_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T)),S) = S ) ).

% Un_Int_eq(1)
tff(fact_833_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,S),T)),S) = S ) ).

% Un_Int_eq(1)
tff(fact_834_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T)),S) = S ) ).

% Un_Int_eq(1)
tff(fact_835_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T)),S) = S ) ).

% Un_Int_eq(1)
tff(fact_836_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T)),T) = T ) ).

% Un_Int_eq(2)
tff(fact_837_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,S),T)),T) = T ) ).

% Un_Int_eq(2)
tff(fact_838_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T)),T) = T ) ).

% Un_Int_eq(2)
tff(fact_839_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T)),T) = T ) ).

% Un_Int_eq(2)
tff(fact_840_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T)) = S ) ).

% Un_Int_eq(3)
tff(fact_841_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,S),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,S),T)) = S ) ).

% Un_Int_eq(3)
tff(fact_842_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T)) = S ) ).

% Un_Int_eq(3)
tff(fact_843_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T)) = S ) ).

% Un_Int_eq(3)
tff(fact_844_Un__Int__eq_I4_J,axiom,
    ! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,T),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T)) = T ) ).

% Un_Int_eq(4)
tff(fact_845_Un__Int__eq_I4_J,axiom,
    ! [T: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,T),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,S),T)) = T ) ).

% Un_Int_eq(4)
tff(fact_846_Un__Int__eq_I4_J,axiom,
    ! [T: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,T),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T)) = T ) ).

% Un_Int_eq(4)
tff(fact_847_Un__Int__eq_I4_J,axiom,
    ! [T: set_nat,S: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,T),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T)) = T ) ).

% Un_Int_eq(4)
tff(fact_848_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T)),S) = S ) ).

% Int_Un_eq(1)
tff(fact_849_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,S),T)),S) = S ) ).

% Int_Un_eq(1)
tff(fact_850_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T)),S) = S ) ).

% Int_Un_eq(1)
tff(fact_851_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T)),S) = S ) ).

% Int_Un_eq(1)
tff(fact_852_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T)),T) = T ) ).

% Int_Un_eq(2)
tff(fact_853_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,S),T)),T) = T ) ).

% Int_Un_eq(2)
tff(fact_854_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T)),T) = T ) ).

% Int_Un_eq(2)
tff(fact_855_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T)),T) = T ) ).

% Int_Un_eq(2)
tff(fact_856_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T)) = S ) ).

% Int_Un_eq(3)
tff(fact_857_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,S),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,S),T)) = S ) ).

% Int_Un_eq(3)
tff(fact_858_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T)) = S ) ).

% Int_Un_eq(3)
tff(fact_859_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T)) = S ) ).

% Int_Un_eq(3)
tff(fact_860_Int__Un__eq_I4_J,axiom,
    ! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,T),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T)) = T ) ).

% Int_Un_eq(4)
tff(fact_861_Int__Un__eq_I4_J,axiom,
    ! [T: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,T),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,S),T)) = T ) ).

% Int_Un_eq(4)
tff(fact_862_Int__Un__eq_I4_J,axiom,
    ! [T: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,T),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T)) = T ) ).

% Int_Un_eq(4)
tff(fact_863_Int__Un__eq_I4_J,axiom,
    ! [T: set_nat,S: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,T),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T)) = T ) ).

% Int_Un_eq(4)
tff(fact_864_Un__empty,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
      ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2) = bot_bo8422036546324065075at_nat )
    <=> ( ( A2 = bot_bo8422036546324065075at_nat )
        & ( B2 = bot_bo8422036546324065075at_nat ) ) ) ).

% Un_empty
tff(fact_865_Un__empty,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = bot_bo228742789529271731at_nat )
    <=> ( ( A2 = bot_bo228742789529271731at_nat )
        & ( B2 = bot_bo228742789529271731at_nat ) ) ) ).

% Un_empty
tff(fact_866_Un__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
    <=> ( ( A2 = bot_bo2099793752762293965at_nat )
        & ( B2 = bot_bo2099793752762293965at_nat ) ) ) ).

% Un_empty
tff(fact_867_Un__empty,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B2) = bot_bot_set_o )
    <=> ( ( A2 = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% Un_empty
tff(fact_868_Un__empty,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = bot_bot_set_nat )
    <=> ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% Un_empty
tff(fact_869_Un__empty,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2) = bot_bot_set_int )
    <=> ( ( A2 = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% Un_empty
tff(fact_870_Compl__disjoint2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,A2)),A2) = bot_bo2099793752762293965at_nat ) ).

% Compl_disjoint2
tff(fact_871_Compl__disjoint2,axiom,
    ! [A2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(uminus_uminus_set_o,A2)),A2) = bot_bot_set_o ) ).

% Compl_disjoint2
tff(fact_872_Compl__disjoint2,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)),A2) = bot_bot_set_nat ) ).

% Compl_disjoint2
tff(fact_873_Compl__disjoint2,axiom,
    ! [A2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,A2)),A2) = bot_bot_set_int ) ).

% Compl_disjoint2
tff(fact_874_empty__iff,axiom,
    ! [C: set_Pr958786334691620121nt_int] : ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),bot_bo1488462491386950373nt_int) ).

% empty_iff
tff(fact_875_empty__iff,axiom,
    ! [C: set_nat] : ~ aa_set_set_nat_o(member_set_nat(C),bot_bot_set_set_nat) ).

% empty_iff
tff(fact_876_empty__iff,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),bot_bo8632255151191677115_nat_o) ).

% empty_iff
tff(fact_877_empty__iff,axiom,
    ! [C: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),bot_bo2099793752762293965at_nat) ).

% empty_iff
tff(fact_878_empty__iff,axiom,
    ! [C: $o] : ~ aa_set_o_o(member_o((C)),bot_bot_set_o) ).

% empty_iff
tff(fact_879_empty__iff,axiom,
    ! [C: nat] : ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),bot_bot_set_nat) ).

% empty_iff
tff(fact_880_empty__iff,axiom,
    ! [C: int] : ~ aa_set_int_o(member_int(C),bot_bot_set_int) ).

% empty_iff
tff(fact_881_all__not__in__conv,axiom,
    ! [A2: set_se6260736226359567993nt_int] :
      ( ! [X4: set_Pr958786334691620121nt_int] : ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X4),A2)
    <=> ( A2 = bot_bo1488462491386950373nt_int ) ) ).

% all_not_in_conv
tff(fact_882_all__not__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ! [X4: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X4),A2)
    <=> ( A2 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_883_all__not__in__conv,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( ! [X4: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),A2)
    <=> ( A2 = bot_bo8632255151191677115_nat_o ) ) ).

% all_not_in_conv
tff(fact_884_all__not__in__conv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A2)
    <=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
tff(fact_885_all__not__in__conv,axiom,
    ! [A2: set_o] :
      ( ! [X4: $o] : ~ aa_set_o_o(member_o((X4)),A2)
    <=> ( A2 = bot_bot_set_o ) ) ).

% all_not_in_conv
tff(fact_886_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ! [X4: nat] : ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A2)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_887_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ! [X4: int] : ~ aa_set_int_o(member_int(X4),A2)
    <=> ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_888_Collect__empty__eq,axiom,
    ! [P: fun_se7249947736802449410_int_o] :
      ( ( collec5210948495886036740nt_int(P) = bot_bo1488462491386950373nt_int )
    <=> ! [X4: set_Pr958786334691620121nt_int] : ~ aa_set6170482117830564987_int_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_889_Collect__empty__eq,axiom,
    ! [P: fun_set_nat_o] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ! [X4: set_nat] : ~ aa_set_nat_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_890_Collect__empty__eq,axiom,
    ! [P: fun_fu5135969500543783852at_o_o] :
      ( ( collec6407654713642994138_nat_o(P) = bot_bo8632255151191677115_nat_o )
    <=> ! [X4: fun_Pr6460944699146269935_nat_o] : ~ aa_fun1675921005893230629at_o_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_891_Collect__empty__eq,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( ( collec3392354462482085612at_nat(P) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_892_Collect__empty__eq,axiom,
    ! [P: fun_o_o] :
      ( ( collect_o(P) = bot_bot_set_o )
    <=> ! [X4: $o] : ~ aa_o_o(P,(X4)) ) ).

% Collect_empty_eq
tff(fact_893_Collect__empty__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ! [X4: nat] : ~ aa_nat_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_894_Collect__empty__eq,axiom,
    ! [P: fun_int_o] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ! [X4: int] : ~ aa_int_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_895_empty__Collect__eq,axiom,
    ! [P: fun_se7249947736802449410_int_o] :
      ( ( bot_bo1488462491386950373nt_int = collec5210948495886036740nt_int(P) )
    <=> ! [X4: set_Pr958786334691620121nt_int] : ~ aa_set6170482117830564987_int_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_896_empty__Collect__eq,axiom,
    ! [P: fun_set_nat_o] :
      ( ( bot_bot_set_set_nat = collect_set_nat(P) )
    <=> ! [X4: set_nat] : ~ aa_set_nat_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_897_empty__Collect__eq,axiom,
    ! [P: fun_fu5135969500543783852at_o_o] :
      ( ( bot_bo8632255151191677115_nat_o = collec6407654713642994138_nat_o(P) )
    <=> ! [X4: fun_Pr6460944699146269935_nat_o] : ~ aa_fun1675921005893230629at_o_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_898_empty__Collect__eq,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( ( bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(P) )
    <=> ! [X4: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_899_empty__Collect__eq,axiom,
    ! [P: fun_o_o] :
      ( ( bot_bot_set_o = collect_o(P) )
    <=> ! [X4: $o] : ~ aa_o_o(P,(X4)) ) ).

% empty_Collect_eq
tff(fact_900_empty__Collect__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( bot_bot_set_nat = collect_nat(P) )
    <=> ! [X4: nat] : ~ aa_nat_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_901_empty__Collect__eq,axiom,
    ! [P: fun_int_o] :
      ( ( bot_bot_set_int = collect_int(P) )
    <=> ! [X4: int] : ~ aa_int_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_902_IntI,axiom,
    ! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),A2)
     => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),B2)
       => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),inf_in8396524679539076455nt_int(A2,B2)) ) ) ).

% IntI
tff(fact_903_IntI,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),A2)
     => ( aa_set_set_nat_o(member_set_nat(C),B2)
       => aa_set_set_nat_o(member_set_nat(C),inf_inf_set_set_nat(A2,B2)) ) ) ).

% IntI
tff(fact_904_IntI,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),A2)
     => ( aa_set_int_o(member_int(C),B2)
       => aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) ) ) ).

% IntI
tff(fact_905_IntI,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B2)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),inf_in3376754634429691197_nat_o(A2,B2)) ) ) ).

% IntI
tff(fact_906_IntI,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A2)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),B2)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ) ) ).

% IntI
tff(fact_907_IntI,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),B2)
       => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ) ).

% IntI
tff(fact_908_Int__iff,axiom,
    ! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),inf_in8396524679539076455nt_int(A2,B2))
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),A2)
        & aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),B2) ) ) ).

% Int_iff
tff(fact_909_Int__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),inf_inf_set_set_nat(A2,B2))
    <=> ( aa_set_set_nat_o(member_set_nat(C),A2)
        & aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% Int_iff
tff(fact_910_Int__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
    <=> ( aa_set_int_o(member_int(C),A2)
        & aa_set_int_o(member_int(C),B2) ) ) ).

% Int_iff
tff(fact_911_Int__iff,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),inf_in3376754634429691197_nat_o(A2,B2))
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A2)
        & aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B2) ) ) ).

% Int_iff
tff(fact_912_Int__iff,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A2)
        & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),B2) ) ) ).

% Int_iff
tff(fact_913_Int__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2)
        & aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),B2) ) ) ).

% Int_iff
tff(fact_914_Int__UNIV,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = top_to4669805908274784177at_nat )
    <=> ( ( A2 = top_to4669805908274784177at_nat )
        & ( B2 = top_to4669805908274784177at_nat ) ) ) ).

% Int_UNIV
tff(fact_915_Int__UNIV,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] :
      ( ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,A2),B2) = top_top_set_list_nat )
    <=> ( ( A2 = top_top_set_list_nat )
        & ( B2 = top_top_set_list_nat ) ) ) ).

% Int_UNIV
tff(fact_916_Int__UNIV,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = top_top_set_o )
    <=> ( ( A2 = top_top_set_o )
        & ( B2 = top_top_set_o ) ) ) ).

% Int_UNIV
tff(fact_917_Int__UNIV,axiom,
    ! [A2: set_rat,B2: set_rat] :
      ( ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,A2),B2) = top_top_set_rat )
    <=> ( ( A2 = top_top_set_rat )
        & ( B2 = top_top_set_rat ) ) ) ).

% Int_UNIV
tff(fact_918_Int__UNIV,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = top_top_set_nat )
    <=> ( ( A2 = top_top_set_nat )
        & ( B2 = top_top_set_nat ) ) ) ).

% Int_UNIV
tff(fact_919_Int__UNIV,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = top_top_set_int )
    <=> ( ( A2 = top_top_set_int )
        & ( B2 = top_top_set_int ) ) ) ).

% Int_UNIV
tff(fact_920_UnCI,axiom,
    ! [C: set_Pr958786334691620121nt_int,B2: set_se6260736226359567993nt_int,A2: set_se6260736226359567993nt_int] :
      ( ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),B2)
       => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),A2) )
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),sup_su2047564715030645325nt_int(A2,B2)) ) ).

% UnCI
tff(fact_921_UnCI,axiom,
    ! [C: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( ~ aa_set_set_nat_o(member_set_nat(C),B2)
       => aa_set_set_nat_o(member_set_nat(C),A2) )
     => aa_set_set_nat_o(member_set_nat(C),sup_sup_set_set_nat(A2,B2)) ) ).

% UnCI
tff(fact_922_UnCI,axiom,
    ! [C: int,B2: set_int,A2: set_int] :
      ( ( ~ aa_set_int_o(member_int(C),B2)
       => aa_set_int_o(member_int(C),A2) )
     => aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) ) ).

% UnCI
tff(fact_923_UnCI,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B2)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A2) )
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),sup_su2896084399781877795_nat_o(A2,B2)) ) ).

% UnCI
tff(fact_924_UnCI,axiom,
    ! [C: produc4166570645942440679at_nat,B2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat] :
      ( ( ~ member6689249552917799696at_nat(C,B2)
       => member6689249552917799696at_nat(C,A2) )
     => member6689249552917799696at_nat(C,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) ) ).

% UnCI
tff(fact_925_UnCI,axiom,
    ! [C: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),B2)
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),A2) )
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ) ).

% UnCI
tff(fact_926_UnCI,axiom,
    ! [C: nat,B2: set_nat,A2: set_nat] :
      ( ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),B2)
       => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2) )
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% UnCI
tff(fact_927_Un__iff,axiom,
    ! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),sup_su2047564715030645325nt_int(A2,B2))
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),A2)
        | aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),B2) ) ) ).

% Un_iff
tff(fact_928_Un__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),sup_sup_set_set_nat(A2,B2))
    <=> ( aa_set_set_nat_o(member_set_nat(C),A2)
        | aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% Un_iff
tff(fact_929_Un__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2))
    <=> ( aa_set_int_o(member_int(C),A2)
        | aa_set_int_o(member_int(C),B2) ) ) ).

% Un_iff
tff(fact_930_Un__iff,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),sup_su2896084399781877795_nat_o(A2,B2))
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A2)
        | aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B2) ) ) ).

% Un_iff
tff(fact_931_Un__iff,axiom,
    ! [C: produc4166570645942440679at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
      ( member6689249552917799696at_nat(C,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2))
    <=> ( member6689249552917799696at_nat(C,A2)
        | member6689249552917799696at_nat(C,B2) ) ) ).

% Un_iff
tff(fact_932_Un__iff,axiom,
    ! [C: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2))
    <=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),A2)
        | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),B2) ) ) ).

% Un_iff
tff(fact_933_Un__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2)
        | aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),B2) ) ) ).

% Un_iff
tff(fact_934_ComplI,axiom,
    ! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),A2)
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),uminus6423885277529793776nt_int(A2)) ) ).

% ComplI
tff(fact_935_ComplI,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(C),A2)
     => aa_set_set_nat_o(member_set_nat(C),uminus613421341184616069et_nat(A2)) ) ).

% ComplI
tff(fact_936_ComplI,axiom,
    ! [C: nat,A2: set_nat] :
      ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2)
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)) ) ).

% ComplI
tff(fact_937_ComplI,axiom,
    ! [C: int,A2: set_int] :
      ( ~ aa_set_int_o(member_int(C),A2)
     => aa_set_int_o(member_int(C),aa_set_int_set_int(uminus1532241313380277803et_int,A2)) ) ).

% ComplI
tff(fact_938_ComplI,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A2)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),uminus2199602881456817606_nat_o(A2)) ) ).

% ComplI
tff(fact_939_Compl__iff,axiom,
    ! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),uminus6423885277529793776nt_int(A2))
    <=> ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),A2) ) ).

% Compl_iff
tff(fact_940_Compl__iff,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),uminus613421341184616069et_nat(A2))
    <=> ~ aa_set_set_nat_o(member_set_nat(C),A2) ) ).

% Compl_iff
tff(fact_941_Compl__iff,axiom,
    ! [C: nat,A2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2))
    <=> ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2) ) ).

% Compl_iff
tff(fact_942_Compl__iff,axiom,
    ! [C: int,A2: set_int] :
      ( aa_set_int_o(member_int(C),aa_set_int_set_int(uminus1532241313380277803et_int,A2))
    <=> ~ aa_set_int_o(member_int(C),A2) ) ).

% Compl_iff
tff(fact_943_Compl__iff,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),uminus2199602881456817606_nat_o(A2))
    <=> ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A2) ) ).

% Compl_iff
tff(fact_944_Collect__const,axiom,
    ! [P: $o] :
      ( collec5210948495886036740nt_int(aTP_Lamm_al((P))) = $ite((P),top_to6034159466715884489nt_int,bot_bo1488462491386950373nt_int) ) ).

% Collect_const
tff(fact_945_Collect__const,axiom,
    ! [P: $o] :
      ( collect_set_nat(aTP_Lamm_am((P))) = $ite((P),top_top_set_set_nat,bot_bot_set_set_nat) ) ).

% Collect_const
tff(fact_946_Collect__const,axiom,
    ! [P: $o] :
      ( collec6407654713642994138_nat_o(aTP_Lamm_an((P))) = $ite((P),top_to9044565752620676511_nat_o,bot_bo8632255151191677115_nat_o) ) ).

% Collect_const
tff(fact_947_Collect__const,axiom,
    ! [P: $o] :
      ( collec3392354462482085612at_nat(aTP_Lamm_ao((P))) = $ite((P),top_to4669805908274784177at_nat,bot_bo2099793752762293965at_nat) ) ).

% Collect_const
tff(fact_948_Collect__const,axiom,
    ! [P: $o] :
      ( collect_list_nat(aTP_Lamm_ap((P))) = $ite((P),top_top_set_list_nat,bot_bot_set_list_nat) ) ).

% Collect_const
tff(fact_949_Collect__const,axiom,
    ! [P: $o] :
      ( collect_o(aa_o_fun_o_o(aTP_Lamm_aq,(P))) = $ite((P),top_top_set_o,bot_bot_set_o) ) ).

% Collect_const
tff(fact_950_Collect__const,axiom,
    ! [P: $o] :
      ( collect_rat(aTP_Lamm_ar((P))) = $ite((P),top_top_set_rat,bot_bot_set_rat) ) ).

% Collect_const
tff(fact_951_Collect__const,axiom,
    ! [P: $o] :
      ( collect_nat(aa_o_fun_nat_o(aTP_Lamm_as,(P))) = $ite((P),top_top_set_nat,bot_bot_set_nat) ) ).

% Collect_const
tff(fact_952_Collect__const,axiom,
    ! [P: $o] :
      ( collect_int(aa_o_fun_int_o(aTP_Lamm_at,(P))) = $ite((P),top_top_set_int,bot_bot_set_int) ) ).

% Collect_const
tff(fact_953_Compl__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,A2)) = bot_bo2099793752762293965at_nat ) ).

% Compl_disjoint
tff(fact_954_Compl__disjoint,axiom,
    ! [A2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),aa_set_o_set_o(uminus_uminus_set_o,A2)) = bot_bot_set_o ) ).

% Compl_disjoint
tff(fact_955_Compl__disjoint,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)) = bot_bot_set_nat ) ).

% Compl_disjoint
tff(fact_956_Compl__disjoint,axiom,
    ! [A2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),aa_set_int_set_int(uminus1532241313380277803et_int,A2)) = bot_bot_set_int ) ).

% Compl_disjoint
tff(fact_957_ComplD,axiom,
    ! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),uminus6423885277529793776nt_int(A2))
     => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),A2) ) ).

% ComplD
tff(fact_958_ComplD,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),uminus613421341184616069et_nat(A2))
     => ~ aa_set_set_nat_o(member_set_nat(C),A2) ) ).

% ComplD
tff(fact_959_ComplD,axiom,
    ! [C: nat,A2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2))
     => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2) ) ).

% ComplD
tff(fact_960_ComplD,axiom,
    ! [C: int,A2: set_int] :
      ( aa_set_int_o(member_int(C),aa_set_int_set_int(uminus1532241313380277803et_int,A2))
     => ~ aa_set_int_o(member_int(C),A2) ) ).

% ComplD
tff(fact_961_ComplD,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),uminus2199602881456817606_nat_o(A2))
     => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A2) ) ).

% ComplD
tff(fact_962_Compl__eq,axiom,
    ! [A2: set_se6260736226359567993nt_int] : ( uminus6423885277529793776nt_int(A2) = collec5210948495886036740nt_int(aTP_Lamm_au(A2)) ) ).

% Compl_eq
tff(fact_963_Compl__eq,axiom,
    ! [A2: set_set_nat] : ( uminus613421341184616069et_nat(A2) = collect_set_nat(aTP_Lamm_av(A2)) ) ).

% Compl_eq
tff(fact_964_Compl__eq,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_aw,A2)) ) ).

% Compl_eq
tff(fact_965_Compl__eq,axiom,
    ! [A2: set_int] : ( aa_set_int_set_int(uminus1532241313380277803et_int,A2) = collect_int(aTP_Lamm_ax(A2)) ) ).

% Compl_eq
tff(fact_966_Compl__eq,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] : ( uminus2199602881456817606_nat_o(A2) = collec6407654713642994138_nat_o(aTP_Lamm_ay(A2)) ) ).

% Compl_eq
tff(fact_967_Collect__neg__eq,axiom,
    ! [P: fun_se7249947736802449410_int_o] : ( collec5210948495886036740nt_int(aTP_Lamm_az(P)) = uminus6423885277529793776nt_int(collec5210948495886036740nt_int(P)) ) ).

% Collect_neg_eq
tff(fact_968_Collect__neg__eq,axiom,
    ! [P: fun_set_nat_o] : ( collect_set_nat(aTP_Lamm_ba(P)) = uminus613421341184616069et_nat(collect_set_nat(P)) ) ).

% Collect_neg_eq
tff(fact_969_Collect__neg__eq,axiom,
    ! [P: fun_nat_o] : ( collect_nat(aTP_Lamm_bb(P)) = aa_set_nat_set_nat(uminus5710092332889474511et_nat,collect_nat(P)) ) ).

% Collect_neg_eq
tff(fact_970_Collect__neg__eq,axiom,
    ! [P: fun_int_o] : ( collect_int(aTP_Lamm_bc(P)) = aa_set_int_set_int(uminus1532241313380277803et_int,collect_int(P)) ) ).

% Collect_neg_eq
tff(fact_971_Collect__neg__eq,axiom,
    ! [P: fun_fu5135969500543783852at_o_o] : ( collec6407654713642994138_nat_o(aTP_Lamm_bd(P)) = uminus2199602881456817606_nat_o(collec6407654713642994138_nat_o(P)) ) ).

% Collect_neg_eq
tff(fact_972_uminus__set__def,axiom,
    ! [A2: set_se6260736226359567993nt_int] : ( uminus6423885277529793776nt_int(A2) = collec5210948495886036740nt_int(uminus3838094161332292555_int_o(aTP_Lamm_ag(A2))) ) ).

% uminus_set_def
tff(fact_973_uminus__set__def,axiom,
    ! [A2: set_set_nat] : ( uminus613421341184616069et_nat(A2) = collect_set_nat(uminus1495274395075166690_nat_o(aTP_Lamm_ah(A2))) ) ).

% uminus_set_def
tff(fact_974_uminus__set__def,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2) = collect_nat(uminus6183504819137795864_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ai,A2))) ) ).

% uminus_set_def
tff(fact_975_uminus__set__def,axiom,
    ! [A2: set_int] : ( aa_set_int_set_int(uminus1532241313380277803et_int,A2) = collect_int(uminus4362462473705107900_int_o(aTP_Lamm_aj(A2))) ) ).

% uminus_set_def
tff(fact_976_uminus__set__def,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] : ( uminus2199602881456817606_nat_o(A2) = collec6407654713642994138_nat_o(uminus7836778930441969781at_o_o(aTP_Lamm_ak(A2))) ) ).

% uminus_set_def
tff(fact_977_emptyE,axiom,
    ! [A: set_Pr958786334691620121nt_int] : ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),bot_bo1488462491386950373nt_int) ).

% emptyE
tff(fact_978_emptyE,axiom,
    ! [A: set_nat] : ~ aa_set_set_nat_o(member_set_nat(A),bot_bot_set_set_nat) ).

% emptyE
tff(fact_979_emptyE,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),bot_bo8632255151191677115_nat_o) ).

% emptyE
tff(fact_980_emptyE,axiom,
    ! [A: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),bot_bo2099793752762293965at_nat) ).

% emptyE
tff(fact_981_emptyE,axiom,
    ! [A: $o] : ~ aa_set_o_o(member_o((A)),bot_bot_set_o) ).

% emptyE
tff(fact_982_emptyE,axiom,
    ! [A: nat] : ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),bot_bot_set_nat) ).

% emptyE
tff(fact_983_emptyE,axiom,
    ! [A: int] : ~ aa_set_int_o(member_int(A),bot_bot_set_int) ).

% emptyE
tff(fact_984_equals0D,axiom,
    ! [A2: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( A2 = bot_bo1488462491386950373nt_int )
     => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A2) ) ).

% equals0D
tff(fact_985_equals0D,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( A2 = bot_bot_set_set_nat )
     => ~ aa_set_set_nat_o(member_set_nat(A),A2) ) ).

% equals0D
tff(fact_986_equals0D,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,A: fun_Pr6460944699146269935_nat_o] :
      ( ( A2 = bot_bo8632255151191677115_nat_o )
     => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2) ) ).

% equals0D
tff(fact_987_equals0D,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( A2 = bot_bo2099793752762293965at_nat )
     => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2) ) ).

% equals0D
tff(fact_988_equals0D,axiom,
    ! [A2: set_o,A: $o] :
      ( ( A2 = bot_bot_set_o )
     => ~ aa_set_o_o(member_o((A)),A2) ) ).

% equals0D
tff(fact_989_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A2) ) ).

% equals0D
tff(fact_990_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ aa_set_int_o(member_int(A),A2) ) ).

% equals0D
tff(fact_991_equals0I,axiom,
    ! [A2: set_se6260736226359567993nt_int] :
      ( ! [Y2: set_Pr958786334691620121nt_int] : ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y2),A2)
     => ( A2 = bot_bo1488462491386950373nt_int ) ) ).

% equals0I
tff(fact_992_equals0I,axiom,
    ! [A2: set_set_nat] :
      ( ! [Y2: set_nat] : ~ aa_set_set_nat_o(member_set_nat(Y2),A2)
     => ( A2 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_993_equals0I,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( ! [Y2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A2)
     => ( A2 = bot_bo8632255151191677115_nat_o ) ) ).

% equals0I
tff(fact_994_equals0I,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [Y2: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y2),A2)
     => ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
tff(fact_995_equals0I,axiom,
    ! [A2: set_o] :
      ( ! [Y2: $o] : ~ aa_set_o_o(member_o((Y2)),A2)
     => ( A2 = bot_bot_set_o ) ) ).

% equals0I
tff(fact_996_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y2: nat] : ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y2),A2)
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_997_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y2: int] : ~ aa_set_int_o(member_int(Y2),A2)
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_998_ex__in__conv,axiom,
    ! [A2: set_se6260736226359567993nt_int] :
      ( ? [X4: set_Pr958786334691620121nt_int] : aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X4),A2)
    <=> ( A2 != bot_bo1488462491386950373nt_int ) ) ).

% ex_in_conv
tff(fact_999_ex__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ? [X4: set_nat] : aa_set_set_nat_o(member_set_nat(X4),A2)
    <=> ( A2 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_1000_ex__in__conv,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( ? [X4: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),A2)
    <=> ( A2 != bot_bo8632255151191677115_nat_o ) ) ).

% ex_in_conv
tff(fact_1001_ex__in__conv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ? [X4: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A2)
    <=> ( A2 != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
tff(fact_1002_ex__in__conv,axiom,
    ! [A2: set_o] :
      ( ? [X4: $o] : aa_set_o_o(member_o((X4)),A2)
    <=> ( A2 != bot_bot_set_o ) ) ).

% ex_in_conv
tff(fact_1003_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ? [X4: nat] : aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A2)
    <=> ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_1004_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ? [X4: int] : aa_set_int_o(member_int(X4),A2)
    <=> ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_1005_bot__set__def,axiom,
    bot_bo1488462491386950373nt_int = collec5210948495886036740nt_int(bot_bo4864556513171353750_int_o) ).

% bot_set_def
tff(fact_1006_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo3967814874310145751_nat_o) ).

% bot_set_def
tff(fact_1007_bot__set__def,axiom,
    bot_bo8632255151191677115_nat_o = collec6407654713642994138_nat_o(bot_bo8793957029302321216at_o_o) ).

% bot_set_def
tff(fact_1008_bot__set__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(bot_bo5074786513356115630_nat_o) ).

% bot_set_def
tff(fact_1009_bot__set__def,axiom,
    bot_bot_set_o = collect_o(bot_bot_fun_o_o) ).

% bot_set_def
tff(fact_1010_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_o) ).

% bot_set_def
tff(fact_1011_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_o) ).

% bot_set_def
tff(fact_1012_Compl__UNIV__eq,axiom,
    aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,top_to4669805908274784177at_nat) = bot_bo2099793752762293965at_nat ).

% Compl_UNIV_eq
tff(fact_1013_Compl__UNIV__eq,axiom,
    aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,top_top_set_list_nat) = bot_bot_set_list_nat ).

% Compl_UNIV_eq
tff(fact_1014_Compl__UNIV__eq,axiom,
    aa_set_o_set_o(uminus_uminus_set_o,top_top_set_o) = bot_bot_set_o ).

% Compl_UNIV_eq
tff(fact_1015_Compl__UNIV__eq,axiom,
    aa_set_rat_set_rat(uminus2201863774496077783et_rat,top_top_set_rat) = bot_bot_set_rat ).

% Compl_UNIV_eq
tff(fact_1016_Compl__UNIV__eq,axiom,
    aa_set_nat_set_nat(uminus5710092332889474511et_nat,top_top_set_nat) = bot_bot_set_nat ).

% Compl_UNIV_eq
tff(fact_1017_Compl__UNIV__eq,axiom,
    aa_set_int_set_int(uminus1532241313380277803et_int,top_top_set_int) = bot_bot_set_int ).

% Compl_UNIV_eq
tff(fact_1018_Compl__empty__eq,axiom,
    aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,bot_bo2099793752762293965at_nat) = top_to4669805908274784177at_nat ).

% Compl_empty_eq
tff(fact_1019_Compl__empty__eq,axiom,
    aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,bot_bot_set_list_nat) = top_top_set_list_nat ).

% Compl_empty_eq
tff(fact_1020_Compl__empty__eq,axiom,
    aa_set_o_set_o(uminus_uminus_set_o,bot_bot_set_o) = top_top_set_o ).

% Compl_empty_eq
tff(fact_1021_Compl__empty__eq,axiom,
    aa_set_rat_set_rat(uminus2201863774496077783et_rat,bot_bot_set_rat) = top_top_set_rat ).

% Compl_empty_eq
tff(fact_1022_Compl__empty__eq,axiom,
    aa_set_nat_set_nat(uminus5710092332889474511et_nat,bot_bot_set_nat) = top_top_set_nat ).

% Compl_empty_eq
tff(fact_1023_Compl__empty__eq,axiom,
    aa_set_int_set_int(uminus1532241313380277803et_int,bot_bot_set_int) = top_top_set_int ).

% Compl_empty_eq
tff(fact_1024_empty__not__UNIV,axiom,
    bot_bo2099793752762293965at_nat != top_to4669805908274784177at_nat ).

% empty_not_UNIV
tff(fact_1025_empty__not__UNIV,axiom,
    bot_bot_set_list_nat != top_top_set_list_nat ).

% empty_not_UNIV
tff(fact_1026_empty__not__UNIV,axiom,
    bot_bot_set_o != top_top_set_o ).

% empty_not_UNIV
tff(fact_1027_empty__not__UNIV,axiom,
    bot_bot_set_rat != top_top_set_rat ).

% empty_not_UNIV
tff(fact_1028_empty__not__UNIV,axiom,
    bot_bot_set_nat != top_top_set_nat ).

% empty_not_UNIV
tff(fact_1029_empty__not__UNIV,axiom,
    bot_bot_set_int != top_top_set_int ).

% empty_not_UNIV
tff(fact_1030_IntE,axiom,
    ! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),inf_in8396524679539076455nt_int(A2,B2))
     => ~ ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),A2)
         => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),B2) ) ) ).

% IntE
tff(fact_1031_IntE,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),inf_inf_set_set_nat(A2,B2))
     => ~ ( aa_set_set_nat_o(member_set_nat(C),A2)
         => ~ aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% IntE
tff(fact_1032_IntE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
     => ~ ( aa_set_int_o(member_int(C),A2)
         => ~ aa_set_int_o(member_int(C),B2) ) ) ).

% IntE
tff(fact_1033_IntE,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),inf_in3376754634429691197_nat_o(A2,B2))
     => ~ ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A2)
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B2) ) ) ).

% IntE
tff(fact_1034_IntE,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
     => ~ ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A2)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),B2) ) ) ).

% IntE
tff(fact_1035_IntE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
     => ~ ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2)
         => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),B2) ) ) ).

% IntE
tff(fact_1036_IntD1,axiom,
    ! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),inf_in8396524679539076455nt_int(A2,B2))
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),A2) ) ).

% IntD1
tff(fact_1037_IntD1,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),inf_inf_set_set_nat(A2,B2))
     => aa_set_set_nat_o(member_set_nat(C),A2) ) ).

% IntD1
tff(fact_1038_IntD1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
     => aa_set_int_o(member_int(C),A2) ) ).

% IntD1
tff(fact_1039_IntD1,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),inf_in3376754634429691197_nat_o(A2,B2))
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A2) ) ).

% IntD1
tff(fact_1040_IntD1,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A2) ) ).

% IntD1
tff(fact_1041_IntD1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2) ) ).

% IntD1
tff(fact_1042_IntD2,axiom,
    ! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),inf_in8396524679539076455nt_int(A2,B2))
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),B2) ) ).

% IntD2
tff(fact_1043_IntD2,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),inf_inf_set_set_nat(A2,B2))
     => aa_set_set_nat_o(member_set_nat(C),B2) ) ).

% IntD2
tff(fact_1044_IntD2,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
     => aa_set_int_o(member_int(C),B2) ) ).

% IntD2
tff(fact_1045_IntD2,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),inf_in3376754634429691197_nat_o(A2,B2))
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B2) ) ).

% IntD2
tff(fact_1046_IntD2,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),B2) ) ).

% IntD2
tff(fact_1047_IntD2,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),B2) ) ).

% IntD2
tff(fact_1048_Int__assoc,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ) ).

% Int_assoc
tff(fact_1049_Int__assoc,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ) ).

% Int_assoc
tff(fact_1050_Int__absorb,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),A2) = A2 ) ).

% Int_absorb
tff(fact_1051_Int__absorb,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),A2) = A2 ) ).

% Int_absorb
tff(fact_1052_Int__commute,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),A2) ) ).

% Int_commute
tff(fact_1053_Int__commute,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A2) ) ).

% Int_commute
tff(fact_1054_Int__UNIV__left,axiom,
    ! [B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,top_to4669805908274784177at_nat),B2) = B2 ) ).

% Int_UNIV_left
tff(fact_1055_Int__UNIV__left,axiom,
    ! [B2: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,top_top_set_list_nat),B2) = B2 ) ).

% Int_UNIV_left
tff(fact_1056_Int__UNIV__left,axiom,
    ! [B2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,top_top_set_o),B2) = B2 ) ).

% Int_UNIV_left
tff(fact_1057_Int__UNIV__left,axiom,
    ! [B2: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,top_top_set_rat),B2) = B2 ) ).

% Int_UNIV_left
tff(fact_1058_Int__UNIV__left,axiom,
    ! [B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,top_top_set_nat),B2) = B2 ) ).

% Int_UNIV_left
tff(fact_1059_Int__UNIV__left,axiom,
    ! [B2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,top_top_set_int),B2) = B2 ) ).

% Int_UNIV_left
tff(fact_1060_Int__UNIV__right,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),top_to4669805908274784177at_nat) = A2 ) ).

% Int_UNIV_right
tff(fact_1061_Int__UNIV__right,axiom,
    ! [A2: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,A2),top_top_set_list_nat) = A2 ) ).

% Int_UNIV_right
tff(fact_1062_Int__UNIV__right,axiom,
    ! [A2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),top_top_set_o) = A2 ) ).

% Int_UNIV_right
tff(fact_1063_Int__UNIV__right,axiom,
    ! [A2: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,A2),top_top_set_rat) = A2 ) ).

% Int_UNIV_right
tff(fact_1064_Int__UNIV__right,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),top_top_set_nat) = A2 ) ).

% Int_UNIV_right
tff(fact_1065_Int__UNIV__right,axiom,
    ! [A2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),top_top_set_int) = A2 ) ).

% Int_UNIV_right
tff(fact_1066_Int__left__absorb,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ) ).

% Int_left_absorb
tff(fact_1067_Int__left__absorb,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ).

% Int_left_absorb
tff(fact_1068_Int__left__commute,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C2)) ) ).

% Int_left_commute
tff(fact_1069_Int__left__commute,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)) ) ).

% Int_left_commute
tff(fact_1070_UnE,axiom,
    ! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),sup_su2047564715030645325nt_int(A2,B2))
     => ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),A2)
       => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),B2) ) ) ).

% UnE
tff(fact_1071_UnE,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),sup_sup_set_set_nat(A2,B2))
     => ( ~ aa_set_set_nat_o(member_set_nat(C),A2)
       => aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% UnE
tff(fact_1072_UnE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2))
     => ( ~ aa_set_int_o(member_int(C),A2)
       => aa_set_int_o(member_int(C),B2) ) ) ).

% UnE
tff(fact_1073_UnE,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),sup_su2896084399781877795_nat_o(A2,B2))
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A2)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B2) ) ) ).

% UnE
tff(fact_1074_UnE,axiom,
    ! [C: produc4166570645942440679at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
      ( member6689249552917799696at_nat(C,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2))
     => ( ~ member6689249552917799696at_nat(C,A2)
       => member6689249552917799696at_nat(C,B2) ) ) ).

% UnE
tff(fact_1075_UnE,axiom,
    ! [C: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2))
     => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),A2)
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),B2) ) ) ).

% UnE
tff(fact_1076_UnE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
     => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2)
       => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),B2) ) ) ).

% UnE
tff(fact_1077_UnI1,axiom,
    ! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),A2)
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),sup_su2047564715030645325nt_int(A2,B2)) ) ).

% UnI1
tff(fact_1078_UnI1,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),A2)
     => aa_set_set_nat_o(member_set_nat(C),sup_sup_set_set_nat(A2,B2)) ) ).

% UnI1
tff(fact_1079_UnI1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),A2)
     => aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) ) ).

% UnI1
tff(fact_1080_UnI1,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A2)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),sup_su2896084399781877795_nat_o(A2,B2)) ) ).

% UnI1
tff(fact_1081_UnI1,axiom,
    ! [C: produc4166570645942440679at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
      ( member6689249552917799696at_nat(C,A2)
     => member6689249552917799696at_nat(C,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) ) ).

% UnI1
tff(fact_1082_UnI1,axiom,
    ! [C: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),A2)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ) ).

% UnI1
tff(fact_1083_UnI1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2)
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% UnI1
tff(fact_1084_UnI2,axiom,
    ! [C: set_Pr958786334691620121nt_int,B2: set_se6260736226359567993nt_int,A2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),B2)
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C),sup_su2047564715030645325nt_int(A2,B2)) ) ).

% UnI2
tff(fact_1085_UnI2,axiom,
    ! [C: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),B2)
     => aa_set_set_nat_o(member_set_nat(C),sup_sup_set_set_nat(A2,B2)) ) ).

% UnI2
tff(fact_1086_UnI2,axiom,
    ! [C: int,B2: set_int,A2: set_int] :
      ( aa_set_int_o(member_int(C),B2)
     => aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) ) ).

% UnI2
tff(fact_1087_UnI2,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B2)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),sup_su2896084399781877795_nat_o(A2,B2)) ) ).

% UnI2
tff(fact_1088_UnI2,axiom,
    ! [C: produc4166570645942440679at_nat,B2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat] :
      ( member6689249552917799696at_nat(C,B2)
     => member6689249552917799696at_nat(C,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) ) ).

% UnI2
tff(fact_1089_UnI2,axiom,
    ! [C: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),B2)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ) ).

% UnI2
tff(fact_1090_UnI2,axiom,
    ! [C: nat,B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),B2)
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% UnI2
tff(fact_1091_bex__Un,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,P: fun_Pr1099658227000185396_nat_o] :
      ( ? [X4: produc4166570645942440679at_nat] :
          ( member6689249552917799696at_nat(X4,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2))
          & aa_Pro8143006071014924973_nat_o(P,X4) )
    <=> ( ? [X4: produc4166570645942440679at_nat] :
            ( member6689249552917799696at_nat(X4,A2)
            & aa_Pro8143006071014924973_nat_o(P,X4) )
        | ? [X4: produc4166570645942440679at_nat] :
            ( member6689249552917799696at_nat(X4,B2)
            & aa_Pro8143006071014924973_nat_o(P,X4) ) ) ) ).

% bex_Un
tff(fact_1092_bex__Un,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,P: fun_Pr3059359313727237556_nat_o] :
      ( ? [X4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2))
          & aa_Pro5947980084102686253_nat_o(P,X4) )
    <=> ( ? [X4: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),A2)
            & aa_Pro5947980084102686253_nat_o(P,X4) )
        | ? [X4: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),B2)
            & aa_Pro5947980084102686253_nat_o(P,X4) ) ) ) ).

% bex_Un
tff(fact_1093_bex__Un,axiom,
    ! [A2: set_nat,B2: set_nat,P: fun_nat_o] :
      ( ? [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
          & aa_nat_o(P,X4) )
    <=> ( ? [X4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A2)
            & aa_nat_o(P,X4) )
        | ? [X4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),B2)
            & aa_nat_o(P,X4) ) ) ) ).

% bex_Un
tff(fact_1094_ball__Un,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,P: fun_Pr1099658227000185396_nat_o] :
      ( ! [X4: produc4166570645942440679at_nat] :
          ( member6689249552917799696at_nat(X4,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2))
         => aa_Pro8143006071014924973_nat_o(P,X4) )
    <=> ( ! [X4: produc4166570645942440679at_nat] :
            ( member6689249552917799696at_nat(X4,A2)
           => aa_Pro8143006071014924973_nat_o(P,X4) )
        & ! [X4: produc4166570645942440679at_nat] :
            ( member6689249552917799696at_nat(X4,B2)
           => aa_Pro8143006071014924973_nat_o(P,X4) ) ) ) ).

% ball_Un
tff(fact_1095_ball__Un,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,P: fun_Pr3059359313727237556_nat_o] :
      ( ! [X4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2))
         => aa_Pro5947980084102686253_nat_o(P,X4) )
    <=> ( ! [X4: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),A2)
           => aa_Pro5947980084102686253_nat_o(P,X4) )
        & ! [X4: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),B2)
           => aa_Pro5947980084102686253_nat_o(P,X4) ) ) ) ).

% ball_Un
tff(fact_1096_ball__Un,axiom,
    ! [A2: set_nat,B2: set_nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
         => aa_nat_o(P,X4) )
    <=> ( ! [X4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A2)
           => aa_nat_o(P,X4) )
        & ! [X4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),B2)
           => aa_nat_o(P,X4) ) ) ) ).

% ball_Un
tff(fact_1097_Un__assoc,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)),C2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B2),C2)) ) ).

% Un_assoc
tff(fact_1098_Un__assoc,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) ) ).

% Un_assoc
tff(fact_1099_Un__assoc,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) ) ).

% Un_assoc
tff(fact_1100_Un__absorb,axiom,
    ! [A2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),A2) = A2 ) ).

% Un_absorb
tff(fact_1101_Un__absorb,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),A2) = A2 ) ).

% Un_absorb
tff(fact_1102_Un__absorb,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),A2) = A2 ) ).

% Un_absorb
tff(fact_1103_Un__commute,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B2),A2) ) ).

% Un_commute
tff(fact_1104_Un__commute,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),A2) ) ).

% Un_commute
tff(fact_1105_Un__commute,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A2) ) ).

% Un_commute
tff(fact_1106_Un__UNIV__left,axiom,
    ! [B2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,top_to8764741339697478167at_nat),B2) = top_to8764741339697478167at_nat ) ).

% Un_UNIV_left
tff(fact_1107_Un__UNIV__left,axiom,
    ! [B2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,top_to6833984726390702231at_nat),B2) = top_to6833984726390702231at_nat ) ).

% Un_UNIV_left
tff(fact_1108_Un__UNIV__left,axiom,
    ! [B2: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,top_top_set_list_nat),B2) = top_top_set_list_nat ) ).

% Un_UNIV_left
tff(fact_1109_Un__UNIV__left,axiom,
    ! [B2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,top_top_set_o),B2) = top_top_set_o ) ).

% Un_UNIV_left
tff(fact_1110_Un__UNIV__left,axiom,
    ! [B2: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,top_top_set_rat),B2) = top_top_set_rat ) ).

% Un_UNIV_left
tff(fact_1111_Un__UNIV__left,axiom,
    ! [B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,top_top_set_nat),B2) = top_top_set_nat ) ).

% Un_UNIV_left
tff(fact_1112_Un__UNIV__left,axiom,
    ! [B2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,top_top_set_int),B2) = top_top_set_int ) ).

% Un_UNIV_left
tff(fact_1113_Un__UNIV__right,axiom,
    ! [A2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),top_to8764741339697478167at_nat) = top_to8764741339697478167at_nat ) ).

% Un_UNIV_right
tff(fact_1114_Un__UNIV__right,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),top_to6833984726390702231at_nat) = top_to6833984726390702231at_nat ) ).

% Un_UNIV_right
tff(fact_1115_Un__UNIV__right,axiom,
    ! [A2: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,A2),top_top_set_list_nat) = top_top_set_list_nat ) ).

% Un_UNIV_right
tff(fact_1116_Un__UNIV__right,axiom,
    ! [A2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),top_top_set_o) = top_top_set_o ) ).

% Un_UNIV_right
tff(fact_1117_Un__UNIV__right,axiom,
    ! [A2: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,A2),top_top_set_rat) = top_top_set_rat ) ).

% Un_UNIV_right
tff(fact_1118_Un__UNIV__right,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),top_top_set_nat) = top_top_set_nat ) ).

% Un_UNIV_right
tff(fact_1119_Un__UNIV__right,axiom,
    ! [A2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),top_top_set_int) = top_top_set_int ) ).

% Un_UNIV_right
tff(fact_1120_Un__left__absorb,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2) ) ).

% Un_left_absorb
tff(fact_1121_Un__left__absorb,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) ) ).

% Un_left_absorb
tff(fact_1122_Un__left__absorb,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) ) ).

% Un_left_absorb
tff(fact_1123_Compl__partition,axiom,
    ! [A2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,A2)) = top_to8764741339697478167at_nat ) ).

% Compl_partition
tff(fact_1124_Compl__partition,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),uminus935396558254630718at_nat(A2)) = top_to6833984726390702231at_nat ) ).

% Compl_partition
tff(fact_1125_Compl__partition,axiom,
    ! [A2: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,A2),aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,A2)) = top_top_set_list_nat ) ).

% Compl_partition
tff(fact_1126_Compl__partition,axiom,
    ! [A2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),aa_set_o_set_o(uminus_uminus_set_o,A2)) = top_top_set_o ) ).

% Compl_partition
tff(fact_1127_Compl__partition,axiom,
    ! [A2: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,A2),aa_set_rat_set_rat(uminus2201863774496077783et_rat,A2)) = top_top_set_rat ) ).

% Compl_partition
tff(fact_1128_Compl__partition,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)) = top_top_set_nat ) ).

% Compl_partition
tff(fact_1129_Compl__partition,axiom,
    ! [A2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),aa_set_int_set_int(uminus1532241313380277803et_int,A2)) = top_top_set_int ) ).

% Compl_partition
tff(fact_1130_Un__left__commute,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B2),C2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),C2)) ) ).

% Un_left_commute
tff(fact_1131_Un__left__commute,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),C2)) ) ).

% Un_left_commute
tff(fact_1132_Un__left__commute,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),C2)) ) ).

% Un_left_commute
tff(fact_1133_Compl__partition2,axiom,
    ! [A2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,A2)),A2) = top_to8764741339697478167at_nat ) ).

% Compl_partition2
tff(fact_1134_Compl__partition2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(A2)),A2) = top_to6833984726390702231at_nat ) ).

% Compl_partition2
tff(fact_1135_Compl__partition2,axiom,
    ! [A2: set_list_nat] : ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,A2)),A2) = top_top_set_list_nat ) ).

% Compl_partition2
tff(fact_1136_Compl__partition2,axiom,
    ! [A2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(uminus_uminus_set_o,A2)),A2) = top_top_set_o ) ).

% Compl_partition2
tff(fact_1137_Compl__partition2,axiom,
    ! [A2: set_rat] : ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,aa_set_rat_set_rat(uminus2201863774496077783et_rat,A2)),A2) = top_top_set_rat ) ).

% Compl_partition2
tff(fact_1138_Compl__partition2,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)),A2) = top_top_set_nat ) ).

% Compl_partition2
tff(fact_1139_Compl__partition2,axiom,
    ! [A2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,A2)),A2) = top_top_set_int ) ).

% Compl_partition2
tff(fact_1140_Set_Oempty__def,axiom,
    bot_bo1488462491386950373nt_int = collec5210948495886036740nt_int(aTP_Lamm_be) ).

% Set.empty_def
tff(fact_1141_Set_Oempty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_bf) ).

% Set.empty_def
tff(fact_1142_Set_Oempty__def,axiom,
    bot_bo8632255151191677115_nat_o = collec6407654713642994138_nat_o(aTP_Lamm_bg) ).

% Set.empty_def
tff(fact_1143_Set_Oempty__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(aTP_Lamm_bh) ).

% Set.empty_def
tff(fact_1144_Set_Oempty__def,axiom,
    bot_bot_set_o = collect_o(aTP_Lamm_bi) ).

% Set.empty_def
tff(fact_1145_Set_Oempty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_bj) ).

% Set.empty_def
tff(fact_1146_Set_Oempty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_bk) ).

% Set.empty_def
tff(fact_1147_Int__def,axiom,
    ! [A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] : ( inf_in8396524679539076455nt_int(A2,B2) = collec5210948495886036740nt_int(aa_set4062574822058255039_int_o(aTP_Lamm_bl(A2),B2)) ) ).

% Int_def
tff(fact_1148_Int__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : ( inf_inf_set_set_nat(A2,B2) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_bm(A2),B2)) ) ).

% Int_def
tff(fact_1149_Int__def,axiom,
    ! [A2: set_int,B2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_bn(A2),B2)) ) ).

% Int_def
tff(fact_1150_Int__def,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] : ( inf_in3376754634429691197_nat_o(A2,B2) = collec6407654713642994138_nat_o(aa_set7516337885093543743at_o_o(aTP_Lamm_bo(A2),B2)) ) ).

% Int_def
tff(fact_1151_Int__def,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_bp(A2),B2)) ) ).

% Int_def
tff(fact_1152_Int__def,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_bq(A2),B2)) ) ).

% Int_def
tff(fact_1153_Int__Collect,axiom,
    ! [X: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,P: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),inf_in8396524679539076455nt_int(A2,collec5210948495886036740nt_int(P)))
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A2)
        & aa_set6170482117830564987_int_o(P,X) ) ) ).

% Int_Collect
tff(fact_1154_Int__Collect,axiom,
    ! [X: set_nat,A2: set_set_nat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(X),inf_inf_set_set_nat(A2,collect_set_nat(P)))
    <=> ( aa_set_set_nat_o(member_set_nat(X),A2)
        & aa_set_nat_o(P,X) ) ) ).

% Int_Collect
tff(fact_1155_Int__Collect,axiom,
    ! [X: int,A2: set_int,P: fun_int_o] :
      ( aa_set_int_o(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P)))
    <=> ( aa_set_int_o(member_int(X),A2)
        & aa_int_o(P,X) ) ) ).

% Int_Collect
tff(fact_1156_Int__Collect,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,P: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),inf_in3376754634429691197_nat_o(A2,collec6407654713642994138_nat_o(P)))
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
        & aa_fun1675921005893230629at_o_o(P,X) ) ) ).

% Int_Collect
tff(fact_1157_Int__Collect,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),collec3392354462482085612at_nat(P)))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2)
        & aa_Pro6150637604445964947_nat_o(P,X) ) ) ).

% Int_Collect
tff(fact_1158_Int__Collect,axiom,
    ! [X: nat,A2: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P)))
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2)
        & aa_nat_o(P,X) ) ) ).

% Int_Collect
tff(fact_1159_inf__set__def,axiom,
    ! [A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] : ( inf_in8396524679539076455nt_int(A2,B2) = collec5210948495886036740nt_int(inf_in9107477634626221460_int_o(aTP_Lamm_ag(A2),aTP_Lamm_ag(B2))) ) ).

% inf_set_def
tff(fact_1160_inf__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : ( inf_inf_set_set_nat(A2,B2) = collect_set_nat(inf_in7602907407060649817_nat_o(aTP_Lamm_ah(A2),aTP_Lamm_ah(B2))) ) ).

% inf_set_def
tff(fact_1161_inf__set__def,axiom,
    ! [A2: set_int,B2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = collect_int(inf_inf_fun_int_o(aTP_Lamm_aj(A2),aTP_Lamm_aj(B2))) ) ).

% inf_set_def
tff(fact_1162_inf__set__def,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] : ( inf_in3376754634429691197_nat_o(A2,B2) = collec6407654713642994138_nat_o(inf_in2686814848456516414at_o_o(aTP_Lamm_ak(A2),aTP_Lamm_ak(B2))) ) ).

% inf_set_def
tff(fact_1163_inf__set__def,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] : ( inf_in2269163501485487111nt_int(A2,B2) = collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aTP_Lamm_br(A2)),aTP_Lamm_br(B2))) ) ).

% inf_set_def
tff(fact_1164_inf__set__def,axiom,
    ! [A2: set_list_char,B2: set_list_char] : ( inf_in6177012701088388009t_char(A2,B2) = collect_list_char(aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aTP_Lamm_bs(A2)),aTP_Lamm_bs(B2))) ) ).

% inf_set_def
tff(fact_1165_inf__set__def,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(inf_in3341167027036195756_nat_o(aTP_Lamm_bt(A2)),aTP_Lamm_bt(B2))) ) ).

% inf_set_def
tff(fact_1166_inf__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = collect_nat(inf_inf_fun_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ai,A2),aa_set_nat_fun_nat_o(aTP_Lamm_ai,B2))) ) ).

% inf_set_def
tff(fact_1167_Collect__conj__eq,axiom,
    ! [P: fun_se7249947736802449410_int_o,Q: fun_se7249947736802449410_int_o] : ( collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_bu(P),Q)) = inf_in8396524679539076455nt_int(collec5210948495886036740nt_int(P),collec5210948495886036740nt_int(Q)) ) ).

% Collect_conj_eq
tff(fact_1168_Collect__conj__eq,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] : ( collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bv(P),Q)) = inf_inf_set_set_nat(collect_set_nat(P),collect_set_nat(Q)) ) ).

% Collect_conj_eq
tff(fact_1169_Collect__conj__eq,axiom,
    ! [P: fun_int_o,Q: fun_int_o] : ( collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bw(P),Q)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,collect_int(P)),collect_int(Q)) ) ).

% Collect_conj_eq
tff(fact_1170_Collect__conj__eq,axiom,
    ! [P: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] : ( collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_bx(P),Q)) = inf_in3376754634429691197_nat_o(collec6407654713642994138_nat_o(P),collec6407654713642994138_nat_o(Q)) ) ).

% Collect_conj_eq
tff(fact_1171_Collect__conj__eq,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] : ( collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_by(P),Q)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,collec3392354462482085612at_nat(P)),collec3392354462482085612at_nat(Q)) ) ).

% Collect_conj_eq
tff(fact_1172_Collect__conj__eq,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] : ( collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bz(P),Q)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,collect_nat(P)),collect_nat(Q)) ) ).

% Collect_conj_eq
tff(fact_1173_Un__def,axiom,
    ! [A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] : ( sup_su2047564715030645325nt_int(A2,B2) = collec5210948495886036740nt_int(aa_set4062574822058255039_int_o(aTP_Lamm_ca(A2),B2)) ) ).

% Un_def
tff(fact_1174_Un__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : ( sup_sup_set_set_nat(A2,B2) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_cb(A2),B2)) ) ).

% Un_def
tff(fact_1175_Un__def,axiom,
    ! [A2: set_int,B2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_cc(A2),B2)) ) ).

% Un_def
tff(fact_1176_Un__def,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] : ( sup_su2896084399781877795_nat_o(A2,B2) = collec6407654713642994138_nat_o(aa_set7516337885093543743at_o_o(aTP_Lamm_cd(A2),B2)) ) ).

% Un_def
tff(fact_1177_Un__def,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2) = collec5204685387357076818at_nat(aa_set1219845518824332607_nat_o(aTP_Lamm_ce(A2),B2)) ) ).

% Un_def
tff(fact_1178_Un__def,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = collec6321179662152712658at_nat(aa_set6353399069419443519_nat_o(aTP_Lamm_cf(A2),B2)) ) ).

% Un_def
tff(fact_1179_Un__def,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_cg(A2),B2)) ) ).

% Un_def
tff(fact_1180_sup__set__def,axiom,
    ! [A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] : ( sup_su2047564715030645325nt_int(A2,B2) = collec5210948495886036740nt_int(sup_su2829768162508406574_int_o(aTP_Lamm_ag(A2),aTP_Lamm_ag(B2))) ) ).

% sup_set_def
tff(fact_1181_sup__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : ( sup_sup_set_set_nat(A2,B2) = collect_set_nat(sup_su7503774895437956671_nat_o(aTP_Lamm_ah(A2),aTP_Lamm_ah(B2))) ) ).

% sup_set_def
tff(fact_1182_sup__set__def,axiom,
    ! [A2: set_int,B2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2) = collect_int(sup_sup_fun_int_o(aTP_Lamm_aj(A2),aTP_Lamm_aj(B2))) ) ).

% sup_set_def
tff(fact_1183_sup__set__def,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] : ( sup_su2896084399781877795_nat_o(A2,B2) = collec6407654713642994138_nat_o(sup_su6907689372094768856at_o_o(aTP_Lamm_ak(A2),aTP_Lamm_ak(B2))) ) ).

% sup_set_def
tff(fact_1184_sup__set__def,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2) = collec5204685387357076818at_nat(sup_su5351388498073598816_nat_o(aTP_Lamm_ch(A2),aTP_Lamm_ch(B2))) ) ).

% sup_set_def
tff(fact_1185_sup__set__def,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = collec6321179662152712658at_nat(sup_su2398265404452057824_nat_o(aTP_Lamm_ci(A2),aTP_Lamm_ci(B2))) ) ).

% sup_set_def
tff(fact_1186_sup__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = collect_nat(sup_sup_fun_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ai,A2),aa_set_nat_fun_nat_o(aTP_Lamm_ai,B2))) ) ).

% sup_set_def
tff(fact_1187_Collect__imp__eq,axiom,
    ! [P: fun_se7249947736802449410_int_o,Q: fun_se7249947736802449410_int_o] : ( collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_cj(P),Q)) = sup_su2047564715030645325nt_int(uminus6423885277529793776nt_int(collec5210948495886036740nt_int(P)),collec5210948495886036740nt_int(Q)) ) ).

% Collect_imp_eq
tff(fact_1188_Collect__imp__eq,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] : ( collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ck(P),Q)) = sup_sup_set_set_nat(uminus613421341184616069et_nat(collect_set_nat(P)),collect_set_nat(Q)) ) ).

% Collect_imp_eq
tff(fact_1189_Collect__imp__eq,axiom,
    ! [P: fun_int_o,Q: fun_int_o] : ( collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cl(P),Q)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,collect_int(P))),collect_int(Q)) ) ).

% Collect_imp_eq
tff(fact_1190_Collect__imp__eq,axiom,
    ! [P: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] : ( collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_cm(P),Q)) = sup_su2896084399781877795_nat_o(uminus2199602881456817606_nat_o(collec6407654713642994138_nat_o(P)),collec6407654713642994138_nat_o(Q)) ) ).

% Collect_imp_eq
tff(fact_1191_Collect__imp__eq,axiom,
    ! [P: fun_Pr1099658227000185396_nat_o,Q: fun_Pr1099658227000185396_nat_o] : ( collec5204685387357076818at_nat(aa_fun9125434449681309094_nat_o(aTP_Lamm_cn(P),Q)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,collec5204685387357076818at_nat(P))),collec5204685387357076818at_nat(Q)) ) ).

% Collect_imp_eq
tff(fact_1192_Collect__imp__eq,axiom,
    ! [P: fun_Pr3059359313727237556_nat_o,Q: fun_Pr3059359313727237556_nat_o] : ( collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_co(P),Q)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(collec6321179662152712658at_nat(P))),collec6321179662152712658at_nat(Q)) ) ).

% Collect_imp_eq
tff(fact_1193_Collect__imp__eq,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] : ( collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cp(P),Q)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,collect_nat(P))),collect_nat(Q)) ) ).

% Collect_imp_eq
tff(fact_1194_Collect__disj__eq,axiom,
    ! [P: fun_se7249947736802449410_int_o,Q: fun_se7249947736802449410_int_o] : ( collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_cq(P),Q)) = sup_su2047564715030645325nt_int(collec5210948495886036740nt_int(P),collec5210948495886036740nt_int(Q)) ) ).

% Collect_disj_eq
tff(fact_1195_Collect__disj__eq,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] : ( collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_cr(P),Q)) = sup_sup_set_set_nat(collect_set_nat(P),collect_set_nat(Q)) ) ).

% Collect_disj_eq
tff(fact_1196_Collect__disj__eq,axiom,
    ! [P: fun_int_o,Q: fun_int_o] : ( collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cs(P),Q)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,collect_int(P)),collect_int(Q)) ) ).

% Collect_disj_eq
tff(fact_1197_Collect__disj__eq,axiom,
    ! [P: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] : ( collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_ct(P),Q)) = sup_su2896084399781877795_nat_o(collec6407654713642994138_nat_o(P),collec6407654713642994138_nat_o(Q)) ) ).

% Collect_disj_eq
tff(fact_1198_Collect__disj__eq,axiom,
    ! [P: fun_Pr1099658227000185396_nat_o,Q: fun_Pr1099658227000185396_nat_o] : ( collec5204685387357076818at_nat(aa_fun9125434449681309094_nat_o(aTP_Lamm_cu(P),Q)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,collec5204685387357076818at_nat(P)),collec5204685387357076818at_nat(Q)) ) ).

% Collect_disj_eq
tff(fact_1199_Collect__disj__eq,axiom,
    ! [P: fun_Pr3059359313727237556_nat_o,Q: fun_Pr3059359313727237556_nat_o] : ( collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_cv(P),Q)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,collec6321179662152712658at_nat(P)),collec6321179662152712658at_nat(Q)) ) ).

% Collect_disj_eq
tff(fact_1200_Collect__disj__eq,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] : ( collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cw(P),Q)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,collect_nat(P)),collect_nat(Q)) ) ).

% Collect_disj_eq
tff(fact_1201_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_1202_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_1203_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_1204_Int__emptyI,axiom,
    ! [A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( ! [X2: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),A2)
         => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),B2) )
     => ( inf_in8396524679539076455nt_int(A2,B2) = bot_bo1488462491386950373nt_int ) ) ).

% Int_emptyI
tff(fact_1205_Int__emptyI,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ! [X2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X2),A2)
         => ~ aa_set_set_nat_o(member_set_nat(X2),B2) )
     => ( inf_inf_set_set_nat(A2,B2) = bot_bot_set_set_nat ) ) ).

% Int_emptyI
tff(fact_1206_Int__emptyI,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A2)
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),B2) )
     => ( inf_in3376754634429691197_nat_o(A2,B2) = bot_bo8632255151191677115_nat_o ) ) ).

% Int_emptyI
tff(fact_1207_Int__emptyI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ! [X2: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),A2)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),B2) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
tff(fact_1208_Int__emptyI,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ! [X2: $o] :
          ( aa_set_o_o(member_o((X2)),A2)
         => ~ aa_set_o_o(member_o((X2)),B2) )
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o ) ) ).

% Int_emptyI
tff(fact_1209_Int__emptyI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
         => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),B2) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat ) ) ).

% Int_emptyI
tff(fact_1210_Int__emptyI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => ~ aa_set_int_o(member_int(X2),B2) )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int ) ) ).

% Int_emptyI
tff(fact_1211_disjoint__iff,axiom,
    ! [A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( ( inf_in8396524679539076455nt_int(A2,B2) = bot_bo1488462491386950373nt_int )
    <=> ! [X4: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X4),A2)
         => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X4),B2) ) ) ).

% disjoint_iff
tff(fact_1212_disjoint__iff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( inf_inf_set_set_nat(A2,B2) = bot_bot_set_set_nat )
    <=> ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X4),A2)
         => ~ aa_set_set_nat_o(member_set_nat(X4),B2) ) ) ).

% disjoint_iff
tff(fact_1213_disjoint__iff,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(A2,B2) = bot_bo8632255151191677115_nat_o )
    <=> ! [X4: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),A2)
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),B2) ) ) ).

% disjoint_iff
tff(fact_1214_disjoint__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A2)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),B2) ) ) ).

% disjoint_iff
tff(fact_1215_disjoint__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o )
    <=> ! [X4: $o] :
          ( aa_set_o_o(member_o((X4)),A2)
         => ~ aa_set_o_o(member_o((X4)),B2) ) ) ).

% disjoint_iff
tff(fact_1216_disjoint__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A2)
         => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),B2) ) ) ).

% disjoint_iff
tff(fact_1217_disjoint__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A2)
         => ~ aa_set_int_o(member_int(X4),B2) ) ) ).

% disjoint_iff
tff(fact_1218_Int__empty__left,axiom,
    ! [B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),B2) = bot_bo2099793752762293965at_nat ) ).

% Int_empty_left
tff(fact_1219_Int__empty__left,axiom,
    ! [B2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,bot_bot_set_o),B2) = bot_bot_set_o ) ).

% Int_empty_left
tff(fact_1220_Int__empty__left,axiom,
    ! [B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),B2) = bot_bot_set_nat ) ).

% Int_empty_left
tff(fact_1221_Int__empty__left,axiom,
    ! [B2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),B2) = bot_bot_set_int ) ).

% Int_empty_left
tff(fact_1222_Int__empty__right,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ) ).

% Int_empty_right
tff(fact_1223_Int__empty__right,axiom,
    ! [A2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),bot_bot_set_o) = bot_bot_set_o ) ).

% Int_empty_right
tff(fact_1224_Int__empty__right,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),bot_bot_set_nat) = bot_bot_set_nat ) ).

% Int_empty_right
tff(fact_1225_Int__empty__right,axiom,
    ! [A2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),bot_bot_set_int) = bot_bot_set_int ) ).

% Int_empty_right
tff(fact_1226_disjoint__iff__not__equal,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A2)
         => ! [Xa2: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa2),B2)
             => ( X4 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_1227_disjoint__iff__not__equal,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o )
    <=> ! [X4: $o] :
          ( aa_set_o_o(member_o((X4)),A2)
         => ! [Xa2: $o] :
              ( aa_set_o_o(member_o((Xa2)),B2)
             => ( (X4)
              <=> ~ (Xa2) ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_1228_disjoint__iff__not__equal,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A2)
         => ! [Xa2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa2),B2)
             => ( X4 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_1229_disjoint__iff__not__equal,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A2)
         => ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),B2)
             => ( X4 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_1230_Un__empty__left,axiom,
    ! [B2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,bot_bo8422036546324065075at_nat),B2) = B2 ) ).

% Un_empty_left
tff(fact_1231_Un__empty__left,axiom,
    ! [B2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),B2) = B2 ) ).

% Un_empty_left
tff(fact_1232_Un__empty__left,axiom,
    ! [B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),B2) = B2 ) ).

% Un_empty_left
tff(fact_1233_Un__empty__left,axiom,
    ! [B2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,bot_bot_set_o),B2) = B2 ) ).

% Un_empty_left
tff(fact_1234_Un__empty__left,axiom,
    ! [B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),B2) = B2 ) ).

% Un_empty_left
tff(fact_1235_Un__empty__left,axiom,
    ! [B2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),B2) = B2 ) ).

% Un_empty_left
tff(fact_1236_Un__empty__right,axiom,
    ! [A2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),bot_bo8422036546324065075at_nat) = A2 ) ).

% Un_empty_right
tff(fact_1237_Un__empty__right,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),bot_bo228742789529271731at_nat) = A2 ) ).

% Un_empty_right
tff(fact_1238_Un__empty__right,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),bot_bo2099793752762293965at_nat) = A2 ) ).

% Un_empty_right
tff(fact_1239_Un__empty__right,axiom,
    ! [A2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),bot_bot_set_o) = A2 ) ).

% Un_empty_right
tff(fact_1240_Un__empty__right,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),bot_bot_set_nat) = A2 ) ).

% Un_empty_right
tff(fact_1241_Un__empty__right,axiom,
    ! [A2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),bot_bot_set_int) = A2 ) ).

% Un_empty_right
tff(fact_1242_Compl__Un,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,A2)),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,B2)) ) ).

% Compl_Un
tff(fact_1243_Compl__Un,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,A2)),aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,B2)) ) ).

% Compl_Un
tff(fact_1244_Compl__Un,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,uminus935396558254630718at_nat(A2)),uminus935396558254630718at_nat(B2)) ) ).

% Compl_Un
tff(fact_1245_Compl__Un,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B2)) ) ).

% Compl_Un
tff(fact_1246_Compl__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,A2)),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,B2)) ) ).

% Compl_Int
tff(fact_1247_Compl__Int,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),B2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,A2)),aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,B2)) ) ).

% Compl_Int
tff(fact_1248_Compl__Int,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(A2)),uminus935396558254630718at_nat(B2)) ) ).

% Compl_Int
tff(fact_1249_Compl__Int,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B2)) ) ).

% Compl_Int
tff(fact_1250_Un__Int__crazy,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C2),A2)) ) ).

% Un_Int_crazy
tff(fact_1251_Un__Int__crazy,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),B2)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B2),C2))),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,C2),A2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B2),C2))),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,C2),A2)) ) ).

% Un_Int_crazy
tff(fact_1252_Un__Int__crazy,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),C2))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,C2),A2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),A2)) ) ).

% Un_Int_crazy
tff(fact_1253_Un__Int__crazy,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),A2)) ) ).

% Un_Int_crazy
tff(fact_1254_Int__Un__distrib,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C2)) ) ).

% Int_Un_distrib
tff(fact_1255_Int__Un__distrib,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B2),C2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),B2)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),C2)) ) ).

% Int_Un_distrib
tff(fact_1256_Int__Un__distrib,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),C2)) ) ).

% Int_Un_distrib
tff(fact_1257_Int__Un__distrib,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)) ) ).

% Int_Un_distrib
tff(fact_1258_Un__Int__distrib,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),C2)) ) ).

% Un_Int_distrib
tff(fact_1259_Un__Int__distrib,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B2),C2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),C2)) ) ).

% Un_Int_distrib
tff(fact_1260_Un__Int__distrib,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),C2)) ) ).

% Un_Int_distrib
tff(fact_1261_Un__Int__distrib,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),C2)) ) ).

% Un_Int_distrib
tff(fact_1262_Int__Un__distrib2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2)),A2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),A2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A2)) ) ).

% Int_Un_distrib2
tff(fact_1263_Int__Un__distrib2,axiom,
    ! [B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B2),C2)),A2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B2),A2)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,C2),A2)) ) ).

% Int_Un_distrib2
tff(fact_1264_Int__Un__distrib2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)),A2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),A2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,C2),A2)) ) ).

% Int_Un_distrib2
tff(fact_1265_Int__Un__distrib2,axiom,
    ! [B2: set_nat,C2: set_nat,A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A2)) ) ).

% Int_Un_distrib2
tff(fact_1266_Un__Int__distrib2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)),A2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),A2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C2),A2)) ) ).

% Un_Int_distrib2
tff(fact_1267_Un__Int__distrib2,axiom,
    ! [B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B2),C2)),A2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B2),A2)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,C2),A2)) ) ).

% Un_Int_distrib2
tff(fact_1268_Un__Int__distrib2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),C2)),A2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),A2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),A2)) ) ).

% Un_Int_distrib2
tff(fact_1269_Un__Int__distrib2,axiom,
    ! [B2: set_nat,C2: set_nat,A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),A2)) ) ).

% Un_Int_distrib2
tff(fact_1270_mult__minus__right,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% mult_minus_right
tff(fact_1271_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ).

% mult_minus_right
tff(fact_1272_mult__minus__right,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ).

% mult_minus_right
tff(fact_1273_minus__mult__minus,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),B) ) ).

% minus_mult_minus
tff(fact_1274_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) ) ).

% minus_mult_minus
tff(fact_1275_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) ) ).

% minus_mult_minus
tff(fact_1276_mult__minus__left,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% mult_minus_left
tff(fact_1277_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ).

% mult_minus_left
tff(fact_1278_mult__minus__left,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ).

% mult_minus_left
tff(fact_1279_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),X) = one_one_int )
    <=> ( ( X = one_one_int )
        | ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_1280_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X) = one_one_Code_integer )
    <=> ( ( X = one_one_Code_integer )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_1281_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X) = one_one_rat )
    <=> ( ( X = one_one_rat )
        | ( X = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_1282_verit__minus__simplify_I4_J,axiom,
    ! [B: int] : ( aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,B)) = B ) ).

% verit_minus_simplify(4)
tff(fact_1283_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = B ) ).

% verit_minus_simplify(4)
tff(fact_1284_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] : ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,B)) = B ) ).

% verit_minus_simplify(4)
tff(fact_1285_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,A3: fun_Pr5098669352780829690e_term,B3: produc8923325533196201883nteger] :
      ( ( aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A3),B3) )
    <=> ( ( A = A3 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_1286_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B) = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A3),B3) )
    <=> ( ( A = A3 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_1287_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B) = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A3),B3) )
    <=> ( ( A = A3 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_1288_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,A3: fun_Pr7763680713840462443e_term,B3: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A3),B3) )
    <=> ( ( A = A3 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_1289_old_Oprod_Oinject,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,A3: fun_in5730379987311621695e_term,B3: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A3),B3) )
    <=> ( ( A = A3 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_1290_prod_Oinject,axiom,
    ! [X1: fun_Pr5098669352780829690e_term,X22: produc8923325533196201883nteger,Y1: fun_Pr5098669352780829690e_term,Y22: produc8923325533196201883nteger] :
      ( ( aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X1),X22) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1291_prod_Oinject,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X22: produc3658429121746597890et_nat,Y1: fun_Pr6460944699146269935_nat_o,Y22: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X1),X22) = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1292_prod_Oinject,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X22: produc8799533983703829792et_nat,Y1: fun_Pr6460944699146269935_nat_o,Y22: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X1),X22) = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1293_prod_Oinject,axiom,
    ! [X1: fun_Pr7763680713840462443e_term,X22: product_prod_int_int,Y1: fun_Pr7763680713840462443e_term,Y22: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X1),X22) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1294_prod_Oinject,axiom,
    ! [X1: fun_in5730379987311621695e_term,X22: product_prod_int_int,Y1: fun_in5730379987311621695e_term,Y22: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X1),X22) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1295_lambda__one,axiom,
    aTP_Lamm_cx = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer) ).

% lambda_one
tff(fact_1296_lambda__one,axiom,
    aTP_Lamm_cy = aa_ass954072293579886259n_assn(times_times_assn,one_one_assn) ).

% lambda_one
tff(fact_1297_lambda__one,axiom,
    aTP_Lamm_cz = aa_rat_fun_rat_rat(times_times_rat,one_one_rat) ).

% lambda_one
tff(fact_1298_lambda__one,axiom,
    aTP_Lamm_da = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_1299_lambda__one,axiom,
    aTP_Lamm_db = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_1300_disjointI,axiom,
    ! [A: set_se6260736226359567993nt_int,B: set_se6260736226359567993nt_int] :
      ( ! [X2: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),A)
         => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),B) )
     => ( inf_in8396524679539076455nt_int(A,B) = bot_bo1488462491386950373nt_int ) ) ).

% disjointI
tff(fact_1301_disjointI,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ! [X2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X2),A)
         => ~ aa_set_set_nat_o(member_set_nat(X2),B) )
     => ( inf_inf_set_set_nat(A,B) = bot_bot_set_set_nat ) ) ).

% disjointI
tff(fact_1302_disjointI,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B: set_fu3555266933746557007_nat_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A)
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),B) )
     => ( inf_in3376754634429691197_nat_o(A,B) = bot_bo8632255151191677115_nat_o ) ) ).

% disjointI
tff(fact_1303_disjointI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ! [X2: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),A)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),B) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ).

% disjointI
tff(fact_1304_disjointI,axiom,
    ! [A: set_o,B: set_o] :
      ( ! [X2: $o] :
          ( aa_set_o_o(member_o((X2)),A)
         => ~ aa_set_o_o(member_o((X2)),B) )
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A),B) = bot_bot_set_o ) ) ).

% disjointI
tff(fact_1305_disjointI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A)
         => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ).

% disjointI
tff(fact_1306_disjointI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
         => ~ aa_set_int_o(member_int(X2),B) )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = bot_bot_set_int ) ) ).

% disjointI
tff(fact_1307_minus__mult__commute,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) ) ).

% minus_mult_commute
tff(fact_1308_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ) ).

% minus_mult_commute
tff(fact_1309_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ) ).

% minus_mult_commute
tff(fact_1310_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),A) = aa_int_int(aa_int_fun_int_int(times_times_int,B),B) )
    <=> ( ( A = B )
        | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_1311_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
    <=> ( ( A = B )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).

% square_eq_iff
tff(fact_1312_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),B) )
    <=> ( ( A = B )
        | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% square_eq_iff
tff(fact_1313_UNIV__I,axiom,
    ! [X: set_Pr958786334691620121nt_int] : aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),top_to6034159466715884489nt_int) ).

% UNIV_I
tff(fact_1314_UNIV__I,axiom,
    ! [X: set_nat] : aa_set_set_nat_o(member_set_nat(X),top_top_set_set_nat) ).

% UNIV_I
tff(fact_1315_UNIV__I,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),top_to9044565752620676511_nat_o) ).

% UNIV_I
tff(fact_1316_UNIV__I,axiom,
    ! [X: list_nat] : member_list_nat(X,top_top_set_list_nat) ).

% UNIV_I
tff(fact_1317_UNIV__I,axiom,
    ! [X: $o] : aa_set_o_o(member_o((X)),top_top_set_o) ).

% UNIV_I
tff(fact_1318_UNIV__I,axiom,
    ! [X: rat] : aa_set_rat_o(member_rat(X),top_top_set_rat) ).

% UNIV_I
tff(fact_1319_UNIV__I,axiom,
    ! [X: nat] : aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),top_top_set_nat) ).

% UNIV_I
tff(fact_1320_UNIV__I,axiom,
    ! [X: int] : aa_set_int_o(member_int(X),top_top_set_int) ).

% UNIV_I
tff(fact_1321_inf1I,axiom,
    ! [A2: fun_Pr3238374608701176162_int_o,X: product_prod_int_int,B2: fun_Pr3238374608701176162_int_o] :
      ( aa_Pro4592068669816069595_int_o(A2,X)
     => ( aa_Pro4592068669816069595_int_o(B2,X)
       => aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A2),B2),X) ) ) ).

% inf1I
tff(fact_1322_inf1I,axiom,
    ! [A2: fun_list_char_o,X: list_char,B2: fun_list_char_o] :
      ( aa_list_char_o(A2,X)
     => ( aa_list_char_o(B2,X)
       => aa_list_char_o(aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A2),B2),X) ) ) ).

% inf1I
tff(fact_1323_UNIV__def,axiom,
    top_to6034159466715884489nt_int = collec5210948495886036740nt_int(aTP_Lamm_dc) ).

% UNIV_def
tff(fact_1324_UNIV__def,axiom,
    top_top_set_set_nat = collect_set_nat(aTP_Lamm_dd) ).

% UNIV_def
tff(fact_1325_UNIV__def,axiom,
    top_to9044565752620676511_nat_o = collec6407654713642994138_nat_o(aTP_Lamm_de) ).

% UNIV_def
tff(fact_1326_UNIV__def,axiom,
    top_top_set_list_nat = collect_list_nat(aTP_Lamm_df) ).

% UNIV_def
tff(fact_1327_UNIV__def,axiom,
    top_top_set_o = collect_o(aTP_Lamm_dg) ).

% UNIV_def
tff(fact_1328_UNIV__def,axiom,
    top_top_set_rat = collect_rat(aTP_Lamm_dh) ).

% UNIV_def
tff(fact_1329_UNIV__def,axiom,
    top_top_set_nat = collect_nat(aTP_Lamm_di) ).

% UNIV_def
tff(fact_1330_UNIV__def,axiom,
    top_top_set_int = collect_int(aTP_Lamm_dj) ).

% UNIV_def
tff(fact_1331_UNIV__eq__I,axiom,
    ! [A2: set_se6260736226359567993nt_int] :
      ( ! [X2: set_Pr958786334691620121nt_int] : aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),A2)
     => ( top_to6034159466715884489nt_int = A2 ) ) ).

% UNIV_eq_I
tff(fact_1332_UNIV__eq__I,axiom,
    ! [A2: set_set_nat] :
      ( ! [X2: set_nat] : aa_set_set_nat_o(member_set_nat(X2),A2)
     => ( top_top_set_set_nat = A2 ) ) ).

% UNIV_eq_I
tff(fact_1333_UNIV__eq__I,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A2)
     => ( top_to9044565752620676511_nat_o = A2 ) ) ).

% UNIV_eq_I
tff(fact_1334_UNIV__eq__I,axiom,
    ! [A2: set_list_nat] :
      ( ! [X2: list_nat] : member_list_nat(X2,A2)
     => ( top_top_set_list_nat = A2 ) ) ).

% UNIV_eq_I
tff(fact_1335_UNIV__eq__I,axiom,
    ! [A2: set_o] :
      ( ! [X2: $o] : aa_set_o_o(member_o((X2)),A2)
     => ( top_top_set_o = A2 ) ) ).

% UNIV_eq_I
tff(fact_1336_UNIV__eq__I,axiom,
    ! [A2: set_rat] :
      ( ! [X2: rat] : aa_set_rat_o(member_rat(X2),A2)
     => ( top_top_set_rat = A2 ) ) ).

% UNIV_eq_I
tff(fact_1337_UNIV__eq__I,axiom,
    ! [A2: set_nat] :
      ( ! [X2: nat] : aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
     => ( top_top_set_nat = A2 ) ) ).

% UNIV_eq_I
tff(fact_1338_UNIV__eq__I,axiom,
    ! [A2: set_int] :
      ( ! [X2: int] : aa_set_int_o(member_int(X2),A2)
     => ( top_top_set_int = A2 ) ) ).

% UNIV_eq_I
tff(fact_1339_top__set__def,axiom,
    top_to6034159466715884489nt_int = collec5210948495886036740nt_int(top_to5472730871871520434_int_o) ).

% top_set_def
tff(fact_1340_top__set__def,axiom,
    top_top_set_set_nat = collect_set_nat(top_to1283867041644559291_nat_o) ).

% top_set_def
tff(fact_1341_top__set__def,axiom,
    top_to9044565752620676511_nat_o = collec6407654713642994138_nat_o(top_to79712303169546844at_o_o) ).

% top_set_def
tff(fact_1342_top__set__def,axiom,
    top_top_set_list_nat = collect_list_nat(top_to317133581726956641_nat_o) ).

% top_set_def
tff(fact_1343_top__set__def,axiom,
    top_top_set_o = collect_o(top_top_fun_o_o) ).

% top_set_def
tff(fact_1344_top__set__def,axiom,
    top_top_set_rat = collect_rat(top_top_fun_rat_o) ).

% top_set_def
tff(fact_1345_top__set__def,axiom,
    top_top_set_nat = collect_nat(top_top_fun_nat_o) ).

% top_set_def
tff(fact_1346_top__set__def,axiom,
    top_top_set_int = collect_int(top_top_fun_int_o) ).

% top_set_def
tff(fact_1347_UNIV__witness,axiom,
    ? [X2: set_Pr958786334691620121nt_int] : aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),top_to6034159466715884489nt_int) ).

% UNIV_witness
tff(fact_1348_UNIV__witness,axiom,
    ? [X2: set_nat] : aa_set_set_nat_o(member_set_nat(X2),top_top_set_set_nat) ).

% UNIV_witness
tff(fact_1349_UNIV__witness,axiom,
    ? [X2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),top_to9044565752620676511_nat_o) ).

% UNIV_witness
tff(fact_1350_UNIV__witness,axiom,
    ? [X2: list_nat] : member_list_nat(X2,top_top_set_list_nat) ).

% UNIV_witness
tff(fact_1351_UNIV__witness,axiom,
    ? [X2: $o] : aa_set_o_o(member_o((X2)),top_top_set_o) ).

% UNIV_witness
tff(fact_1352_UNIV__witness,axiom,
    ? [X2: rat] : aa_set_rat_o(member_rat(X2),top_top_set_rat) ).

% UNIV_witness
tff(fact_1353_UNIV__witness,axiom,
    ? [X2: nat] : aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),top_top_set_nat) ).

% UNIV_witness
tff(fact_1354_UNIV__witness,axiom,
    ? [X2: int] : aa_set_int_o(member_int(X2),top_top_set_int) ).

% UNIV_witness
tff(fact_1355_inf1D2,axiom,
    ! [A2: fun_Pr3238374608701176162_int_o,B2: fun_Pr3238374608701176162_int_o,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A2),B2),X)
     => aa_Pro4592068669816069595_int_o(B2,X) ) ).

% inf1D2
tff(fact_1356_inf1D2,axiom,
    ! [A2: fun_list_char_o,B2: fun_list_char_o,X: list_char] :
      ( aa_list_char_o(aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A2),B2),X)
     => aa_list_char_o(B2,X) ) ).

% inf1D2
tff(fact_1357_inf1D1,axiom,
    ! [A2: fun_Pr3238374608701176162_int_o,B2: fun_Pr3238374608701176162_int_o,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A2),B2),X)
     => aa_Pro4592068669816069595_int_o(A2,X) ) ).

% inf1D1
tff(fact_1358_inf1D1,axiom,
    ! [A2: fun_list_char_o,B2: fun_list_char_o,X: list_char] :
      ( aa_list_char_o(aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A2),B2),X)
     => aa_list_char_o(A2,X) ) ).

% inf1D1
tff(fact_1359_inf1E,axiom,
    ! [A2: fun_Pr3238374608701176162_int_o,B2: fun_Pr3238374608701176162_int_o,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A2),B2),X)
     => ~ ( aa_Pro4592068669816069595_int_o(A2,X)
         => ~ aa_Pro4592068669816069595_int_o(B2,X) ) ) ).

% inf1E
tff(fact_1360_inf1E,axiom,
    ! [A2: fun_list_char_o,B2: fun_list_char_o,X: list_char] :
      ( aa_list_char_o(aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A2),B2),X)
     => ~ ( aa_list_char_o(A2,X)
         => ~ aa_list_char_o(B2,X) ) ) ).

% inf1E
tff(fact_1361_times__assn__raw_Ocases,axiom,
    ! [X: produc3112576180687729086et_nat] :
      ~ ! [P2: fun_Pr6460944699146269935_nat_o,Q2: fun_Pr6460944699146269935_nat_o,H: heap_e7401611519738050253t_unit,As4: set_nat] : ( X != aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,P2),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Q2),produc7507926704131184380et_nat(H,As4))) ) ).

% times_assn_raw.cases
tff(fact_1362_pairself_Ocases,axiom,
    ! [X: produc4788131091073775919nt_int] :
      ~ ! [F2: fun_in5730379987311621695e_term,A4: int,B4: int] : ( X != aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,F2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B4)) ) ).

% pairself.cases
tff(fact_1363_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2453742332425399446nteger] :
      ~ ! [A4: fun_Pr5098669352780829690e_term,B4: produc8923325533196201883nteger] : ( Y != aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A4),B4) ) ).

% old.prod.exhaust
tff(fact_1364_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8799533983703829792et_nat] :
      ~ ! [A4: fun_Pr6460944699146269935_nat_o,B4: produc3658429121746597890et_nat] : ( Y != aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A4),B4) ) ).

% old.prod.exhaust
tff(fact_1365_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A4: fun_Pr6460944699146269935_nat_o,B4: produc8799533983703829792et_nat] : ( Y != aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A4),B4) ) ).

% old.prod.exhaust
tff(fact_1366_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3043004577601930755nt_int] :
      ~ ! [A4: fun_Pr7763680713840462443e_term,B4: product_prod_int_int] : ( Y != aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),B4) ) ).

% old.prod.exhaust
tff(fact_1367_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4788131091073775919nt_int] :
      ~ ! [A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] : ( Y != aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4) ) ).

% old.prod.exhaust
tff(fact_1368_bex2I,axiom,
    ! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,S: set_Pr3486312109254157132nteger,P: fun_fu7695273699059480439eger_o] :
      ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B),S)
     => ( ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B),S)
         => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(P,A),B) )
       => ? [A4: fun_Pr5098669352780829690e_term,B4: produc8923325533196201883nteger] :
            ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A4),B4),S)
            & aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(P,A4),B4) ) ) ) ).

% bex2I
tff(fact_1369_bex2I,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,S: set_Pr1312406793078187350et_nat,P: fun_fu8419763957888848259_nat_o] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B),S)
     => ( ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B),S)
         => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,A),B) )
       => ? [A4: fun_Pr6460944699146269935_nat_o,B4: produc3658429121746597890et_nat] :
            ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A4),B4),S)
            & aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,A4),B4) ) ) ) ).

% bex2I
tff(fact_1370_bex2I,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,S: set_Pr4040394263813967220et_nat,P: fun_fu2611741329380808165_nat_o] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B),S)
     => ( ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B),S)
         => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,A),B) )
       => ? [A4: fun_Pr6460944699146269935_nat_o,B4: produc8799533983703829792et_nat] :
            ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A4),B4),S)
            & aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,A4),B4) ) ) ) ).

% bex2I
tff(fact_1371_bex2I,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,S: set_Pr6700895561065273913nt_int,P: fun_fu6865642474282033866_int_o] :
      ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B),S)
     => ( ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B),S)
         => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(P,A),B) )
       => ? [A4: fun_Pr7763680713840462443e_term,B4: product_prod_int_int] :
            ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),B4),S)
            & aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(P,A4),B4) ) ) ) ).

% bex2I
tff(fact_1372_bex2I,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,S: set_Pr4166680992137330533nt_int,P: fun_fu2978539155105838238_int_o] :
      ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B),S)
     => ( ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B),S)
         => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(P,A),B) )
       => ? [A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
            ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4),S)
            & aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(P,A4),B4) ) ) ) ).

% bex2I
tff(fact_1373_surj__pair,axiom,
    ! [P3: produc2453742332425399446nteger] :
    ? [X2: fun_Pr5098669352780829690e_term,Y2: produc8923325533196201883nteger] : ( P3 = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X2),Y2) ) ).

% surj_pair
tff(fact_1374_surj__pair,axiom,
    ! [P3: produc8799533983703829792et_nat] :
    ? [X2: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] : ( P3 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X2),Y2) ) ).

% surj_pair
tff(fact_1375_surj__pair,axiom,
    ! [P3: produc3112576180687729086et_nat] :
    ? [X2: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat] : ( P3 = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X2),Y2) ) ).

% surj_pair
tff(fact_1376_surj__pair,axiom,
    ! [P3: produc3043004577601930755nt_int] :
    ? [X2: fun_Pr7763680713840462443e_term,Y2: product_prod_int_int] : ( P3 = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X2),Y2) ) ).

% surj_pair
tff(fact_1377_surj__pair,axiom,
    ! [P3: produc4788131091073775919nt_int] :
    ? [X2: fun_in5730379987311621695e_term,Y2: product_prod_int_int] : ( P3 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X2),Y2) ) ).

% surj_pair
tff(fact_1378_prod__cases,axiom,
    ! [P: fun_Pr7015759342434529115eger_o,P3: produc2453742332425399446nteger] :
      ( ! [A4: fun_Pr5098669352780829690e_term,B4: produc8923325533196201883nteger] : aa_Pro4148064003102580676eger_o(P,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A4),B4))
     => aa_Pro4148064003102580676eger_o(P,P3) ) ).

% prod_cases
tff(fact_1379_prod__cases,axiom,
    ! [P: fun_Pr2620521676128474833_nat_o,P3: produc8799533983703829792et_nat] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: produc3658429121746597890et_nat] : aa_Pro5643009357358181178_nat_o(P,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A4),B4))
     => aa_Pro5643009357358181178_nat_o(P,P3) ) ).

% prod_cases
tff(fact_1380_prod__cases,axiom,
    ! [P: fun_Pr1227177680593602867_nat_o,P3: produc3112576180687729086et_nat] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: produc8799533983703829792et_nat] : aa_Pro2671238023309943196_nat_o(P,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A4),B4))
     => aa_Pro2671238023309943196_nat_o(P,P3) ) ).

% prod_cases
tff(fact_1381_prod__cases,axiom,
    ! [P: fun_Pr3540825087398332590_int_o,P3: produc3043004577601930755nt_int] :
      ( ! [A4: fun_Pr7763680713840462443e_term,B4: product_prod_int_int] : aa_Pro2601676399929874711_int_o(P,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),B4))
     => aa_Pro2601676399929874711_int_o(P,P3) ) ).

% prod_cases
tff(fact_1382_prod__cases,axiom,
    ! [P: fun_Pr1196100989263358594_int_o,P3: produc4788131091073775919nt_int] :
      ( ! [A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] : aa_Pro5451197003417746155_int_o(P,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4))
     => aa_Pro5451197003417746155_int_o(P,P3) ) ).

% prod_cases
tff(fact_1383_Pair__inject,axiom,
    ! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,A3: fun_Pr5098669352780829690e_term,B3: produc8923325533196201883nteger] :
      ( ( aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A3),B3) )
     => ~ ( ( A = A3 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_1384_Pair__inject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B) = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A3),B3) )
     => ~ ( ( A = A3 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_1385_Pair__inject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B) = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A3),B3) )
     => ~ ( ( A = A3 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_1386_Pair__inject,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,A3: fun_Pr7763680713840462443e_term,B3: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A3),B3) )
     => ~ ( ( A = A3 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_1387_Pair__inject,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,A3: fun_in5730379987311621695e_term,B3: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A3),B3) )
     => ~ ( ( A = A3 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_1388_prod__cases3,axiom,
    ! [Y: produc2453742332425399446nteger] :
      ~ ! [A4: fun_Pr5098669352780829690e_term,B4: code_integer,C3: code_integer] : ( Y != aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A4),produc1086072967326762835nteger(B4,C3)) ) ).

% prod_cases3
tff(fact_1389_prod__cases3,axiom,
    ! [Y: produc8799533983703829792et_nat] :
      ~ ! [A4: fun_Pr6460944699146269935_nat_o,B4: heap_e7401611519738050253t_unit,C3: set_nat] : ( Y != aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A4),produc7507926704131184380et_nat(B4,C3)) ) ).

% prod_cases3
tff(fact_1390_prod__cases3,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A4: fun_Pr6460944699146269935_nat_o,B4: fun_Pr6460944699146269935_nat_o,C3: produc3658429121746597890et_nat] : ( Y != aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A4),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,B4),C3)) ) ).

% prod_cases3
tff(fact_1391_prod__cases3,axiom,
    ! [Y: produc3043004577601930755nt_int] :
      ~ ! [A4: fun_Pr7763680713840462443e_term,B4: int,C3: int] : ( Y != aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B4),C3)) ) ).

% prod_cases3
tff(fact_1392_prod__cases3,axiom,
    ! [Y: produc4788131091073775919nt_int] :
      ~ ! [A4: fun_in5730379987311621695e_term,B4: int,C3: int] : ( Y != aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B4),C3)) ) ).

% prod_cases3
tff(fact_1393_prod__cases4,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A4: fun_Pr6460944699146269935_nat_o,B4: fun_Pr6460944699146269935_nat_o,C3: heap_e7401611519738050253t_unit,D: set_nat] : ( Y != aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A4),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,B4),produc7507926704131184380et_nat(C3,D))) ) ).

% prod_cases4
tff(fact_1394_prod__induct3,axiom,
    ! [P: fun_Pr7015759342434529115eger_o,X: produc2453742332425399446nteger] :
      ( ! [A4: fun_Pr5098669352780829690e_term,B4: code_integer,C3: code_integer] : aa_Pro4148064003102580676eger_o(P,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A4),produc1086072967326762835nteger(B4,C3)))
     => aa_Pro4148064003102580676eger_o(P,X) ) ).

% prod_induct3
tff(fact_1395_prod__induct3,axiom,
    ! [P: fun_Pr2620521676128474833_nat_o,X: produc8799533983703829792et_nat] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: heap_e7401611519738050253t_unit,C3: set_nat] : aa_Pro5643009357358181178_nat_o(P,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A4),produc7507926704131184380et_nat(B4,C3)))
     => aa_Pro5643009357358181178_nat_o(P,X) ) ).

% prod_induct3
tff(fact_1396_prod__induct3,axiom,
    ! [P: fun_Pr1227177680593602867_nat_o,X: produc3112576180687729086et_nat] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: fun_Pr6460944699146269935_nat_o,C3: produc3658429121746597890et_nat] : aa_Pro2671238023309943196_nat_o(P,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A4),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,B4),C3)))
     => aa_Pro2671238023309943196_nat_o(P,X) ) ).

% prod_induct3
tff(fact_1397_prod__induct3,axiom,
    ! [P: fun_Pr3540825087398332590_int_o,X: produc3043004577601930755nt_int] :
      ( ! [A4: fun_Pr7763680713840462443e_term,B4: int,C3: int] : aa_Pro2601676399929874711_int_o(P,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B4),C3)))
     => aa_Pro2601676399929874711_int_o(P,X) ) ).

% prod_induct3
tff(fact_1398_prod__induct3,axiom,
    ! [P: fun_Pr1196100989263358594_int_o,X: produc4788131091073775919nt_int] :
      ( ! [A4: fun_in5730379987311621695e_term,B4: int,C3: int] : aa_Pro5451197003417746155_int_o(P,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B4),C3)))
     => aa_Pro5451197003417746155_int_o(P,X) ) ).

% prod_induct3
tff(fact_1399_prod__induct4,axiom,
    ! [P: fun_Pr1227177680593602867_nat_o,X: produc3112576180687729086et_nat] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: fun_Pr6460944699146269935_nat_o,C3: heap_e7401611519738050253t_unit,D: set_nat] : aa_Pro2671238023309943196_nat_o(P,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A4),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,B4),produc7507926704131184380et_nat(C3,D))))
     => aa_Pro2671238023309943196_nat_o(P,X) ) ).

% prod_induct4
tff(fact_1400_memb__imp__not__empty,axiom,
    ! [X: set_Pr958786334691620121nt_int,S: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),S)
     => ( S != bot_bo1488462491386950373nt_int ) ) ).

% memb_imp_not_empty
tff(fact_1401_memb__imp__not__empty,axiom,
    ! [X: set_nat,S: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(X),S)
     => ( S != bot_bot_set_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_1402_memb__imp__not__empty,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,S: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),S)
     => ( S != bot_bo8632255151191677115_nat_o ) ) ).

% memb_imp_not_empty
tff(fact_1403_memb__imp__not__empty,axiom,
    ! [X: product_prod_nat_nat,S: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),S)
     => ( S != bot_bo2099793752762293965at_nat ) ) ).

% memb_imp_not_empty
tff(fact_1404_memb__imp__not__empty,axiom,
    ! [X: $o,S: set_o] :
      ( aa_set_o_o(member_o((X)),S)
     => ( S != bot_bot_set_o ) ) ).

% memb_imp_not_empty
tff(fact_1405_memb__imp__not__empty,axiom,
    ! [X: nat,S: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),S)
     => ( S != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_1406_memb__imp__not__empty,axiom,
    ! [X: int,S: set_int] :
      ( aa_set_int_o(member_int(X),S)
     => ( S != bot_bot_set_int ) ) ).

% memb_imp_not_empty
tff(fact_1407_set__notEmptyE,axiom,
    ! [S: set_se6260736226359567993nt_int] :
      ( ( S != bot_bo1488462491386950373nt_int )
     => ~ ! [X2: set_Pr958786334691620121nt_int] : ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),S) ) ).

% set_notEmptyE
tff(fact_1408_set__notEmptyE,axiom,
    ! [S: set_set_nat] :
      ( ( S != bot_bot_set_set_nat )
     => ~ ! [X2: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X2),S) ) ).

% set_notEmptyE
tff(fact_1409_set__notEmptyE,axiom,
    ! [S: set_fu3555266933746557007_nat_o] :
      ( ( S != bot_bo8632255151191677115_nat_o )
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),S) ) ).

% set_notEmptyE
tff(fact_1410_set__notEmptyE,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( S != bot_bo2099793752762293965at_nat )
     => ~ ! [X2: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),S) ) ).

% set_notEmptyE
tff(fact_1411_set__notEmptyE,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ~ ! [X2: $o] : ~ aa_set_o_o(member_o((X2)),S) ) ).

% set_notEmptyE
tff(fact_1412_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X2: nat] : ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S) ) ).

% set_notEmptyE
tff(fact_1413_set__notEmptyE,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ~ ! [X2: int] : ~ aa_set_int_o(member_int(X2),S) ) ).

% set_notEmptyE
tff(fact_1414_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_1415_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_1416_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_1417_sup__Un__eq,axiom,
    ! [R: set_se6260736226359567993nt_int,S: set_se6260736226359567993nt_int,X3: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(sup_su2829768162508406574_int_o(aTP_Lamm_ag(R),aTP_Lamm_ag(S)),X3)
    <=> aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),sup_su2047564715030645325nt_int(R,S)) ) ).

% sup_Un_eq
tff(fact_1418_sup__Un__eq,axiom,
    ! [R: set_set_nat,S: set_set_nat,X3: set_nat] :
      ( aa_set_nat_o(sup_su7503774895437956671_nat_o(aTP_Lamm_ah(R),aTP_Lamm_ah(S)),X3)
    <=> aa_set_set_nat_o(member_set_nat(X3),sup_sup_set_set_nat(R,S)) ) ).

% sup_Un_eq
tff(fact_1419_sup__Un__eq,axiom,
    ! [R: set_int,S: set_int,X3: int] :
      ( aa_int_o(sup_sup_fun_int_o(aTP_Lamm_aj(R),aTP_Lamm_aj(S)),X3)
    <=> aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,R),S)) ) ).

% sup_Un_eq
tff(fact_1420_sup__Un__eq,axiom,
    ! [R: set_fu3555266933746557007_nat_o,S: set_fu3555266933746557007_nat_o,X3: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(sup_su6907689372094768856at_o_o(aTP_Lamm_ak(R),aTP_Lamm_ak(S)),X3)
    <=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),sup_su2896084399781877795_nat_o(R,S)) ) ).

% sup_Un_eq
tff(fact_1421_sup__Un__eq,axiom,
    ! [R: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat,X3: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(sup_su5351388498073598816_nat_o(aTP_Lamm_ch(R),aTP_Lamm_ch(S)),X3)
    <=> member6689249552917799696at_nat(X3,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,R),S)) ) ).

% sup_Un_eq
tff(fact_1422_sup__Un__eq,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(sup_su2398265404452057824_nat_o(aTP_Lamm_ci(R),aTP_Lamm_ci(S)),X3)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,R),S)) ) ).

% sup_Un_eq
tff(fact_1423_sup__Un__eq,axiom,
    ! [R: set_nat,S: set_nat,X3: nat] :
      ( aa_nat_o(sup_sup_fun_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ai,R),aa_set_nat_fun_nat_o(aTP_Lamm_ai,S)),X3)
    <=> aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,R),S)) ) ).

% sup_Un_eq
tff(fact_1424_inf__Int__eq,axiom,
    ! [R: set_se6260736226359567993nt_int,S: set_se6260736226359567993nt_int,X3: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(inf_in9107477634626221460_int_o(aTP_Lamm_ag(R),aTP_Lamm_ag(S)),X3)
    <=> aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),inf_in8396524679539076455nt_int(R,S)) ) ).

% inf_Int_eq
tff(fact_1425_inf__Int__eq,axiom,
    ! [R: set_set_nat,S: set_set_nat,X3: set_nat] :
      ( aa_set_nat_o(inf_in7602907407060649817_nat_o(aTP_Lamm_ah(R),aTP_Lamm_ah(S)),X3)
    <=> aa_set_set_nat_o(member_set_nat(X3),inf_inf_set_set_nat(R,S)) ) ).

% inf_Int_eq
tff(fact_1426_inf__Int__eq,axiom,
    ! [R: set_int,S: set_int,X3: int] :
      ( aa_int_o(inf_inf_fun_int_o(aTP_Lamm_aj(R),aTP_Lamm_aj(S)),X3)
    <=> aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,R),S)) ) ).

% inf_Int_eq
tff(fact_1427_inf__Int__eq,axiom,
    ! [R: set_fu3555266933746557007_nat_o,S: set_fu3555266933746557007_nat_o,X3: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(inf_in2686814848456516414at_o_o(aTP_Lamm_ak(R),aTP_Lamm_ak(S)),X3)
    <=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),inf_in3376754634429691197_nat_o(R,S)) ) ).

% inf_Int_eq
tff(fact_1428_inf__Int__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(inf_in3341167027036195756_nat_o(aTP_Lamm_bt(R)),aTP_Lamm_bt(S)),X3)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,R),S)) ) ).

% inf_Int_eq
tff(fact_1429_inf__Int__eq,axiom,
    ! [R: set_nat,S: set_nat,X3: nat] :
      ( aa_nat_o(inf_inf_fun_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ai,R),aa_set_nat_fun_nat_o(aTP_Lamm_ai,S)),X3)
    <=> aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,R),S)) ) ).

% inf_Int_eq
tff(fact_1430_inf__Int__eq,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int,X3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aTP_Lamm_br(R)),aTP_Lamm_br(S)),X3)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(X3),inf_in2269163501485487111nt_int(R,S)) ) ).

% inf_Int_eq
tff(fact_1431_inf__Int__eq,axiom,
    ! [R: set_list_char,S: set_list_char,X3: list_char] :
      ( aa_list_char_o(aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aTP_Lamm_bs(R)),aTP_Lamm_bs(S)),X3)
    <=> member_list_char(X3,inf_in6177012701088388009t_char(R,S)) ) ).

% inf_Int_eq
tff(fact_1432_iso__tuple__UNIV__I,axiom,
    ! [X: set_Pr958786334691620121nt_int] : aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),top_to6034159466715884489nt_int) ).

% iso_tuple_UNIV_I
tff(fact_1433_iso__tuple__UNIV__I,axiom,
    ! [X: set_nat] : aa_set_set_nat_o(member_set_nat(X),top_top_set_set_nat) ).

% iso_tuple_UNIV_I
tff(fact_1434_iso__tuple__UNIV__I,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),top_to9044565752620676511_nat_o) ).

% iso_tuple_UNIV_I
tff(fact_1435_iso__tuple__UNIV__I,axiom,
    ! [X: list_nat] : member_list_nat(X,top_top_set_list_nat) ).

% iso_tuple_UNIV_I
tff(fact_1436_iso__tuple__UNIV__I,axiom,
    ! [X: $o] : aa_set_o_o(member_o((X)),top_top_set_o) ).

% iso_tuple_UNIV_I
tff(fact_1437_iso__tuple__UNIV__I,axiom,
    ! [X: rat] : aa_set_rat_o(member_rat(X),top_top_set_rat) ).

% iso_tuple_UNIV_I
tff(fact_1438_iso__tuple__UNIV__I,axiom,
    ! [X: nat] : aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),top_top_set_nat) ).

% iso_tuple_UNIV_I
tff(fact_1439_iso__tuple__UNIV__I,axiom,
    ! [X: int] : aa_set_int_o(member_int(X),top_top_set_int) ).

% iso_tuple_UNIV_I
tff(fact_1440_Collect__empty__eq__bot,axiom,
    ! [P: fun_se7249947736802449410_int_o] :
      ( ( collec5210948495886036740nt_int(P) = bot_bo1488462491386950373nt_int )
    <=> ( P = bot_bo4864556513171353750_int_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1441_Collect__empty__eq__bot,axiom,
    ! [P: fun_set_nat_o] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ( P = bot_bo3967814874310145751_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1442_Collect__empty__eq__bot,axiom,
    ! [P: fun_fu5135969500543783852at_o_o] :
      ( ( collec6407654713642994138_nat_o(P) = bot_bo8632255151191677115_nat_o )
    <=> ( P = bot_bo8793957029302321216at_o_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1443_Collect__empty__eq__bot,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( ( collec3392354462482085612at_nat(P) = bot_bo2099793752762293965at_nat )
    <=> ( P = bot_bo5074786513356115630_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1444_Collect__empty__eq__bot,axiom,
    ! [P: fun_o_o] :
      ( ( collect_o(P) = bot_bot_set_o )
    <=> ( P = bot_bot_fun_o_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1445_Collect__empty__eq__bot,axiom,
    ! [P: fun_nat_o] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ( P = bot_bot_fun_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1446_Collect__empty__eq__bot,axiom,
    ! [P: fun_int_o] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ( P = bot_bot_fun_int_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1447_bot__empty__eq,axiom,
    ! [X3: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(bot_bo4864556513171353750_int_o,X3)
    <=> aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),bot_bo1488462491386950373nt_int) ) ).

% bot_empty_eq
tff(fact_1448_bot__empty__eq,axiom,
    ! [X3: set_nat] :
      ( aa_set_nat_o(bot_bo3967814874310145751_nat_o,X3)
    <=> aa_set_set_nat_o(member_set_nat(X3),bot_bot_set_set_nat) ) ).

% bot_empty_eq
tff(fact_1449_bot__empty__eq,axiom,
    ! [X3: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(bot_bo8793957029302321216at_o_o,X3)
    <=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),bot_bo8632255151191677115_nat_o) ) ).

% bot_empty_eq
tff(fact_1450_bot__empty__eq,axiom,
    ! [X3: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(bot_bo5074786513356115630_nat_o,X3)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq
tff(fact_1451_bot__empty__eq,axiom,
    ! [X3: $o] :
      ( aa_o_o(bot_bot_fun_o_o,(X3))
    <=> aa_set_o_o(member_o((X3)),bot_bot_set_o) ) ).

% bot_empty_eq
tff(fact_1452_bot__empty__eq,axiom,
    ! [X3: nat] :
      ( aa_nat_o(bot_bot_fun_nat_o,X3)
    <=> aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),bot_bot_set_nat) ) ).

% bot_empty_eq
tff(fact_1453_bot__empty__eq,axiom,
    ! [X3: int] :
      ( aa_int_o(bot_bot_fun_int_o,X3)
    <=> aa_set_int_o(member_int(X3),bot_bot_set_int) ) ).

% bot_empty_eq
tff(fact_1454_type__copy__ex__RepI,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,F3: fun_o_o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( ? [X_12: $o] : aa_o_o(F3,(X_12))
      <=> ? [B5: product_unit] : aa_o_o(F3,aa_Product_unit_o(Rep,B5)) ) ) ).

% type_copy_ex_RepI
tff(fact_1455_type__copy__ex__RepI,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,F3: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),top_to9044565752620676511_nat_o)
     => ( ? [X_12: fun_Pr6460944699146269935_nat_o] : aa_fun1675921005893230629at_o_o(F3,X_12)
      <=> ? [B5: assn] : aa_fun1675921005893230629at_o_o(F3,aa_ass4243947775541419017_nat_o(Rep,B5)) ) ) ).

% type_copy_ex_RepI
tff(fact_1456_type__copy__ex__RepI,axiom,
    ! [Rep: fun_li7473841323753414984t_char,Abs: fun_li5857714827575272844iteral,F3: fun_list_char_o] :
      ( type_d4752411451802217481t_char(Rep,Abs,top_to3190553582279535303t_char)
     => ( ? [X_12: list_char] : aa_list_char_o(F3,X_12)
      <=> ? [B5: literal] : aa_list_char_o(F3,aa_literal_list_char(Rep,B5)) ) ) ).

% type_copy_ex_RepI
tff(fact_1457_type__copy__ex__RepI,axiom,
    ! [Rep: fun_ra7314870381713115802nt_int,Abs: fun_se593059424582005380nt_rat,F3: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(Rep,Abs),top_to6034159466715884489nt_int)
     => ( ? [X_12: set_Pr958786334691620121nt_int] : aa_set6170482117830564987_int_o(F3,X_12)
      <=> ? [B5: rat] : aa_set6170482117830564987_int_o(F3,aa_rat8624586542854104963nt_int(Rep,B5)) ) ) ).

% type_copy_ex_RepI
tff(fact_1458_type__copy__ex__RepI,axiom,
    ! [Rep: fun_in8804710076054006414at_nat,Abs: fun_se9142667792015812128at_int,F3: fun_se4782206882984618170_nat_o] :
      ( type_d8752208705193531015at_nat(Rep,Abs,top_to7629004291339433233at_nat)
     => ( ? [X_12: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(F3,X_12)
      <=> ? [B5: int] : aa_set3702741264012733747_nat_o(F3,aa_int8822742982810151799at_nat(Rep,B5)) ) ) ).

% type_copy_ex_RepI
tff(fact_1459_type__copy__obj__one__point__absE,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,S2: product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ~ ! [X2: $o] : ( S2 != aa_o_Product_unit(Abs,(X2)) ) ) ).

% type_copy_obj_one_point_absE
tff(fact_1460_type__copy__obj__one__point__absE,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,S2: assn] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),top_to9044565752620676511_nat_o)
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o] : ( S2 != aa_fun5526454687433275355o_assn(Abs,X2) ) ) ).

% type_copy_obj_one_point_absE
tff(fact_1461_type__copy__obj__one__point__absE,axiom,
    ! [Rep: fun_li7473841323753414984t_char,Abs: fun_li5857714827575272844iteral,S2: literal] :
      ( type_d4752411451802217481t_char(Rep,Abs,top_to3190553582279535303t_char)
     => ~ ! [X2: list_char] : ( S2 != aa_list_char_literal(Abs,X2) ) ) ).

% type_copy_obj_one_point_absE
tff(fact_1462_type__copy__obj__one__point__absE,axiom,
    ! [Rep: fun_ra7314870381713115802nt_int,Abs: fun_se593059424582005380nt_rat,S2: rat] :
      ( aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(Rep,Abs),top_to6034159466715884489nt_int)
     => ~ ! [X2: set_Pr958786334691620121nt_int] : ( S2 != aa_set4118085860027418549nt_rat(Abs,X2) ) ) ).

% type_copy_obj_one_point_absE
tff(fact_1463_type__copy__obj__one__point__absE,axiom,
    ! [Rep: fun_in8804710076054006414at_nat,Abs: fun_se9142667792015812128at_int,S2: int] :
      ( type_d8752208705193531015at_nat(Rep,Abs,top_to7629004291339433233at_nat)
     => ~ ! [X2: set_Pr1261947904930325089at_nat] : ( S2 != aa_set4746584241299694929at_int(Abs,X2) ) ) ).

% type_copy_obj_one_point_absE
tff(fact_1464_dbl__inc__simps_I4_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_inc_simps(4)
tff(fact_1465_dbl__inc__simps_I4_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_inc_simps(4)
tff(fact_1466_dbl__inc__simps_I4_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_1467_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    aa_Product_unit_o(boolea7165431671056489859t_unit(inf_inf_Product_unit,sup_sup_Product_unit,uminus2952777764628376836t_unit,bot_bot_Product_unit),top_top_Product_unit) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_1468_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    aa_set_o_o(boolea379910186789422830_set_o(inf_inf_set_o,sup_sup_set_o,uminus_uminus_set_o,bot_bot_set_o),top_top_set_o) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_1469_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    aa_set_rat_o(boolea6493995471900120728et_rat(inf_inf_set_rat,sup_sup_set_rat,uminus2201863774496077783et_rat,bot_bot_set_rat),top_top_set_rat) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_1470_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    aa_set_nat_o(aa_set7861427066675100071_nat_o(boolea778851993438741648et_nat(inf_inf_set_nat,sup_sup_set_nat,uminus5710092332889474511et_nat),bot_bot_set_nat),top_top_set_nat) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_1471_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    aa_set_int_o(boolea5824373010784320748et_int(inf_inf_set_int,sup_sup_set_int,uminus1532241313380277803et_int,bot_bot_set_int),top_top_set_int) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_1472_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea9074081411689916704st_nat(inf_inf_set_list_nat,sup_sup_set_list_nat,uminus3195874150345416415st_nat,bot_bot_set_list_nat,top_top_set_list_nat) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_1473_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea7067051528696211142char_o(inf_in94156054473216830char_o,sup_su7322457434961926436char_o,uminus8184685084194811975char_o,bot_bo8860746597864711612char_o,top_to7404587909000090784char_o) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_1474_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    aa_set3702741264012733747_nat_o(boolea949959532490082519at_nat(inf_in2572325071724192079at_nat,sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat,bot_bo2099793752762293965at_nat),top_to4669805908274784177at_nat) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_1475_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea44587999064921452_int_o(inf_in1782598092406300404_int_o,sup_su4047618219528032398_int_o,uminus103585356917985579_int_o,bot_bo3516217578726220278_int_o,top_to8487252020945707026_int_o) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_1476_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea51845251928480829at_nat(inf_in1697001100524423349at_nat,sup_su3035147773818789531at_nat,uminus2330091110623919550at_nat,bot_bo8422036546324065075at_nat,top_to8764741339697478167at_nat) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_1477_top__empty__eq,axiom,
    ! [X3: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(top_to5472730871871520434_int_o,X3)
    <=> aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),top_to6034159466715884489nt_int) ) ).

% top_empty_eq
tff(fact_1478_top__empty__eq,axiom,
    ! [X3: set_nat] :
      ( aa_set_nat_o(top_to1283867041644559291_nat_o,X3)
    <=> aa_set_set_nat_o(member_set_nat(X3),top_top_set_set_nat) ) ).

% top_empty_eq
tff(fact_1479_top__empty__eq,axiom,
    ! [X3: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(top_to79712303169546844at_o_o,X3)
    <=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),top_to9044565752620676511_nat_o) ) ).

% top_empty_eq
tff(fact_1480_top__empty__eq,axiom,
    ! [X3: list_nat] :
      ( aa_list_nat_o(top_to317133581726956641_nat_o,X3)
    <=> member_list_nat(X3,top_top_set_list_nat) ) ).

% top_empty_eq
tff(fact_1481_top__empty__eq,axiom,
    ! [X3: $o] :
      ( aa_o_o(top_top_fun_o_o,(X3))
    <=> aa_set_o_o(member_o((X3)),top_top_set_o) ) ).

% top_empty_eq
tff(fact_1482_top__empty__eq,axiom,
    ! [X3: rat] :
      ( aa_rat_o(top_top_fun_rat_o,X3)
    <=> aa_set_rat_o(member_rat(X3),top_top_set_rat) ) ).

% top_empty_eq
tff(fact_1483_top__empty__eq,axiom,
    ! [X3: nat] :
      ( aa_nat_o(top_top_fun_nat_o,X3)
    <=> aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),top_top_set_nat) ) ).

% top_empty_eq
tff(fact_1484_top__empty__eq,axiom,
    ! [X3: int] :
      ( aa_int_o(top_top_fun_int_o,X3)
    <=> aa_set_int_o(member_int(X3),top_top_set_int) ) ).

% top_empty_eq
tff(fact_1485_sup__Un__eq2,axiom,
    ! [R: set_Pr3486312109254157132nteger,S: set_Pr3486312109254157132nteger,X3: fun_Pr5098669352780829690e_term,Xa3: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(sup_su8446958762992054435eger_o(aTP_Lamm_dk(R),aTP_Lamm_dk(S)),X3),Xa3)
    <=> member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X3),Xa3),sup_su4503389074446631032nteger(R,S)) ) ).

% sup_Un_eq2
tff(fact_1486_sup__Un__eq2,axiom,
    ! [R: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat,X3: fun_Pr6460944699146269935_nat_o,Xa3: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(sup_su4970993554803434671_nat_o(aTP_Lamm_dl(R),aTP_Lamm_dl(S)),X3),Xa3)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X3),Xa3),sup_su2387659076810923138et_nat(R,S)) ) ).

% sup_Un_eq2
tff(fact_1487_sup__Un__eq2,axiom,
    ! [R: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat,X3: fun_Pr6460944699146269935_nat_o,Xa3: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(sup_su6780828561269846801_nat_o(aTP_Lamm_dm(R),aTP_Lamm_dm(S)),X3),Xa3)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X3),Xa3),sup_su5858552540197408416et_nat(R,S)) ) ).

% sup_Un_eq2
tff(fact_1488_sup__Un__eq2,axiom,
    ! [R: set_Pr6700895561065273913nt_int,S: set_Pr6700895561065273913nt_int,X3: fun_Pr7763680713840462443e_term,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(sup_su2175524065394371574_int_o(aTP_Lamm_dn(R),aTP_Lamm_dn(S)),X3),Xa3)
    <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X3),Xa3),sup_su2017857570560100197nt_int(R,S)) ) ).

% sup_Un_eq2
tff(fact_1489_sup__Un__eq2,axiom,
    ! [R: set_Pr4166680992137330533nt_int,S: set_Pr4166680992137330533nt_int,X3: fun_in5730379987311621695e_term,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(sup_su7230269426179251658_int_o(aTP_Lamm_do(R),aTP_Lamm_do(S)),X3),Xa3)
    <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Xa3),sup_su7513280395484302481nt_int(R,S)) ) ).

% sup_Un_eq2
tff(fact_1490_sup__Un__eq2,axiom,
    ! [R: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat,X3: multis2468970476368604999at_nat,Xa3: multis2468970476368604999at_nat] :
      ( aa_mul869302825175964237_nat_o(aa_mul3916465897682844511_nat_o(sup_su9034197199115871746_nat_o(aTP_Lamm_dp(R),aTP_Lamm_dp(S)),X3),Xa3)
    <=> member6689249552917799696at_nat(produc4348348721325984599at_nat(X3,Xa3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,R),S)) ) ).

% sup_Un_eq2
tff(fact_1491_sup__Un__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat,X3: set_Pr1261947904930325089at_nat,Xa3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(sup_su5424986631308460290_nat_o(aTP_Lamm_dq(R),aTP_Lamm_dq(S)),X3),Xa3)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X3,Xa3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,R),S)) ) ).

% sup_Un_eq2
tff(fact_1492_inf__Int__eq2,axiom,
    ! [R: set_Pr3486312109254157132nteger,S: set_Pr3486312109254157132nteger,X3: fun_Pr5098669352780829690e_term,Xa3: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(inf_in7430926233015486473eger_o(aTP_Lamm_dk(R),aTP_Lamm_dk(S)),X3),Xa3)
    <=> member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X3),Xa3),inf_in6300790993181727198nteger(R,S)) ) ).

% inf_Int_eq2
tff(fact_1493_inf__Int__eq2,axiom,
    ! [R: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat,X3: fun_Pr6460944699146269935_nat_o,Xa3: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(inf_in3476382694143140117_nat_o(aTP_Lamm_dl(R),aTP_Lamm_dl(S)),X3),Xa3)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X3),Xa3),inf_in4791168559842171624et_nat(R,S)) ) ).

% inf_Int_eq2
tff(fact_1494_inf__Int__eq2,axiom,
    ! [R: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat,X3: fun_Pr6460944699146269935_nat_o,Xa3: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(inf_in6333540121576287351_nat_o(aTP_Lamm_dm(R),aTP_Lamm_dm(S)),X3),Xa3)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X3),Xa3),inf_in6821596156192049670et_nat(R,S)) ) ).

% inf_Int_eq2
tff(fact_1495_inf__Int__eq2,axiom,
    ! [R: set_Pr6700895561065273913nt_int,S: set_Pr6700895561065273913nt_int,X3: fun_Pr7763680713840462443e_term,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(inf_in132845422080707420_int_o(aTP_Lamm_dn(R),aTP_Lamm_dn(S)),X3),Xa3)
    <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X3),Xa3),inf_in3653535833467300043nt_int(R,S)) ) ).

% inf_Int_eq2
tff(fact_1496_inf__Int__eq2,axiom,
    ! [R: set_Pr4166680992137330533nt_int,S: set_Pr4166680992137330533nt_int,X3: fun_in5730379987311621695e_term,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(inf_in4973581681881435440_int_o(aTP_Lamm_do(R),aTP_Lamm_do(S)),X3),Xa3)
    <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Xa3),inf_in8549327947507203575nt_int(R,S)) ) ).

% inf_Int_eq2
tff(fact_1497_inf__Int__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X3: nat,Xa3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(inf_in375002649321709592_nat_o(aTP_Lamm_dr(R),aTP_Lamm_dr(S)),X3),Xa3)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,R),S)) ) ).

% inf_Int_eq2
tff(fact_1498_bot__empty__eq2,axiom,
    ! [X3: fun_Pr5098669352780829690e_term,Xa3: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(bot_bo4567088895781559819eger_o,X3),Xa3)
    <=> member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X3),Xa3),bot_bo9050247954524366304nteger) ) ).

% bot_empty_eq2
tff(fact_1499_bot__empty__eq2,axiom,
    ! [X3: fun_Pr6460944699146269935_nat_o,Xa3: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(bot_bo3509460380234399255_nat_o,X3),Xa3)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X3),Xa3),bot_bo1493351289776401898et_nat) ) ).

% bot_empty_eq2
tff(fact_1500_bot__empty__eq2,axiom,
    ! [X3: fun_Pr6460944699146269935_nat_o,Xa3: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(bot_bo5031374944739956857_nat_o,X3),Xa3)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X3),Xa3),bot_bo6121529163330120712et_nat) ) ).

% bot_empty_eq2
tff(fact_1501_bot__empty__eq2,axiom,
    ! [X3: fun_Pr7763680713840462443e_term,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(bot_bo1117279750869432670_int_o,X3),Xa3)
    <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X3),Xa3),bot_bo2501116328598523085nt_int) ) ).

% bot_empty_eq2
tff(fact_1502_bot__empty__eq2,axiom,
    ! [X3: fun_in5730379987311621695e_term,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(bot_bo6524246659364023090_int_o,X3),Xa3)
    <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Xa3),bot_bo5189594446080201209nt_int) ) ).

% bot_empty_eq2
tff(fact_1503_bot__empty__eq2,axiom,
    ! [X3: nat,Xa3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X3),Xa3)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa3)),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_1504_pred__equals__eq2,axiom,
    ! [R: set_Pr3486312109254157132nteger,S: set_Pr3486312109254157132nteger] :
      ( ! [X4: fun_Pr5098669352780829690e_term,Xa2: produc8923325533196201883nteger] :
          ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X4),Xa2),R)
        <=> member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X4),Xa2),S) )
    <=> ( R = S ) ) ).

% pred_equals_eq2
tff(fact_1505_pred__equals__eq2,axiom,
    ! [R: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat] :
      ( ! [X4: fun_Pr6460944699146269935_nat_o,Xa2: produc3658429121746597890et_nat] :
          ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Xa2),R)
        <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Xa2),S) )
    <=> ( R = S ) ) ).

% pred_equals_eq2
tff(fact_1506_pred__equals__eq2,axiom,
    ! [R: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat] :
      ( ! [X4: fun_Pr6460944699146269935_nat_o,Xa2: produc8799533983703829792et_nat] :
          ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Xa2),R)
        <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Xa2),S) )
    <=> ( R = S ) ) ).

% pred_equals_eq2
tff(fact_1507_pred__equals__eq2,axiom,
    ! [R: set_Pr6700895561065273913nt_int,S: set_Pr6700895561065273913nt_int] :
      ( ! [X4: fun_Pr7763680713840462443e_term,Xa2: product_prod_int_int] :
          ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Xa2),R)
        <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Xa2),S) )
    <=> ( R = S ) ) ).

% pred_equals_eq2
tff(fact_1508_pred__equals__eq2,axiom,
    ! [R: set_Pr4166680992137330533nt_int,S: set_Pr4166680992137330533nt_int] :
      ( ! [X4: fun_in5730379987311621695e_term,Xa2: product_prod_int_int] :
          ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Xa2),R)
        <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Xa2),S) )
    <=> ( R = S ) ) ).

% pred_equals_eq2
tff(fact_1509_top__empty__eq2,axiom,
    ! [X3: fun_Pr5098669352780829690e_term,Xa3: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(top_to2784585791485011495eger_o,X3),Xa3)
    <=> member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X3),Xa3),top_to7525899545246753276nteger) ) ).

% top_empty_eq2
tff(fact_1510_top__empty__eq2,axiom,
    ! [X3: fun_Pr6460944699146269935_nat_o,Xa3: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(top_to2712223222597888051_nat_o,X3),Xa3)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X3),Xa3),top_to2987642378616949766et_nat) ) ).

% top_empty_eq2
tff(fact_1511_top__empty__eq2,axiom,
    ! [X3: fun_Pr6460944699146269935_nat_o,Xa3: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(top_to3067105749023872149_nat_o,X3),Xa3)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X3),Xa3),top_to2648389616742571044et_nat) ) ).

% top_empty_eq2
tff(fact_1512_top__empty__eq2,axiom,
    ! [X3: fun_Pr7763680713840462443e_term,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(top_to1218662295448195450_int_o,X3),Xa3)
    <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X3),Xa3),top_to6107225058432391401nt_int) ) ).

% top_empty_eq2
tff(fact_1513_top__empty__eq2,axiom,
    ! [X3: fun_in5730379987311621695e_term,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(top_to1089409542825796430_int_o,X3),Xa3)
    <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Xa3),top_to1212741241995312661nt_int) ) ).

% top_empty_eq2
tff(fact_1514_times__assn__raw_Opelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As4) )
             => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),produc7507926704131184380et_nat(H,As4))))
               => ? [As13: set_nat,As23: set_nat] :
                    ( ( As4 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As13),As23) )
                    & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As13),As23) = bot_bot_set_nat )
                    & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H,As13))
                    & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As23)) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
tff(fact_1515_times__assn__raw_Opelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As4) )
             => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),produc7507926704131184380et_nat(H,As4))))
               => ~ ? [As12: set_nat,As22: set_nat] :
                      ( ( As4 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As12),As22) )
                      & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As12),As22) = bot_bot_set_nat )
                      & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H,As12))
                      & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As22)) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
tff(fact_1516_times__assn__raw_Opelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
      <=> (Y) )
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As4) )
             => ( ( (Y)
                <=> ? [As1: set_nat,As2: set_nat] :
                      ( ( As4 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As2) )
                      & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As2) = bot_bot_set_nat )
                      & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H,As1))
                      & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As2)) ) )
               => ~ aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),produc7507926704131184380et_nat(H,As4)))) ) ) ) ) ).

% times_assn_raw.pelims(1)
tff(fact_1517_Set_Ois__empty__def,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(is_emp1662574758705540307at_nat,A2)
    <=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% Set.is_empty_def
tff(fact_1518_Set_Ois__empty__def,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_empty_o,A2)
    <=> ( A2 = bot_bot_set_o ) ) ).

% Set.is_empty_def
tff(fact_1519_Set_Ois__empty__def,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_empty_nat,A2)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% Set.is_empty_def
tff(fact_1520_Set_Ois__empty__def,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(is_empty_int,A2)
    <=> ( A2 = bot_bot_set_int ) ) ).

% Set.is_empty_def
tff(fact_1521_wand__assnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As: set_nat,Q: assn,R: assn] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As))
     => ( ! [H4: heap_e7401611519738050253t_unit,As5: set_nat] :
            ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As5) = bot_bot_set_nat )
           => ( relH(As,H2,H4)
             => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As))
               => ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),produc7507926704131184380et_nat(H4,As5))
                 => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,R),produc7507926704131184380et_nat(H4,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As5))) ) ) ) )
       => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,wand_assn(Q,R)),produc7507926704131184380et_nat(H2,As)) ) ) ).

% wand_assnI
tff(fact_1522_vimage__if,axiom,
    ! [B2: set_o,C: nat,D2: nat,A2: set_nat] :
      ( vimage_o_nat(aa_nat_fun_o_nat(aa_nat7267195842670305526_o_nat(aTP_Lamm_ds(B2),C),D2),A2) = $ite(
          aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2),
          $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,D2),A2),top_top_set_o,B2),
          $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,D2),A2),aa_set_o_set_o(uminus_uminus_set_o,B2),bot_bot_set_o) ) ) ).

% vimage_if
tff(fact_1523_vimage__if,axiom,
    ! [B2: set_o,C: int,D2: int,A2: set_int] :
      ( vimage_o_int(aa_int_fun_o_int(aa_int1171789168926488730_o_int(aTP_Lamm_dt(B2),C),D2),A2) = $ite(
          aa_set_int_o(member_int(C),A2),
          $ite(aa_set_int_o(member_int(D2),A2),top_top_set_o,B2),
          $ite(aa_set_int_o(member_int(D2),A2),aa_set_o_set_o(uminus_uminus_set_o,B2),bot_bot_set_o) ) ) ).

% vimage_if
tff(fact_1524_vimage__if,axiom,
    ! [B2: set_rat,C: nat,D2: nat,A2: set_nat] :
      ( vimage_rat_nat(aa_nat_fun_rat_nat(aa_nat8605836375616589056at_nat(aTP_Lamm_du(B2),C),D2),A2) = $ite(
          aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2),
          $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,D2),A2),top_top_set_rat,B2),
          $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,D2),A2),aa_set_rat_set_rat(uminus2201863774496077783et_rat,B2),bot_bot_set_rat) ) ) ).

% vimage_if
tff(fact_1525_vimage__if,axiom,
    ! [B2: set_rat,C: int,D2: int,A2: set_int] :
      ( vimage_rat_int(aa_int_fun_rat_int(aa_int1442183312183061524at_int(aTP_Lamm_dv(B2),C),D2),A2) = $ite(
          aa_set_int_o(member_int(C),A2),
          $ite(aa_set_int_o(member_int(D2),A2),top_top_set_rat,B2),
          $ite(aa_set_int_o(member_int(D2),A2),aa_set_rat_set_rat(uminus2201863774496077783et_rat,B2),bot_bot_set_rat) ) ) ).

% vimage_if
tff(fact_1526_vimage__if,axiom,
    ! [B2: set_nat,C: nat,D2: nat,A2: set_nat] :
      ( aa_set_nat_set_nat(vimage_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_dw(B2),C),D2)),A2) = $ite(
          aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2),
          $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,D2),A2),top_top_set_nat,B2),
          $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,D2),A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B2),bot_bot_set_nat) ) ) ).

% vimage_if
tff(fact_1527_vimage__if,axiom,
    ! [B2: set_nat,C: int,D2: int,A2: set_int] :
      ( vimage_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_dx(B2),C),D2),A2) = $ite(
          aa_set_int_o(member_int(C),A2),
          $ite(aa_set_int_o(member_int(D2),A2),top_top_set_nat,B2),
          $ite(aa_set_int_o(member_int(D2),A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B2),bot_bot_set_nat) ) ) ).

% vimage_if
tff(fact_1528_vimage__if,axiom,
    ! [B2: set_int,C: nat,D2: nat,A2: set_nat] :
      ( vimage_int_nat(aa_nat_fun_int_nat(aa_nat5970070103963837268nt_nat(aTP_Lamm_dy(B2),C),D2),A2) = $ite(
          aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2),
          $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,D2),A2),top_top_set_int,B2),
          $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,D2),A2),aa_set_int_set_int(uminus1532241313380277803et_int,B2),bot_bot_set_int) ) ) ).

% vimage_if
tff(fact_1529_vimage__if,axiom,
    ! [B2: set_int,C: int,D2: int,A2: set_int] :
      ( vimage_int_int(aa_int_fun_int_int(aa_int8029789077385085544nt_int(aTP_Lamm_dz(B2),C),D2),A2) = $ite(
          aa_set_int_o(member_int(C),A2),
          $ite(aa_set_int_o(member_int(D2),A2),top_top_set_int,B2),
          $ite(aa_set_int_o(member_int(D2),A2),aa_set_int_set_int(uminus1532241313380277803et_int,B2),bot_bot_set_int) ) ) ).

% vimage_if
tff(fact_1530_vimage__if,axiom,
    ! [B2: set_set_nat,C: nat,D2: nat,A2: set_nat] :
      ( vimage_set_nat_nat(aa_nat6910189468943142497at_nat(aa_nat8373145576387461038at_nat(aTP_Lamm_ea(B2),C),D2),A2) = $ite(
          aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2),
          $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,D2),A2),top_top_set_set_nat,B2),
          $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,D2),A2),uminus613421341184616069et_nat(B2),bot_bot_set_set_nat) ) ) ).

% vimage_if
tff(fact_1531_vimage__if,axiom,
    ! [B2: set_set_nat,C: int,D2: int,A2: set_int] :
      ( vimage_set_nat_int(aa_int2448287375063835929at_int(aa_int7007698265767296706at_int(aTP_Lamm_eb(B2),C),D2),A2) = $ite(
          aa_set_int_o(member_int(C),A2),
          $ite(aa_set_int_o(member_int(D2),A2),top_top_set_set_nat,B2),
          $ite(aa_set_int_o(member_int(D2),A2),uminus613421341184616069et_nat(B2),bot_bot_set_set_nat) ) ) ).

% vimage_if
tff(fact_1532_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
    type_d5972188401318659438st_nat(bNF_id_bnf_list_nat,bNF_id_bnf_list_nat,top_top_set_list_nat) ).

% BNF_Composition.type_definition_id_bnf_UNIV
tff(fact_1533_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
    aa_set_o_o(type_definition_o_o(bNF_id_bnf_o,bNF_id_bnf_o),top_top_set_o) ).

% BNF_Composition.type_definition_id_bnf_UNIV
tff(fact_1534_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
    aa_set_rat_o(type_d5298809244756387038at_rat(bNF_id_bnf_rat,bNF_id_bnf_rat),top_top_set_rat) ).

% BNF_Composition.type_definition_id_bnf_UNIV
tff(fact_1535_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
    aa_set_nat_o(type_d6250493948777748686at_nat(bNF_id_bnf_nat,bNF_id_bnf_nat),top_top_set_nat) ).

% BNF_Composition.type_definition_id_bnf_UNIV
tff(fact_1536_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
    aa_set_int_o(type_d7247357190169752966nt_int(bNF_id_bnf_int,bNF_id_bnf_int),top_top_set_int) ).

% BNF_Composition.type_definition_id_bnf_UNIV
tff(fact_1537_dbl__dec__simps_I3_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,one_one_Code_integer) = one_one_Code_integer ).

% dbl_dec_simps(3)
tff(fact_1538_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_1539_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_1540_wand__raw_Oelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
     => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As4) )
           => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As4))
              & ! [H4: heap_e7401611519738050253t_unit,As5: set_nat] :
                  ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As4),As5) = bot_bot_set_nat )
                    & relH(As4,H,H4)
                    & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As4))
                    & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H4,As5)) )
                 => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H4,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As4),As5))) ) ) ) ) ).

% wand_raw.elims(3)
tff(fact_1541_wand__raw_Oelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
     => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As4) )
           => ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As4))
                & ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
                    ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As4),As6) = bot_bot_set_nat )
                      & relH(As4,H,H5)
                      & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As4))
                      & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H5,As6)) )
                   => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As4),As6))) ) ) ) ) ).

% wand_raw.elims(2)
tff(fact_1542_vimage__Collect__eq,axiom,
    ! [F: fun_nat_nat,P: fun_nat_o] : ( aa_set_nat_set_nat(vimage_nat_nat(F),collect_nat(P)) = collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ec(F),P)) ) ).

% vimage_Collect_eq
tff(fact_1543_vimage__Collect__eq,axiom,
    ! [F: fun_int_nat,P: fun_nat_o] : ( vimage_int_nat(F,collect_nat(P)) = collect_int(aa_fun4116151057475255200_int_o(aTP_Lamm_ed(F),P)) ) ).

% vimage_Collect_eq
tff(fact_1544_vimage__Collect__eq,axiom,
    ! [F: fun_nat_int,P: fun_int_o] : ( vimage_nat_int(F,collect_int(P)) = collect_nat(aa_fun8064560132910005592_nat_o(aTP_Lamm_ee(F),P)) ) ).

% vimage_Collect_eq
tff(fact_1545_vimage__Collect__eq,axiom,
    ! [F: fun_int_int,P: fun_int_o] : ( vimage_int_int(F,collect_int(P)) = collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ef(F),P)) ) ).

% vimage_Collect_eq
tff(fact_1546_vimage__Collect__eq,axiom,
    ! [F: fun_nat_set_nat,P: fun_set_nat_o] : ( vimage_nat_set_nat(F,collect_set_nat(P)) = collect_nat(aa_fun7902669250853151666_nat_o(aTP_Lamm_eg(F),P)) ) ).

% vimage_Collect_eq
tff(fact_1547_vimage__Collect__eq,axiom,
    ! [F: fun_int_set_nat,P: fun_set_nat_o] : ( vimage_int_set_nat(F,collect_set_nat(P)) = collect_int(aa_fun6081626905420463702_int_o(aTP_Lamm_eh(F),P)) ) ).

% vimage_Collect_eq
tff(fact_1548_vimage__Collect__eq,axiom,
    ! [F: fun_set_nat_nat,P: fun_nat_o] : ( vimage_set_nat_nat(F,collect_nat(P)) = collect_set_nat(aa_fun5113018963486751686_nat_o(aTP_Lamm_ei(F),P)) ) ).

% vimage_Collect_eq
tff(fact_1549_vimage__Collect__eq,axiom,
    ! [F: fun_set_nat_int,P: fun_int_o] : ( vimage_set_nat_int(F,collect_int(P)) = collect_set_nat(aa_fun6321136096415131682_nat_o(aTP_Lamm_ej(F),P)) ) ).

% vimage_Collect_eq
tff(fact_1550_vimage__Collect__eq,axiom,
    ! [F: fun_set_nat_set_nat,P: fun_set_nat_o] : ( vimage4765135879290611145et_nat(F,collect_set_nat(P)) = collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ek(F),P)) ) ).

% vimage_Collect_eq
tff(fact_1551_vimage__Collect__eq,axiom,
    ! [F: fun_na5449828587677346466nt_int,P: fun_se7249947736802449410_int_o] : ( vimage2935056327143036492nt_int(F,collec5210948495886036740nt_int(P)) = collect_nat(aa_fun7485545044466630625_nat_o(aTP_Lamm_el(F),P)) ) ).

% vimage_Collect_eq
tff(fact_1552_vimage__empty,axiom,
    ! [F: fun_o_o] : ( vimage_o_o(F,bot_bot_set_o) = bot_bot_set_o ) ).

% vimage_empty
tff(fact_1553_vimage__empty,axiom,
    ! [F: fun_nat_o] : ( vimage_nat_o(F,bot_bot_set_o) = bot_bot_set_nat ) ).

% vimage_empty
tff(fact_1554_vimage__empty,axiom,
    ! [F: fun_int_o] : ( vimage_int_o(F,bot_bot_set_o) = bot_bot_set_int ) ).

% vimage_empty
tff(fact_1555_vimage__empty,axiom,
    ! [F: fun_o_nat] : ( vimage_o_nat(F,bot_bot_set_nat) = bot_bot_set_o ) ).

% vimage_empty
tff(fact_1556_vimage__empty,axiom,
    ! [F: fun_nat_nat] : ( aa_set_nat_set_nat(vimage_nat_nat(F),bot_bot_set_nat) = bot_bot_set_nat ) ).

% vimage_empty
tff(fact_1557_vimage__empty,axiom,
    ! [F: fun_int_nat] : ( vimage_int_nat(F,bot_bot_set_nat) = bot_bot_set_int ) ).

% vimage_empty
tff(fact_1558_vimage__empty,axiom,
    ! [F: fun_o_int] : ( vimage_o_int(F,bot_bot_set_int) = bot_bot_set_o ) ).

% vimage_empty
tff(fact_1559_vimage__empty,axiom,
    ! [F: fun_nat_int] : ( vimage_nat_int(F,bot_bot_set_int) = bot_bot_set_nat ) ).

% vimage_empty
tff(fact_1560_vimage__empty,axiom,
    ! [F: fun_int_int] : ( vimage_int_int(F,bot_bot_set_int) = bot_bot_set_int ) ).

% vimage_empty
tff(fact_1561_vimage__empty,axiom,
    ! [F: fun_o_67843798866261420at_nat] : ( vimage2606541041029201316at_nat(F,bot_bo2099793752762293965at_nat) = bot_bot_set_o ) ).

% vimage_empty
tff(fact_1562_vimage__UNIV,axiom,
    ! [F: fun_o_o] : ( vimage_o_o(F,top_top_set_o) = top_top_set_o ) ).

% vimage_UNIV
tff(fact_1563_vimage__UNIV,axiom,
    ! [F: fun_rat_o] : ( vimage_rat_o(F,top_top_set_o) = top_top_set_rat ) ).

% vimage_UNIV
tff(fact_1564_vimage__UNIV,axiom,
    ! [F: fun_nat_o] : ( vimage_nat_o(F,top_top_set_o) = top_top_set_nat ) ).

% vimage_UNIV
tff(fact_1565_vimage__UNIV,axiom,
    ! [F: fun_int_o] : ( vimage_int_o(F,top_top_set_o) = top_top_set_int ) ).

% vimage_UNIV
tff(fact_1566_vimage__UNIV,axiom,
    ! [F: fun_o_rat] : ( vimage_o_rat(F,top_top_set_rat) = top_top_set_o ) ).

% vimage_UNIV
tff(fact_1567_vimage__UNIV,axiom,
    ! [F: fun_rat_rat] : ( vimage_rat_rat(F,top_top_set_rat) = top_top_set_rat ) ).

% vimage_UNIV
tff(fact_1568_vimage__UNIV,axiom,
    ! [F: fun_nat_rat] : ( vimage_nat_rat(F,top_top_set_rat) = top_top_set_nat ) ).

% vimage_UNIV
tff(fact_1569_vimage__UNIV,axiom,
    ! [F: fun_int_rat] : ( vimage_int_rat(F,top_top_set_rat) = top_top_set_int ) ).

% vimage_UNIV
tff(fact_1570_vimage__UNIV,axiom,
    ! [F: fun_o_nat] : ( vimage_o_nat(F,top_top_set_nat) = top_top_set_o ) ).

% vimage_UNIV
tff(fact_1571_vimage__UNIV,axiom,
    ! [F: fun_rat_nat] : ( vimage_rat_nat(F,top_top_set_nat) = top_top_set_rat ) ).

% vimage_UNIV
tff(fact_1572_vimage__Int,axiom,
    ! [F: fun_Pr2118491707810114563at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( vimage2449269961533847803at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,vimage2449269961533847803at_nat(F,A2)),vimage2449269961533847803at_nat(F,B2)) ) ).

% vimage_Int
tff(fact_1573_vimage__Int,axiom,
    ! [F: fun_na1476630412504255114at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( vimage8013328719654469172at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage8013328719654469172at_nat(F,A2)),vimage8013328719654469172at_nat(F,B2)) ) ).

% vimage_Int
tff(fact_1574_vimage__Int,axiom,
    ! [F: fun_Pr2602117396471556196at_nat,A2: set_nat,B2: set_nat] : ( vimage4653281326611754070at_nat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,vimage4653281326611754070at_nat(F,A2)),vimage4653281326611754070at_nat(F,B2)) ) ).

% vimage_Int
tff(fact_1575_vimage__Int,axiom,
    ! [F: fun_nat_nat,A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(vimage_nat_nat(F),A2)),aa_set_nat_set_nat(vimage_nat_nat(F),B2)) ) ).

% vimage_Int
tff(fact_1576_vimage__Un,axiom,
    ! [F: fun_Pr4459400599278340739at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( vimage3529983231074640251at_nat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,vimage3529983231074640251at_nat(F,A2)),vimage3529983231074640251at_nat(F,B2)) ) ).

% vimage_Un
tff(fact_1577_vimage__Un,axiom,
    ! [F: fun_Pr5779791428465206787at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( vimage2716263466903273467at_nat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,vimage2716263466903273467at_nat(F,A2)),vimage2716263466903273467at_nat(F,B2)) ) ).

% vimage_Un
tff(fact_1578_vimage__Un,axiom,
    ! [F: fun_na8407688532849143408at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( vimage7262694317370708122at_nat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vimage7262694317370708122at_nat(F,A2)),vimage7262694317370708122at_nat(F,B2)) ) ).

% vimage_Un
tff(fact_1579_vimage__Un,axiom,
    ! [F: fun_Pr6766502751620754691at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( vimage5412736291816116987at_nat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,vimage5412736291816116987at_nat(F,A2)),vimage5412736291816116987at_nat(F,B2)) ) ).

% vimage_Un
tff(fact_1580_vimage__Un,axiom,
    ! [F: fun_Pr9034069120582568067at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( vimage6896080417876799867at_nat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,vimage6896080417876799867at_nat(F,A2)),vimage6896080417876799867at_nat(F,B2)) ) ).

% vimage_Un
tff(fact_1581_vimage__Un,axiom,
    ! [F: fun_na367238361360896496at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( vimage6435164912253009178at_nat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vimage6435164912253009178at_nat(F,A2)),vimage6435164912253009178at_nat(F,B2)) ) ).

% vimage_Un
tff(fact_1582_vimage__Un,axiom,
    ! [F: fun_Pr7574179848310716106at_nat,A2: set_nat,B2: set_nat] : ( vimage7952471809483217980at_nat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,vimage7952471809483217980at_nat(F,A2)),vimage7952471809483217980at_nat(F,B2)) ) ).

% vimage_Un
tff(fact_1583_vimage__Un,axiom,
    ! [F: fun_Pr1253225528153422666at_nat,A2: set_nat,B2: set_nat] : ( vimage7134676753939176892at_nat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,vimage7134676753939176892at_nat(F,A2)),vimage7134676753939176892at_nat(F,B2)) ) ).

% vimage_Un
tff(fact_1584_vimage__Un,axiom,
    ! [F: fun_nat_nat,A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(vimage_nat_nat(F),A2)),aa_set_nat_set_nat(vimage_nat_nat(F),B2)) ) ).

% vimage_Un
tff(fact_1585_bool__assn__proper_I1_J,axiom,
    aa_fun1675921005893230629at_o_o(proper,in_range) ).

% bool_assn_proper(1)
tff(fact_1586_relH__dist__union,axiom,
    ! [As: set_nat,As3: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( relH(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As3),H2,H3)
    <=> ( relH(As,H2,H3)
        & relH(As3,H2,H3) ) ) ).

% relH_dist_union
tff(fact_1587_bool__assn__proper_I5_J,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
     => aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aTP_Lamm_em,P)) ) ).

% bool_assn_proper(5)
tff(fact_1588_in__range__empty,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ).

% in_range_empty
tff(fact_1589_vimage__const,axiom,
    ! [C: nat,A2: set_nat] :
      ( vimage_o_nat(aTP_Lamm_en(C),A2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2),top_top_set_o,bot_bot_set_o) ) ).

% vimage_const
tff(fact_1590_vimage__const,axiom,
    ! [C: int,A2: set_int] :
      ( vimage_o_int(aTP_Lamm_eo(C),A2) = $ite(aa_set_int_o(member_int(C),A2),top_top_set_o,bot_bot_set_o) ) ).

% vimage_const
tff(fact_1591_vimage__const,axiom,
    ! [C: nat,A2: set_nat] :
      ( vimage_rat_nat(aTP_Lamm_ep(C),A2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2),top_top_set_rat,bot_bot_set_rat) ) ).

% vimage_const
tff(fact_1592_vimage__const,axiom,
    ! [C: int,A2: set_int] :
      ( vimage_rat_int(aTP_Lamm_eq(C),A2) = $ite(aa_set_int_o(member_int(C),A2),top_top_set_rat,bot_bot_set_rat) ) ).

% vimage_const
tff(fact_1593_vimage__const,axiom,
    ! [C: nat,A2: set_nat] :
      ( aa_set_nat_set_nat(vimage_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_er,C)),A2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2),top_top_set_nat,bot_bot_set_nat) ) ).

% vimage_const
tff(fact_1594_vimage__const,axiom,
    ! [C: int,A2: set_int] :
      ( vimage_nat_int(aa_int_fun_nat_int(aTP_Lamm_es,C),A2) = $ite(aa_set_int_o(member_int(C),A2),top_top_set_nat,bot_bot_set_nat) ) ).

% vimage_const
tff(fact_1595_vimage__const,axiom,
    ! [C: nat,A2: set_nat] :
      ( vimage_int_nat(aa_nat_fun_int_nat(aTP_Lamm_et,C),A2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2),top_top_set_int,bot_bot_set_int) ) ).

% vimage_const
tff(fact_1596_vimage__const,axiom,
    ! [C: int,A2: set_int] :
      ( vimage_int_int(aa_int_fun_int_int(aTP_Lamm_eu,C),A2) = $ite(aa_set_int_o(member_int(C),A2),top_top_set_int,bot_bot_set_int) ) ).

% vimage_const
tff(fact_1597_vimage__const,axiom,
    ! [C: nat,A2: set_nat] :
      ( vimage_list_nat_nat(aTP_Lamm_ev(C),A2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,C),A2),top_top_set_list_nat,bot_bot_set_list_nat) ) ).

% vimage_const
tff(fact_1598_vimage__const,axiom,
    ! [C: int,A2: set_int] :
      ( vimage_list_nat_int(aTP_Lamm_ew(C),A2) = $ite(aa_set_int_o(member_int(C),A2),top_top_set_list_nat,bot_bot_set_list_nat) ) ).

% vimage_const
tff(fact_1599_in__range__dist__union,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As: set_nat,As3: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As3)))
    <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As))
        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As3)) ) ) ).

% in_range_dist_union
tff(fact_1600_relH__sym,axiom,
    ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( relH(As,H2,H3)
     => relH(As,H3,H2) ) ).

% relH_sym
tff(fact_1601_relH__trans,axiom,
    ! [As: set_nat,H12: heap_e7401611519738050253t_unit,H23: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
      ( relH(As,H12,H23)
     => ( relH(As,H23,H32)
       => relH(As,H12,H32) ) ) ).

% relH_trans
tff(fact_1602_vimage__def,axiom,
    ! [F: fun_nat_nat,B2: set_nat] : ( aa_set_nat_set_nat(vimage_nat_nat(F),B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ex(F),B2)) ) ).

% vimage_def
tff(fact_1603_vimage__def,axiom,
    ! [F: fun_nat_int,B2: set_int] : ( vimage_nat_int(F,B2) = collect_nat(aa_set_int_fun_nat_o(aTP_Lamm_ey(F),B2)) ) ).

% vimage_def
tff(fact_1604_vimage__def,axiom,
    ! [F: fun_int_nat,B2: set_nat] : ( vimage_int_nat(F,B2) = collect_int(aa_set_nat_fun_int_o(aTP_Lamm_ez(F),B2)) ) ).

% vimage_def
tff(fact_1605_vimage__def,axiom,
    ! [F: fun_int_int,B2: set_int] : ( vimage_int_int(F,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_fa(F),B2)) ) ).

% vimage_def
tff(fact_1606_vimage__def,axiom,
    ! [F: fun_set_nat_nat,B2: set_nat] : ( vimage_set_nat_nat(F,B2) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_fb(F),B2)) ) ).

% vimage_def
tff(fact_1607_vimage__def,axiom,
    ! [F: fun_set_nat_int,B2: set_int] : ( vimage_set_nat_int(F,B2) = collect_set_nat(aa_set7670721151442836043_nat_o(aTP_Lamm_fc(F),B2)) ) ).

% vimage_def
tff(fact_1608_vimage__def,axiom,
    ! [F: fun_nat_set_nat,B2: set_set_nat] : ( vimage_nat_set_nat(F,B2) = collect_nat(aa_set1169776424217490599_nat_o(aTP_Lamm_fd(F),B2)) ) ).

% vimage_def
tff(fact_1609_vimage__def,axiom,
    ! [F: fun_int_set_nat,B2: set_set_nat] : ( vimage_int_set_nat(F,B2) = collect_int(aa_set8572106115639578443_int_o(aTP_Lamm_fe(F),B2)) ) ).

% vimage_def
tff(fact_1610_vimage__def,axiom,
    ! [F: fun_set_nat_set_nat,B2: set_set_nat] : ( vimage4765135879290611145et_nat(F,B2) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_ff(F),B2)) ) ).

% vimage_def
tff(fact_1611_vimage__def,axiom,
    ! [F: fun_se4101287982975402108nt_nat,B2: set_nat] : ( vimage928727498438615150nt_nat(F,B2) = collec5210948495886036740nt_int(aa_set5909024289875323334_int_o(aTP_Lamm_fg(F),B2)) ) ).

% vimage_def
tff(fact_1612_relH__in__rangeI_I2_J,axiom,
    ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( relH(As,H2,H3)
     => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As)) ) ).

% relH_in_rangeI(2)
tff(fact_1613_relH__in__rangeI_I1_J,axiom,
    ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( relH(As,H2,H3)
     => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As)) ) ).

% relH_in_rangeI(1)
tff(fact_1614_relH__refl,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As))
     => relH(As,H2,H2) ) ).

% relH_refl
tff(fact_1615_proper__iff,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
     => ( relH(As,H2,H3)
       => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As))
         => ( aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H2,As))
          <=> aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H3,As)) ) ) ) ) ).

% proper_iff
tff(fact_1616_proper__def,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
    <=> ! [H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,As7: set_nat] :
          ( ( aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H6,As7))
           => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H6,As7)) )
          & ( ( aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H6,As7))
              & relH(As7,H6,H7)
              & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As7)) )
           => aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H7,As7)) ) ) ) ).

% proper_def
tff(fact_1617_properD2,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,H2: heap_e7401611519738050253t_unit,As: set_nat,H3: heap_e7401611519738050253t_unit] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
     => ( aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H2,As))
       => ( relH(As,H2,H3)
         => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As))
           => aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H3,As)) ) ) ) ) ).

% properD2
tff(fact_1618_properI,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o] :
      ( ! [As4: set_nat,H: heap_e7401611519738050253t_unit] :
          ( aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H,As4))
         => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As4)) )
     => ( ! [As4: set_nat,H: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
            ( aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H,As4))
           => ( relH(As4,H,H4)
             => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As4))
               => aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H4,As4)) ) ) )
       => aa_fun1675921005893230629at_o_o(proper,P) ) ) ).

% properI
tff(fact_1619_models__in__range,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H2)
     => aa_Pro653082453696054616_nat_o(in_range,H2) ) ).

% models_in_range
tff(fact_1620_top__assn__def,axiom,
    top_top_assn = aa_fun5526454687433275355o_assn(abs_assn,in_range) ).

% top_assn_def
tff(fact_1621_properD1,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
     => ( aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H2,As))
       => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As)) ) ) ).

% properD1
tff(fact_1622_mod__relH,axiom,
    ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,P: assn] :
      ( relH(As,H2,H3)
     => ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),produc7507926704131184380et_nat(H2,As))
      <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),produc7507926704131184380et_nat(H3,As)) ) ) ).

% mod_relH
tff(fact_1623_uminus__assn__def,axiom,
    ! [P: assn] : ( uminus_uminus_assn(P) = aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(aTP_Lamm_fh,P)) ) ).

% uminus_assn_def
tff(fact_1624_wand__raw_Osimps,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(P),Q),produc7507926704131184380et_nat(H2,As))
    <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As))
        & ! [H7: heap_e7401611519738050253t_unit,As8: set_nat] :
            ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As8) = bot_bot_set_nat )
              & relH(As,H2,H7)
              & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As))
              & aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H7,As8)) )
           => aa_Pro653082453696054616_nat_o(Q,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As8))) ) ) ) ).

% wand_raw.simps
tff(fact_1625_wand__raw_Oelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As4) )
           => ( (Y)
            <=> ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As4))
                  & ! [H7: heap_e7401611519738050253t_unit,As8: set_nat] :
                      ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As4),As8) = bot_bot_set_nat )
                        & relH(As4,H,H7)
                        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As4))
                        & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H7,As8)) )
                     => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As4),As8))) ) ) ) ) ) ).

% wand_raw.elims(1)
tff(fact_1626_wand__raw_Opelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
      <=> (Y) )
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As4) )
             => ( ( (Y)
                <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As4))
                    & ! [H7: heap_e7401611519738050253t_unit,As8: set_nat] :
                        ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As4),As8) = bot_bot_set_nat )
                          & relH(As4,H,H7)
                          & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As4))
                          & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H7,As8)) )
                       => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As4),As8))) ) ) )
               => ~ aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),produc7507926704131184380et_nat(H,As4)))) ) ) ) ) ).

% wand_raw.pelims(1)
tff(fact_1627_wand__raw_Opelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As4) )
             => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),produc7507926704131184380et_nat(H,As4))))
               => ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As4))
                    & ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
                        ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As4),As6) = bot_bot_set_nat )
                          & relH(As4,H,H5)
                          & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As4))
                          & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H5,As6)) )
                       => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As4),As6))) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
tff(fact_1628_wand__raw_Opelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As4) )
             => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),produc7507926704131184380et_nat(H,As4))))
               => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As4))
                  & ! [H4: heap_e7401611519738050253t_unit,As5: set_nat] :
                      ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As4),As5) = bot_bot_set_nat )
                        & relH(As4,H,H4)
                        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As4))
                        & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H4,As5)) )
                     => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H4,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As4),As5))) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
tff(fact_1629_bijective__Empty,axiom,
    aa_set3702741264012733747_nat_o(bijective_nat_nat,bot_bo2099793752762293965at_nat) ).

% bijective_Empty
tff(fact_1630_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : ( aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K))) ) ).

% dbl_inc_simps(1)
tff(fact_1631_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : ( aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) ) ).

% dbl_inc_simps(1)
tff(fact_1632_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : ( aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu3179335615603231917ec_rat,numeral_numeral_rat(K))) ) ).

% dbl_inc_simps(1)
tff(fact_1633_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : ( aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K))) ) ).

% dbl_dec_simps(1)
tff(fact_1634_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : ( aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) ) ).

% dbl_dec_simps(1)
tff(fact_1635_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : ( aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu5219082963157363817nc_rat,numeral_numeral_rat(K))) ) ).

% dbl_dec_simps(1)
tff(fact_1636_dbl__dec__simps_I2_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_dec_simps(2)
tff(fact_1637_dbl__dec__simps_I2_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_1638_dbl__dec__simps_I2_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,zero_zero_rat) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_1639_pairself_Opelims,axiom,
    ! [X: fun_in5730379987311621695e_term,Xa: product_prod_int_int,Y: produc6576344331059438605e_term] :
      ( ( pairse3534876335208825186e_term(X,Xa) = Y )
     => ( aa_Pro5451197003417746155_int_o(accp_P3253464921323968230nt_int(pairse6848479906795794847e_term),aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X),Xa))
       => ~ ! [A4: int,B4: int] :
              ( ( Xa = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B4) )
             => ( ( Y = produc5950683997804057413e_term(aa_int3262262799336221824e_term(X,A4),aa_int3262262799336221824e_term(X,B4)) )
               => ~ aa_Pro5451197003417746155_int_o(accp_P3253464921323968230nt_int(pairse6848479906795794847e_term),aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B4))) ) ) ) ) ).

% pairself.pelims
tff(fact_1640_rel__restrict__compl,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,rel_restrict_nat(R,A2)),rel_restrict_nat(R,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2))) = bot_bo2099793752762293965at_nat ) ).

% rel_restrict_compl
tff(fact_1641_Id__on__empty,axiom,
    id_on_2554058798563519774at_nat(bot_bo2099793752762293965at_nat) = bot_bo5327735625951526323at_nat ).

% Id_on_empty
tff(fact_1642_Id__on__empty,axiom,
    id_on_o(bot_bot_set_o) = bot_bo7073875226086086771od_o_o ).

% Id_on_empty
tff(fact_1643_Id__on__empty,axiom,
    id_on_nat(bot_bot_set_nat) = bot_bo2099793752762293965at_nat ).

% Id_on_empty
tff(fact_1644_Id__on__empty,axiom,
    id_on_int(bot_bot_set_int) = bot_bo1796632182523588997nt_int ).

% Id_on_empty
tff(fact_1645_mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1646_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1647_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1648_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1649_mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1650_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1651_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1652_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1653_mult__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = zero_z3403309356797280102nteger )
    <=> ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% mult_eq_0_iff
tff(fact_1654_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_1655_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_1656_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_1657_mult__zero__right,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ) ).

% mult_zero_right
tff(fact_1658_mult__zero__right,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),zero_zero_rat) = zero_zero_rat ) ).

% mult_zero_right
tff(fact_1659_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_1660_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_1661_mult__zero__left,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ) ).

% mult_zero_left
tff(fact_1662_mult__zero__left,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A) = zero_zero_rat ) ).

% mult_zero_left
tff(fact_1663_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_1664_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_1665_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(times_times_num(V,W))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_1666_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: 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(times_times_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,times_times_num(V,W))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_1667_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,times_times_num(V,W))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_1668_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(times_times_num(V,W))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_1669_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: code_natural] : ( times_2397367101498566445atural(numera5444537566228673987atural(V),times_2397367101498566445atural(numera5444537566228673987atural(W),Z)) = times_2397367101498566445atural(numera5444537566228673987atural(times_times_num(V,W)),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_1670_numeral__times__numeral,axiom,
    ! [M: num,N: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(M)),numeral_numeral_nat(N)) = numeral_numeral_nat(times_times_num(M,N)) ) ).

% numeral_times_numeral
tff(fact_1671_numeral__times__numeral,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,times_times_num(M,N)) ) ).

% numeral_times_numeral
tff(fact_1672_numeral__times__numeral,axiom,
    ! [M: num,N: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_num_Code_integer(numera6620942414471956472nteger,times_times_num(M,N)) ) ).

% numeral_times_numeral
tff(fact_1673_numeral__times__numeral,axiom,
    ! [M: num,N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(M)),numeral_numeral_rat(N)) = numeral_numeral_rat(times_times_num(M,N)) ) ).

% numeral_times_numeral
tff(fact_1674_numeral__times__numeral,axiom,
    ! [M: num,N: num] : ( times_2397367101498566445atural(numera5444537566228673987atural(M),numera5444537566228673987atural(N)) = numera5444537566228673987atural(times_times_num(M,N)) ) ).

% numeral_times_numeral
tff(fact_1675_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_1676_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_1677_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_1678_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_1679_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_1680_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_1681_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_1682_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_1683_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_1684_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_1685_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_1686_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_1687_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_1688_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_1689_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_1690_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_1691_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_1692_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_1693_rel__restrict__empty,axiom,
    ! [R: set_Pr8693737435421807431at_nat] : ( rel_re36662636952045539at_nat(R,bot_bo2099793752762293965at_nat) = R ) ).

% rel_restrict_empty
tff(fact_1694_rel__restrict__empty,axiom,
    ! [R: set_Product_prod_o_o] : ( rel_restrict_o(R,bot_bot_set_o) = R ) ).

% rel_restrict_empty
tff(fact_1695_rel__restrict__empty,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : ( rel_restrict_nat(R,bot_bot_set_nat) = R ) ).

% rel_restrict_empty
tff(fact_1696_rel__restrict__empty,axiom,
    ! [R: set_Pr958786334691620121nt_int] : ( rel_restrict_int(R,bot_bot_set_int) = R ) ).

% rel_restrict_empty
tff(fact_1697_Id__onI,axiom,
    ! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A2)
     => member5325734588016868240nt_int(produc782388162306416855nt_int(A,A),id_on_6410342593070439734nt_int(A2)) ) ).

% Id_onI
tff(fact_1698_Id__onI,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => member8277197624267554838et_nat(produc4532415448927165861et_nat(A,A),id_on_set_nat(A2)) ) ).

% Id_onI
tff(fact_1699_Id__onI,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A2)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),A)),id_on_nat(A2)) ) ).

% Id_onI
tff(fact_1700_Id__onI,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),A)),id_on_int(A2)) ) ).

% Id_onI
tff(fact_1701_Id__onI,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2)
     => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A,A),id_on_7912660205148244492_nat_o(A2)) ) ).

% Id_onI
tff(fact_1702_mult__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = C )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
tff(fact_1703_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_1704_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_1705_mult__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
tff(fact_1706_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_1707_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_1708_mult__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = C )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
tff(fact_1709_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_1710_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_1711_mult__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
tff(fact_1712_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_1713_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_1714_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,times_times_num(M,N)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_1715_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,times_times_num(M,N)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_1716_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = numeral_numeral_rat(times_times_num(M,N)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_1717_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,times_times_num(M,N))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_1718_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,times_times_num(M,N))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_1719_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),numeral_numeral_rat(N)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(times_times_num(M,N))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_1720_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,times_times_num(M,N))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_1721_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,times_times_num(M,N))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_1722_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(M)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(times_times_num(M,N))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_1723_dbl__inc__simps_I2_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,zero_z3403309356797280102nteger) = one_one_Code_integer ).

% dbl_inc_simps(2)
tff(fact_1724_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_1725_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_1726_zero__reorient,axiom,
    ! [X: code_integer] :
      ( ( zero_z3403309356797280102nteger = X )
    <=> ( X = zero_z3403309356797280102nteger ) ) ).

% zero_reorient
tff(fact_1727_zero__reorient,axiom,
    ! [X: multis2468970476368604999at_nat] :
      ( ( zero_z1048942125864253310at_nat = X )
    <=> ( X = zero_z1048942125864253310at_nat ) ) ).

% zero_reorient
tff(fact_1728_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_1729_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_1730_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_1731_zero__neq__neg__numeral,axiom,
    ! [N: num] : ( zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ) ).

% zero_neq_neg_numeral
tff(fact_1732_zero__neq__neg__numeral,axiom,
    ! [N: num] : ( zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ) ).

% zero_neq_neg_numeral
tff(fact_1733_zero__neq__neg__numeral,axiom,
    ! [N: num] : ( zero_zero_rat != aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)) ) ).

% zero_neq_neg_numeral
tff(fact_1734_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : ( aa_num_int(numeral_numeral_int,M) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ) ).

% numeral_neq_neg_numeral
tff(fact_1735_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : ( aa_num_Code_integer(numera6620942414471956472nteger,M) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ) ).

% numeral_neq_neg_numeral
tff(fact_1736_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : ( numeral_numeral_rat(M) != aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)) ) ).

% numeral_neq_neg_numeral
tff(fact_1737_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) != aa_num_int(numeral_numeral_int,N) ) ).

% neg_numeral_neq_numeral
tff(fact_1738_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)) != aa_num_Code_integer(numera6620942414471956472nteger,N) ) ).

% neg_numeral_neq_numeral
tff(fact_1739_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : ( aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M)) != numeral_numeral_rat(N) ) ).

% neg_numeral_neq_numeral
tff(fact_1740_mult__right__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1741_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1742_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1743_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1744_mult__left__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1745_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1746_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1747_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1748_no__zero__divisors,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( B != zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) != zero_z3403309356797280102nteger ) ) ) ).

% no_zero_divisors
tff(fact_1749_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_1750_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_1751_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_1752_divisors__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = zero_z3403309356797280102nteger )
     => ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% divisors_zero
tff(fact_1753_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_1754_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_1755_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_1756_mult__not__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) != zero_z3403309356797280102nteger )
     => ( ( A != zero_z3403309356797280102nteger )
        & ( B != zero_z3403309356797280102nteger ) ) ) ).

% mult_not_zero
tff(fact_1757_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_1758_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_1759_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_1760_zero__neq__one,axiom,
    zero_z3403309356797280102nteger != one_one_Code_integer ).

% zero_neq_one
tff(fact_1761_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1762_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1763_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1764_rel__restrictI,axiom,
    ! [X: set_Pr958786334691620121nt_int,R: set_se6260736226359567993nt_int,Y: set_Pr958786334691620121nt_int,E: set_Pr8057934254988874055nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),R)
     => ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y),R)
       => ( member5325734588016868240nt_int(produc782388162306416855nt_int(X,Y),E)
         => member5325734588016868240nt_int(produc782388162306416855nt_int(X,Y),rel_re4349003428253852667nt_int(E,R)) ) ) ) ).

% rel_restrictI
tff(fact_1765_rel__restrictI,axiom,
    ! [X: set_nat,R: set_set_nat,Y: set_nat,E: set_Pr5488025237498180813et_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(X),R)
     => ( ~ aa_set_set_nat_o(member_set_nat(Y),R)
       => ( member8277197624267554838et_nat(produc4532415448927165861et_nat(X,Y),E)
         => member8277197624267554838et_nat(produc4532415448927165861et_nat(X,Y),rel_restrict_set_nat(E,R)) ) ) ) ).

% rel_restrictI
tff(fact_1766_rel__restrictI,axiom,
    ! [X: nat,R: set_nat,Y: nat,E: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),R)
     => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y),R)
       => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),E)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),rel_restrict_nat(E,R)) ) ) ) ).

% rel_restrictI
tff(fact_1767_rel__restrictI,axiom,
    ! [X: int,R: set_int,Y: int,E: set_Pr958786334691620121nt_int] :
      ( ~ aa_set_int_o(member_int(X),R)
     => ( ~ aa_set_int_o(member_int(Y),R)
       => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),E)
         => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),rel_restrict_int(E,R)) ) ) ) ).

% rel_restrictI
tff(fact_1768_rel__restrictI,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,R: set_fu3555266933746557007_nat_o,Y: fun_Pr6460944699146269935_nat_o,E: set_Pr2890355626087285447_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),R)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),R)
       => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),E)
         => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),rel_re4406848170174997969_nat_o(E,R)) ) ) ) ).

% rel_restrictI
tff(fact_1769_rel__restrict__notR_I1_J,axiom,
    ! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,A2: set_Pr8057934254988874055nt_int,R: set_se6260736226359567993nt_int] :
      ( member5325734588016868240nt_int(produc782388162306416855nt_int(X,Y),rel_re4349003428253852667nt_int(A2,R))
     => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),R) ) ).

% rel_restrict_notR(1)
tff(fact_1770_rel__restrict__notR_I1_J,axiom,
    ! [X: set_nat,Y: set_nat,A2: set_Pr5488025237498180813et_nat,R: set_set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(X,Y),rel_restrict_set_nat(A2,R))
     => ~ aa_set_set_nat_o(member_set_nat(X),R) ) ).

% rel_restrict_notR(1)
tff(fact_1771_rel__restrict__notR_I1_J,axiom,
    ! [X: nat,Y: nat,A2: set_Pr1261947904930325089at_nat,R: set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),rel_restrict_nat(A2,R))
     => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),R) ) ).

% rel_restrict_notR(1)
tff(fact_1772_rel__restrict__notR_I1_J,axiom,
    ! [X: int,Y: int,A2: set_Pr958786334691620121nt_int,R: set_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),rel_restrict_int(A2,R))
     => ~ aa_set_int_o(member_int(X),R) ) ).

% rel_restrict_notR(1)
tff(fact_1773_rel__restrict__notR_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o,A2: set_Pr2890355626087285447_nat_o,R: set_fu3555266933746557007_nat_o] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),rel_re4406848170174997969_nat_o(A2,R))
     => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),R) ) ).

% rel_restrict_notR(1)
tff(fact_1774_rel__restrict__notR_I2_J,axiom,
    ! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,A2: set_Pr8057934254988874055nt_int,R: set_se6260736226359567993nt_int] :
      ( member5325734588016868240nt_int(produc782388162306416855nt_int(X,Y),rel_re4349003428253852667nt_int(A2,R))
     => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y),R) ) ).

% rel_restrict_notR(2)
tff(fact_1775_rel__restrict__notR_I2_J,axiom,
    ! [X: set_nat,Y: set_nat,A2: set_Pr5488025237498180813et_nat,R: set_set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(X,Y),rel_restrict_set_nat(A2,R))
     => ~ aa_set_set_nat_o(member_set_nat(Y),R) ) ).

% rel_restrict_notR(2)
tff(fact_1776_rel__restrict__notR_I2_J,axiom,
    ! [X: nat,Y: nat,A2: set_Pr1261947904930325089at_nat,R: set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),rel_restrict_nat(A2,R))
     => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y),R) ) ).

% rel_restrict_notR(2)
tff(fact_1777_rel__restrict__notR_I2_J,axiom,
    ! [X: int,Y: int,A2: set_Pr958786334691620121nt_int,R: set_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),rel_restrict_int(A2,R))
     => ~ aa_set_int_o(member_int(Y),R) ) ).

% rel_restrict_notR(2)
tff(fact_1778_rel__restrict__notR_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o,A2: set_Pr2890355626087285447_nat_o,R: set_fu3555266933746557007_nat_o] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),rel_re4406848170174997969_nat_o(A2,R))
     => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),R) ) ).

% rel_restrict_notR(2)
tff(fact_1779_rel__restrict__union,axiom,
    ! [R: set_Pr4759807026041935047at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( rel_re2197483123211154633at_nat(R,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) = rel_re2197483123211154633at_nat(rel_re2197483123211154633at_nat(R,A2),B2) ) ).

% rel_restrict_union
tff(fact_1780_rel__restrict__union,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( rel_re2767227651283840073at_nat(R,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) = rel_re2767227651283840073at_nat(rel_re2767227651283840073at_nat(R,A2),B2) ) ).

% rel_restrict_union
tff(fact_1781_rel__restrict__union,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_nat] : ( rel_restrict_nat(R,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = rel_restrict_nat(rel_restrict_nat(R,A2),B2) ) ).

% rel_restrict_union
tff(fact_1782_Id__on__iff,axiom,
    ! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
      ( member5325734588016868240nt_int(produc782388162306416855nt_int(X,Y),id_on_6410342593070439734nt_int(A2))
    <=> ( ( X = Y )
        & aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A2) ) ) ).

% Id_on_iff
tff(fact_1783_Id__on__iff,axiom,
    ! [X: set_nat,Y: set_nat,A2: set_set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(X,Y),id_on_set_nat(A2))
    <=> ( ( X = Y )
        & aa_set_set_nat_o(member_set_nat(X),A2) ) ) ).

% Id_on_iff
tff(fact_1784_Id__on__iff,axiom,
    ! [X: nat,Y: nat,A2: set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),id_on_nat(A2))
    <=> ( ( X = Y )
        & aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2) ) ) ).

% Id_on_iff
tff(fact_1785_Id__on__iff,axiom,
    ! [X: int,Y: int,A2: set_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),id_on_int(A2))
    <=> ( ( X = Y )
        & aa_set_int_o(member_int(X),A2) ) ) ).

% Id_on_iff
tff(fact_1786_Id__on__iff,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),id_on_7912660205148244492_nat_o(A2))
    <=> ( ( X = Y )
        & aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2) ) ) ).

% Id_on_iff
tff(fact_1787_Id__on__eqI,axiom,
    ! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
      ( ( A = B )
     => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A2)
       => member5325734588016868240nt_int(produc782388162306416855nt_int(A,B),id_on_6410342593070439734nt_int(A2)) ) ) ).

% Id_on_eqI
tff(fact_1788_Id__on__eqI,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( ( A = B )
     => ( aa_set_set_nat_o(member_set_nat(A),A2)
       => member8277197624267554838et_nat(produc4532415448927165861et_nat(A,B),id_on_set_nat(A2)) ) ) ).

% Id_on_eqI
tff(fact_1789_Id__on__eqI,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( ( A = B )
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A2)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),id_on_nat(A2)) ) ) ).

% Id_on_eqI
tff(fact_1790_Id__on__eqI,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( ( A = B )
     => ( aa_set_int_o(member_int(A),A2)
       => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),id_on_int(A2)) ) ) ).

% Id_on_eqI
tff(fact_1791_Id__on__eqI,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( ( A = B )
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2)
       => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A,B),id_on_7912660205148244492_nat_o(A2)) ) ) ).

% Id_on_eqI
tff(fact_1792_Id__onE,axiom,
    ! [C: produc412284730452459111nt_int,A2: set_se6260736226359567993nt_int] :
      ( member5325734588016868240nt_int(C,id_on_6410342593070439734nt_int(A2))
     => ~ ! [X2: set_Pr958786334691620121nt_int] :
            ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),A2)
           => ( C != produc782388162306416855nt_int(X2,X2) ) ) ) ).

% Id_onE
tff(fact_1793_Id__onE,axiom,
    ! [C: produc7819656566062154093et_nat,A2: set_set_nat] :
      ( member8277197624267554838et_nat(C,id_on_set_nat(A2))
     => ~ ! [X2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X2),A2)
           => ( C != produc4532415448927165861et_nat(X2,X2) ) ) ) ).

% Id_onE
tff(fact_1794_Id__onE,axiom,
    ! [C: product_prod_nat_nat,A2: set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),id_on_nat(A2))
     => ~ ! [X2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
           => ( C != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),X2) ) ) ) ).

% Id_onE
tff(fact_1795_Id__onE,axiom,
    ! [C: product_prod_int_int,A2: set_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(C),id_on_int(A2))
     => ~ ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
           => ( C != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),X2) ) ) ) ).

% Id_onE
tff(fact_1796_Id__onE,axiom,
    ! [C: produc6671579610312156647_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( member3012145478770660624_nat_o(C,id_on_7912660205148244492_nat_o(A2))
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A2)
           => ( C != produc4714251849521319255_nat_o(X2,X2) ) ) ) ).

% Id_onE
tff(fact_1797_lambda__zero,axiom,
    aTP_Lamm_fi = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger) ).

% lambda_zero
tff(fact_1798_lambda__zero,axiom,
    aTP_Lamm_fj = aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat) ).

% lambda_zero
tff(fact_1799_lambda__zero,axiom,
    aTP_Lamm_fk = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_1800_lambda__zero,axiom,
    aTP_Lamm_fl = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_1801_one__neq__neg__numeral,axiom,
    ! [N: num] : ( one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ) ).

% one_neq_neg_numeral
tff(fact_1802_one__neq__neg__numeral,axiom,
    ! [N: num] : ( one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ) ).

% one_neq_neg_numeral
tff(fact_1803_one__neq__neg__numeral,axiom,
    ! [N: num] : ( one_one_rat != aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)) ) ).

% one_neq_neg_numeral
tff(fact_1804_numeral__neq__neg__one,axiom,
    ! [N: num] : ( aa_num_int(numeral_numeral_int,N) != aa_int_int(uminus_uminus_int,one_one_int) ) ).

% numeral_neq_neg_one
tff(fact_1805_numeral__neq__neg__one,axiom,
    ! [N: num] : ( aa_num_Code_integer(numera6620942414471956472nteger,N) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ).

% numeral_neq_neg_one
tff(fact_1806_numeral__neq__neg__one,axiom,
    ! [N: num] : ( numeral_numeral_rat(N) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ).

% numeral_neq_neg_one
tff(fact_1807_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_1808_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_1809_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_1810_bijective__def,axiom,
    ! [R: set_Pr3486312109254157132nteger] :
      ( biject6040901034283248210nteger(R)
    <=> ( ! [X4: fun_Pr5098669352780829690e_term,Y3: produc8923325533196201883nteger,Z3: produc8923325533196201883nteger] :
            ( ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X4),Y3),R)
              & member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X4),Z3),R) )
           => ( Y3 = Z3 ) )
        & ! [X4: fun_Pr5098669352780829690e_term,Y3: fun_Pr5098669352780829690e_term,Z3: produc8923325533196201883nteger] :
            ( ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X4),Z3),R)
              & member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Y3),Z3),R) )
           => ( X4 = Y3 ) ) ) ) ).

% bijective_def
tff(fact_1811_bijective__def,axiom,
    ! [R: set_Pr1312406793078187350et_nat] :
      ( biject6574446902887564708et_nat(R)
    <=> ( ! [X4: fun_Pr6460944699146269935_nat_o,Y3: produc3658429121746597890et_nat,Z3: produc3658429121746597890et_nat] :
            ( ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Y3),R)
              & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Z3),R) )
           => ( Y3 = Z3 ) )
        & ! [X4: fun_Pr6460944699146269935_nat_o,Y3: fun_Pr6460944699146269935_nat_o,Z3: produc3658429121746597890et_nat] :
            ( ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Z3),R)
              & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Y3),Z3),R) )
           => ( X4 = Y3 ) ) ) ) ).

% bijective_def
tff(fact_1812_bijective__def,axiom,
    ! [R: set_Pr4040394263813967220et_nat] :
      ( biject4066639966817638850et_nat(R)
    <=> ( ! [X4: fun_Pr6460944699146269935_nat_o,Y3: produc8799533983703829792et_nat,Z3: produc8799533983703829792et_nat] :
            ( ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Y3),R)
              & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Z3),R) )
           => ( Y3 = Z3 ) )
        & ! [X4: fun_Pr6460944699146269935_nat_o,Y3: fun_Pr6460944699146269935_nat_o,Z3: produc8799533983703829792et_nat] :
            ( ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Z3),R)
              & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Y3),Z3),R) )
           => ( X4 = Y3 ) ) ) ) ).

% bijective_def
tff(fact_1813_bijective__def,axiom,
    ! [R: set_Pr6700895561065273913nt_int] :
      ( biject1579208791189821119nt_int(R)
    <=> ( ! [X4: fun_Pr7763680713840462443e_term,Y3: product_prod_int_int,Z3: product_prod_int_int] :
            ( ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Y3),R)
              & member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Z3),R) )
           => ( Y3 = Z3 ) )
        & ! [X4: fun_Pr7763680713840462443e_term,Y3: fun_Pr7763680713840462443e_term,Z3: product_prod_int_int] :
            ( ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Z3),R)
              & member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Y3),Z3),R) )
           => ( X4 = Y3 ) ) ) ) ).

% bijective_def
tff(fact_1814_bijective__def,axiom,
    ! [R: set_Pr4166680992137330533nt_int] :
      ( biject9165800892491502059nt_int(R)
    <=> ( ! [X4: fun_in5730379987311621695e_term,Y3: product_prod_int_int,Z3: product_prod_int_int] :
            ( ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Y3),R)
              & member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Z3),R) )
           => ( Y3 = Z3 ) )
        & ! [X4: fun_in5730379987311621695e_term,Y3: fun_in5730379987311621695e_term,Z3: product_prod_int_int] :
            ( ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Z3),R)
              & member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Y3),Z3),R) )
           => ( X4 = Y3 ) ) ) ) ).

% bijective_def
tff(fact_1815_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,times_times_num(V,W)))),Y) ) ).

% semiring_norm(170)
tff(fact_1816_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,times_times_num(V,W)))),Y) ) ).

% semiring_norm(170)
tff(fact_1817_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(times_times_num(V,W)))),Y) ) ).

% semiring_norm(170)
tff(fact_1818_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: 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(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,times_times_num(V,W)))),Y) ) ).

% semiring_norm(171)
tff(fact_1819_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,times_times_num(V,W)))),Y) ) ).

% semiring_norm(171)
tff(fact_1820_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(times_times_num(V,W)))),Y) ) ).

% semiring_norm(171)
tff(fact_1821_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,times_times_num(V,W))),Y) ) ).

% semiring_norm(172)
tff(fact_1822_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,times_times_num(V,W))),Y) ) ).

% semiring_norm(172)
tff(fact_1823_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(times_times_num(V,W))),Y) ) ).

% semiring_norm(172)
tff(fact_1824_numeral__times__minus__swap,axiom,
    ! [W: num,X: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,X)) = aa_int_int(aa_int_fun_int_int(times_times_int,X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ) ).

% numeral_times_minus_swap
tff(fact_1825_numeral__times__minus__swap,axiom,
    ! [W: num,X: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))) ) ).

% numeral_times_minus_swap
tff(fact_1826_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),aa_rat_rat(uminus_uminus_rat,X)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) ) ).

% numeral_times_minus_swap
tff(fact_1827_dbl__simps_I1_J,axiom,
    ! [K: num] : ( aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K))) ) ).

% dbl_simps(1)
tff(fact_1828_dbl__simps_I1_J,axiom,
    ! [K: num] : ( aa_Cod1783974652533396728nteger(neg_nu8804712462038260780nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(neg_nu8804712462038260780nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) ) ).

% dbl_simps(1)
tff(fact_1829_dbl__simps_I1_J,axiom,
    ! [K: num] : ( aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_numeral_dbl_rat,numeral_numeral_rat(K))) ) ).

% dbl_simps(1)
tff(fact_1830_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_1831_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) = A )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_1832_eq__numeral__iff__iszero_I12_J,axiom,
    ! [Y: num] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Y)) )
    <=> aa_int_o(ring_1_iszero_int,aa_num_int(numeral_numeral_int,Y)) ) ).

% eq_numeral_iff_iszero(12)
tff(fact_1833_eq__numeral__iff__iszero_I12_J,axiom,
    ! [Y: num] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Y)) )
    <=> aa_Code_integer_o(ring_16219924574208605041nteger,aa_num_Code_integer(numera6620942414471956472nteger,Y)) ) ).

% eq_numeral_iff_iszero(12)
tff(fact_1834_eq__numeral__iff__iszero_I12_J,axiom,
    ! [Y: num] :
      ( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(Y)) )
    <=> aa_rat_o(ring_1_iszero_rat,numeral_numeral_rat(Y)) ) ).

% eq_numeral_iff_iszero(12)
tff(fact_1835_eq__numeral__iff__iszero_I11_J,axiom,
    ! [X: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X)) = zero_zero_int )
    <=> aa_int_o(ring_1_iszero_int,aa_num_int(numeral_numeral_int,X)) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_1836_eq__numeral__iff__iszero_I11_J,axiom,
    ! [X: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(ring_16219924574208605041nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_1837_eq__numeral__iff__iszero_I11_J,axiom,
    ! [X: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(X)) = zero_zero_rat )
    <=> aa_rat_o(ring_1_iszero_rat,numeral_numeral_rat(X)) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_1838_divides__aux__eq,axiom,
    ! [Q3: code_integer,R2: code_integer] :
      ( unique5706413561485394159nteger(produc1086072967326762835nteger(Q3,R2))
    <=> ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_1839_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_1840_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_1841_minus__assn__def,axiom,
    ! [A: assn,B: assn] : ( minus_minus_assn(A,B) = inf_inf_assn(A,uminus_uminus_assn(B)) ) ).

% minus_assn_def
tff(fact_1842_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),A) = zero_z3403309356797280102nteger ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1843_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: multis2468970476368604999at_nat] : ( minus_4286766774447292334at_nat(A,A) = zero_z1048942125864253310at_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1844_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),A) = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1845_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_1846_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_1847_diff__zero,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),zero_z3403309356797280102nteger) = A ) ).

% diff_zero
tff(fact_1848_diff__zero,axiom,
    ! [A: multis2468970476368604999at_nat] : ( minus_4286766774447292334at_nat(A,zero_z1048942125864253310at_nat) = A ) ).

% diff_zero
tff(fact_1849_diff__zero,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),zero_zero_rat) = A ) ).

% diff_zero
tff(fact_1850_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_1851_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_1852_zero__diff,axiom,
    ! [A: multis2468970476368604999at_nat] : ( minus_4286766774447292334at_nat(zero_z1048942125864253310at_nat,A) = zero_z1048942125864253310at_nat ) ).

% zero_diff
tff(fact_1853_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_1854_diff__0__right,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),zero_z3403309356797280102nteger) = A ) ).

% diff_0_right
tff(fact_1855_diff__0__right,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),zero_zero_rat) = A ) ).

% diff_0_right
tff(fact_1856_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_1857_diff__self,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),A) = zero_z3403309356797280102nteger ) ).

% diff_self
tff(fact_1858_diff__self,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),A) = zero_zero_rat ) ).

% diff_self
tff(fact_1859_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_1860_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_1861_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A) ) ).

% minus_diff_eq
tff(fact_1862_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A) ) ).

% minus_diff_eq
tff(fact_1863_div__by__1,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ) ).

% div_by_1
tff(fact_1864_div__by__1,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ) ).

% div_by_1
tff(fact_1865_div__by__1,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),one_one_Code_integer) = A ) ).

% div_by_1
tff(fact_1866_div__by__1,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),one_one_rat) = A ) ).

% div_by_1
tff(fact_1867_div__by__1,axiom,
    ! [A: code_natural] : ( divide5121882707175180666atural(A,one_one_Code_natural) = A ) ).

% div_by_1
tff(fact_1868_diff__numeral__special_I9_J,axiom,
    aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% diff_numeral_special(9)
tff(fact_1869_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_1870_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_1871_left__diff__distrib__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(minus_minus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(minus_minus_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))) ) ).

% left_diff_distrib_numeral
tff(fact_1872_left__diff__distrib__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),aa_num_Code_integer(numera6620942414471956472nteger,V)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_num_Code_integer(numera6620942414471956472nteger,V))) ) ).

% left_diff_distrib_numeral
tff(fact_1873_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),numeral_numeral_rat(V)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),numeral_numeral_rat(V))) ) ).

% left_diff_distrib_numeral
tff(fact_1874_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: 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(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,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)),C)) ) ).

% right_diff_distrib_numeral
tff(fact_1875_right__diff__distrib__numeral,axiom,
    ! [V: num,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),C)) ) ).

% right_diff_distrib_numeral
tff(fact_1876_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),C)) ) ).

% right_diff_distrib_numeral
tff(fact_1877_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_1878_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(zero_z3403309356797280102nteger),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ).

% verit_minus_simplify(3)
tff(fact_1879_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ) ).

% verit_minus_simplify(3)
tff(fact_1880_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_1881_diff__0,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(zero_z3403309356797280102nteger),A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ).

% diff_0
tff(fact_1882_diff__0,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ) ).

% diff_0
tff(fact_1883_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_1884_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_1885_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1886_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1887_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(times_2397367101498566445atural(A,B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1888_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_1889_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_1890_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1891_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1892_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(times_2397367101498566445atural(A,B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1893_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_1894_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_1895_div__self,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),A) = one_one_Code_integer ) ) ).

% div_self
tff(fact_1896_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = one_one_rat ) ) ).

% div_self
tff(fact_1897_div__self,axiom,
    ! [A: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(A,A) = one_one_Code_natural ) ) ).

% div_self
tff(fact_1898_iszero__neg__numeral,axiom,
    ! [W: num] :
      ( aa_int_o(ring_1_iszero_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W)))
    <=> aa_int_o(ring_1_iszero_int,aa_num_int(numeral_numeral_int,W)) ) ).

% iszero_neg_numeral
tff(fact_1899_iszero__neg__numeral,axiom,
    ! [W: num] :
      ( aa_Code_integer_o(ring_16219924574208605041nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)))
    <=> aa_Code_integer_o(ring_16219924574208605041nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)) ) ).

% iszero_neg_numeral
tff(fact_1900_iszero__neg__numeral,axiom,
    ! [W: num] :
      ( aa_rat_o(ring_1_iszero_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))
    <=> aa_rat_o(ring_1_iszero_rat,numeral_numeral_rat(W)) ) ).

% iszero_neg_numeral
tff(fact_1901_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),numeral_numeral_rat(W)) )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1902_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),numeral_numeral_rat(W)) = A )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1903_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_1904_diff__numeral__special_I12_J,axiom,
    aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_1905_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_1906_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1907_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1908_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1909_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D2) )
     => ( ( A = B )
      <=> ( C = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_1910_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D2) )
     => ( ( A = B )
      <=> ( C = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_1911_eq__iff__diff__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
    <=> ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B) = zero_z3403309356797280102nteger ) ) ).

% eq_iff_diff_eq_0
tff(fact_1912_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_1913_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_1914_left__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% left_diff_distrib
tff(fact_1915_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% left_diff_distrib
tff(fact_1916_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% left_diff_distrib
tff(fact_1917_right__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ).

% right_diff_distrib
tff(fact_1918_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ).

% right_diff_distrib
tff(fact_1919_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ).

% right_diff_distrib
tff(fact_1920_left__diff__distrib_H,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)),A) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)) ) ).

% left_diff_distrib'
tff(fact_1921_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)),A) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)) ) ).

% left_diff_distrib'
tff(fact_1922_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ) ).

% left_diff_distrib'
tff(fact_1923_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) ) ).

% left_diff_distrib'
tff(fact_1924_right__diff__distrib_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ).

% right_diff_distrib'
tff(fact_1925_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ).

% right_diff_distrib'
tff(fact_1926_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ) ).

% right_diff_distrib'
tff(fact_1927_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ).

% right_diff_distrib'
tff(fact_1928_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_1929_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).

% minus_diff_commute
tff(fact_1930_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ).

% minus_diff_commute
tff(fact_1931_not__iszero__1,axiom,
    ~ aa_Code_integer_o(ring_16219924574208605041nteger,one_one_Code_integer) ).

% not_iszero_1
tff(fact_1932_not__iszero__1,axiom,
    ~ aa_rat_o(ring_1_iszero_rat,one_one_rat) ).

% not_iszero_1
tff(fact_1933_not__iszero__1,axiom,
    ~ aa_int_o(ring_1_iszero_int,one_one_int) ).

% not_iszero_1
tff(fact_1934_diff__eq,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Y)) ) ).

% diff_eq
tff(fact_1935_diff__eq,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(minus_827889751674334651_int_o,X),Y) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,Y)) ) ).

% diff_eq
tff(fact_1936_diff__eq,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(minus_790609929918799799char_o,X),Y) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,Y)) ) ).

% diff_eq
tff(fact_1937_diff__eq,axiom,
    ! [X: set_nat,Y: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ) ).

% diff_eq
tff(fact_1938_diff__eq,axiom,
    ! [X: product_unit,Y: product_unit] : ( minus_3524152463667985524t_unit(X,Y) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)) ) ).

% diff_eq
tff(fact_1939_not__iszero__neg__1,axiom,
    ~ aa_int_o(ring_1_iszero_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_iszero_neg_1
tff(fact_1940_not__iszero__neg__1,axiom,
    ~ aa_Code_integer_o(ring_16219924574208605041nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_iszero_neg_1
tff(fact_1941_not__iszero__neg__1,axiom,
    ~ aa_rat_o(ring_1_iszero_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_iszero_neg_1
tff(fact_1942_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( numeral_numeral_rat(W) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C) = B,numeral_numeral_rat(W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1943_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = numeral_numeral_rat(W) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C),numeral_numeral_rat(W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_1944_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_1945_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C) = B,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_1946_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1947_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1948_divide__minus1,axiom,
    ! [X: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,X) ) ).

% divide_minus1
tff(fact_1949_div__minus1__right,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ) ).

% div_minus1_right
tff(fact_1950_div__minus1__right,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ).

% div_minus1_right
tff(fact_1951_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_1952_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_1953_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_1954_divide__self__if,axiom,
    ! [A: rat] :
      ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ) ).

% divide_self_if
tff(fact_1955_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_1956_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_1957_Diff__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),bot_bo2099793752762293965at_nat) = A2 ) ).

% Diff_empty
tff(fact_1958_Diff__empty,axiom,
    ! [A2: set_o] : ( minus_minus_set_o(A2,bot_bot_set_o) = A2 ) ).

% Diff_empty
tff(fact_1959_Diff__empty,axiom,
    ! [A2: set_int] : ( minus_minus_set_int(A2,bot_bot_set_int) = A2 ) ).

% Diff_empty
tff(fact_1960_Diff__empty,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),bot_bot_set_nat) = A2 ) ).

% Diff_empty
tff(fact_1961_empty__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,bot_bo2099793752762293965at_nat),A2) = bot_bo2099793752762293965at_nat ) ).

% empty_Diff
tff(fact_1962_empty__Diff,axiom,
    ! [A2: set_o] : ( minus_minus_set_o(bot_bot_set_o,A2) = bot_bot_set_o ) ).

% empty_Diff
tff(fact_1963_empty__Diff,axiom,
    ! [A2: set_int] : ( minus_minus_set_int(bot_bot_set_int,A2) = bot_bot_set_int ) ).

% empty_Diff
tff(fact_1964_empty__Diff,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,bot_bot_set_nat),A2) = bot_bot_set_nat ) ).

% empty_Diff
tff(fact_1965_Diff__cancel,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),A2) = bot_bo2099793752762293965at_nat ) ).

% Diff_cancel
tff(fact_1966_Diff__cancel,axiom,
    ! [A2: set_o] : ( minus_minus_set_o(A2,A2) = bot_bot_set_o ) ).

% Diff_cancel
tff(fact_1967_Diff__cancel,axiom,
    ! [A2: set_int] : ( minus_minus_set_int(A2,A2) = bot_bot_set_int ) ).

% Diff_cancel
tff(fact_1968_Diff__cancel,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),A2) = bot_bot_set_nat ) ).

% Diff_cancel
tff(fact_1969_Un__Diff__cancel,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,B2),A2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2) ) ).

% Un_Diff_cancel
tff(fact_1970_Un__Diff__cancel,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B2),A2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) ) ).

% Un_Diff_cancel
tff(fact_1971_Un__Diff__cancel,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) ) ).

% Un_Diff_cancel
tff(fact_1972_Un__Diff__cancel2,axiom,
    ! [B2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,B2),A2)),A2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B2),A2) ) ).

% Un_Diff_cancel2
tff(fact_1973_Un__Diff__cancel2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B2),A2)),A2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),A2) ) ).

% Un_Diff_cancel2
tff(fact_1974_Un__Diff__cancel2,axiom,
    ! [B2: set_nat,A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A2) ) ).

% Un_Diff_cancel2
tff(fact_1975_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) ) ).

% times_divide_eq_right
tff(fact_1976_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ).

% divide_divide_eq_right
tff(fact_1977_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% divide_divide_eq_left
tff(fact_1978_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),C) ) ).

% times_divide_eq_left
tff(fact_1979_Diff__UNIV,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),top_to4669805908274784177at_nat) = bot_bo2099793752762293965at_nat ) ).

% Diff_UNIV
tff(fact_1980_Diff__UNIV,axiom,
    ! [A2: set_list_nat] : ( minus_7954133019191499631st_nat(A2,top_top_set_list_nat) = bot_bot_set_list_nat ) ).

% Diff_UNIV
tff(fact_1981_Diff__UNIV,axiom,
    ! [A2: set_o] : ( minus_minus_set_o(A2,top_top_set_o) = bot_bot_set_o ) ).

% Diff_UNIV
tff(fact_1982_Diff__UNIV,axiom,
    ! [A2: set_rat] : ( minus_minus_set_rat(A2,top_top_set_rat) = bot_bot_set_rat ) ).

% Diff_UNIV
tff(fact_1983_Diff__UNIV,axiom,
    ! [A2: set_int] : ( minus_minus_set_int(A2,top_top_set_int) = bot_bot_set_int ) ).

% Diff_UNIV
tff(fact_1984_Diff__UNIV,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),top_top_set_nat) = bot_bot_set_nat ) ).

% Diff_UNIV
tff(fact_1985_div__minus__minus,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ).

% div_minus_minus
tff(fact_1986_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) ) ).

% div_minus_minus
tff(fact_1987_Diff__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),A2)) = bot_bo2099793752762293965at_nat ) ).

% Diff_disjoint
tff(fact_1988_Diff__disjoint,axiom,
    ! [A2: set_o,B2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),minus_minus_set_o(B2,A2)) = bot_bot_set_o ) ).

% Diff_disjoint
tff(fact_1989_Diff__disjoint,axiom,
    ! [A2: set_int,B2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),minus_minus_set_int(B2,A2)) = bot_bot_set_int ) ).

% Diff_disjoint
tff(fact_1990_Diff__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)) = bot_bot_set_nat ) ).

% Diff_disjoint
tff(fact_1991_Diff__Compl,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ) ).

% Diff_Compl
tff(fact_1992_Diff__Compl,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ).

% Diff_Compl
tff(fact_1993_inter__compl__diff__conv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2) ) ).

% inter_compl_diff_conv
tff(fact_1994_inter__compl__diff__conv,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) ) ).

% inter_compl_diff_conv
tff(fact_1995_Compl__Diff__eq,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),B2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,A2)),B2) ) ).

% Compl_Diff_eq
tff(fact_1996_Compl__Diff__eq,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(A2)),B2) ) ).

% Compl_Diff_eq
tff(fact_1997_Compl__Diff__eq,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)),B2) ) ).

% Compl_Diff_eq
tff(fact_1998_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ).

% div_mult_mult1_if
tff(fact_1999_div__mult__mult1__if,axiom,
    ! [C: 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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = $ite(C = 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_2000_div__mult__mult1__if,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = $ite(C = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ).

% div_mult_mult1_if
tff(fact_2001_div__mult__mult1__if,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( divide5121882707175180666atural(times_2397367101498566445atural(C,A),times_2397367101498566445atural(C,B)) = $ite(C = zero_z2226904508553997617atural,zero_z2226904508553997617atural,divide5121882707175180666atural(A,B)) ) ).

% div_mult_mult1_if
tff(fact_2002_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_2003_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult2
tff(fact_2004_div__mult__mult2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) ) ) ).

% div_mult_mult2
tff(fact_2005_div__mult__mult2,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(times_2397367101498566445atural(A,C),times_2397367101498566445atural(B,C)) = divide5121882707175180666atural(A,B) ) ) ).

% div_mult_mult2
tff(fact_2006_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_2007_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult1
tff(fact_2008_div__mult__mult1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) ) ) ).

% div_mult_mult1
tff(fact_2009_div__mult__mult1,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(times_2397367101498566445atural(C,A),times_2397367101498566445atural(C,B)) = divide5121882707175180666atural(A,B) ) ) ).

% div_mult_mult1
tff(fact_2010_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2011_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2012_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2013_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2014_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2015_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_2016_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_2017_Int__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),C2)) ) ).

% Int_Diff
tff(fact_2018_Int__Diff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),C2)) ) ).

% Int_Diff
tff(fact_2019_Diff__Int2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C2)),B2) ) ).

% Diff_Int2
tff(fact_2020_Diff__Int2,axiom,
    ! [A2: set_nat,C2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)),B2) ) ).

% Diff_Int2
tff(fact_2021_Diff__Diff__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ) ).

% Diff_Diff_Int
tff(fact_2022_Diff__Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ).

% Diff_Diff_Int
tff(fact_2023_Diff__Int__distrib,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),B2)) ) ).

% Diff_Int_distrib
tff(fact_2024_Diff__Int__distrib,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),B2)) ) ).

% Diff_Int_distrib
tff(fact_2025_Diff__Int__distrib2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ) ).

% Diff_Int_distrib2
tff(fact_2026_Diff__Int__distrib2,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ) ).

% Diff_Int_distrib2
tff(fact_2027_Un__Diff,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)),C2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),C2)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,B2),C2)) ) ).

% Un_Diff
tff(fact_2028_Un__Diff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),C2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B2),C2)) ) ).

% Un_Diff
tff(fact_2029_Un__Diff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),C2)) ) ).

% Un_Diff
tff(fact_2030_set__diff__diff__left,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),B2)),C2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B2),C2)) ) ).

% set_diff_diff_left
tff(fact_2031_set__diff__diff__left,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),B2)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) ) ).

% set_diff_diff_left
tff(fact_2032_set__diff__diff__left,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) ) ).

% set_diff_diff_left
tff(fact_2033_Diff__triv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2) = A2 ) ) ).

% Diff_triv
tff(fact_2034_Diff__triv,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o )
     => ( minus_minus_set_o(A2,B2) = A2 ) ) ).

% Diff_triv
tff(fact_2035_Diff__triv,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
     => ( minus_minus_set_int(A2,B2) = A2 ) ) ).

% Diff_triv
tff(fact_2036_Diff__triv,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = A2 ) ) ).

% Diff_triv
tff(fact_2037_Int__Diff__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)) = bot_bo2099793752762293965at_nat ) ).

% Int_Diff_disjoint
tff(fact_2038_Int__Diff__disjoint,axiom,
    ! [A2: set_o,B2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2)),minus_minus_set_o(A2,B2)) = bot_bot_set_o ) ).

% Int_Diff_disjoint
tff(fact_2039_Int__Diff__disjoint,axiom,
    ! [A2: set_int,B2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)),minus_minus_set_int(A2,B2)) = bot_bot_set_int ) ).

% Int_Diff_disjoint
tff(fact_2040_Int__Diff__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = bot_bot_set_nat ) ).

% Int_Diff_disjoint
tff(fact_2041_disjoint__alt__simp1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2) = A2 )
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp1
tff(fact_2042_disjoint__alt__simp1,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( minus_minus_set_o(A2,B2) = A2 )
    <=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o ) ) ).

% disjoint_alt_simp1
tff(fact_2043_disjoint__alt__simp1,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( minus_minus_set_int(A2,B2) = A2 )
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int ) ) ).

% disjoint_alt_simp1
tff(fact_2044_disjoint__alt__simp1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = A2 )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat ) ) ).

% disjoint_alt_simp1
tff(fact_2045_disjoint__alt__simp2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2) != A2 )
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp2
tff(fact_2046_disjoint__alt__simp2,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( minus_minus_set_o(A2,B2) != A2 )
    <=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) != bot_bot_set_o ) ) ).

% disjoint_alt_simp2
tff(fact_2047_disjoint__alt__simp2,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( minus_minus_set_int(A2,B2) != A2 )
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) != bot_bot_set_int ) ) ).

% disjoint_alt_simp2
tff(fact_2048_disjoint__alt__simp2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) != A2 )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp2
tff(fact_2049_Diff__Un,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),C2)) ) ).

% Diff_Un
tff(fact_2050_Diff__Un,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B2),C2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),B2)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),C2)) ) ).

% Diff_Un
tff(fact_2051_Diff__Un,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),C2)) ) ).

% Diff_Un
tff(fact_2052_Diff__Un,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),C2)) ) ).

% Diff_Un
tff(fact_2053_Diff__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),C2)) ) ).

% Diff_Int
tff(fact_2054_Diff__Int,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B2),C2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),B2)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),C2)) ) ).

% Diff_Int
tff(fact_2055_Diff__Int,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),C2)) ) ).

% Diff_Int
tff(fact_2056_Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),C2)) ) ).

% Diff_Int
tff(fact_2057_Int__Diff__Un,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)) = A2 ) ).

% Int_Diff_Un
tff(fact_2058_Int__Diff__Un,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),B2)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),B2)) = A2 ) ).

% Int_Diff_Un
tff(fact_2059_Int__Diff__Un,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),B2)) = A2 ) ).

% Int_Diff_Un
tff(fact_2060_Int__Diff__Un,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = A2 ) ).

% Int_Diff_Un
tff(fact_2061_Un__Diff__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) = A2 ) ).

% Un_Diff_Int
tff(fact_2062_Un__Diff__Int,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),B2)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),B2)) = A2 ) ).

% Un_Diff_Int
tff(fact_2063_Un__Diff__Int,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2)) = A2 ) ).

% Un_Diff_Int
tff(fact_2064_Un__Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = A2 ) ).

% Un_Diff_Int
tff(fact_2065_Compl__eq__Diff__UNIV,axiom,
    ! [A2: set_list_nat] : ( aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,A2) = minus_7954133019191499631st_nat(top_top_set_list_nat,A2) ) ).

% Compl_eq_Diff_UNIV
tff(fact_2066_Compl__eq__Diff__UNIV,axiom,
    ! [A2: set_o] : ( aa_set_o_set_o(uminus_uminus_set_o,A2) = minus_minus_set_o(top_top_set_o,A2) ) ).

% Compl_eq_Diff_UNIV
tff(fact_2067_Compl__eq__Diff__UNIV,axiom,
    ! [A2: set_rat] : ( aa_set_rat_set_rat(uminus2201863774496077783et_rat,A2) = minus_minus_set_rat(top_top_set_rat,A2) ) ).

% Compl_eq_Diff_UNIV
tff(fact_2068_Compl__eq__Diff__UNIV,axiom,
    ! [A2: set_int] : ( aa_set_int_set_int(uminus1532241313380277803et_int,A2) = minus_minus_set_int(top_top_set_int,A2) ) ).

% Compl_eq_Diff_UNIV
tff(fact_2069_Compl__eq__Diff__UNIV,axiom,
    ! [A2: set_nat] : ( aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,top_top_set_nat),A2) ) ).

% Compl_eq_Diff_UNIV
tff(fact_2070_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ).

% divide_divide_eq_left'
tff(fact_2071_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Z),W)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),W)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ).

% divide_divide_times_eq
tff(fact_2072_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Z),W)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),W)) ) ).

% times_divide_times_eq
tff(fact_2073_div__minus__right,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% div_minus_right
tff(fact_2074_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).

% div_minus_right
tff(fact_2075_minus__divide__left,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ).

% minus_divide_left
tff(fact_2076_minus__divide__divide,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ).

% minus_divide_divide
tff(fact_2077_minus__divide__right,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ) ).

% minus_divide_right
tff(fact_2078_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2079_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = A )
      <=> ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2080_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B )
       => ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2081_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2082_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_2083_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = A )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_2084_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z) )
        <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2085_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2086_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_2087_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_2088_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2089_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2090_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2091_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B)),Z)) ) ).

% add_divide_eq_if_simps(4)
tff(fact_2092_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2093_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2094_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) = A )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_2095_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(uminus_uminus_rat,B),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_2096_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_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_2097_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z))),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2098_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z)),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z)) ) ).

% add_divide_eq_if_simps(5)
tff(fact_2099_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z))),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z)) ) ).

% add_divide_eq_if_simps(6)
tff(fact_2100_bits__div__by__1,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ) ).

% bits_div_by_1
tff(fact_2101_bits__div__by__1,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ) ).

% bits_div_by_1
tff(fact_2102_bits__div__by__1,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),one_one_Code_integer) = A ) ).

% bits_div_by_1
tff(fact_2103_bits__div__by__1,axiom,
    ! [A: code_natural] : ( divide5121882707175180666atural(A,one_one_Code_natural) = A ) ).

% bits_div_by_1
tff(fact_2104_inf__period_I1_J,axiom,
    ! [P: fun_Code_integer_o,D3: code_integer,Q: fun_Code_integer_o] :
      ( ! [X2: code_integer,K2: code_integer] :
          ( aa_Code_integer_o(P,X2)
        <=> aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D3))) )
     => ( ! [X2: code_integer,K2: code_integer] :
            ( aa_Code_integer_o(Q,X2)
          <=> aa_Code_integer_o(Q,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D3))) )
       => ! [X3: code_integer,K3: code_integer] :
            ( ( aa_Code_integer_o(P,X3)
              & aa_Code_integer_o(Q,X3) )
          <=> ( aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3)))
              & aa_Code_integer_o(Q,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_2105_inf__period_I1_J,axiom,
    ! [P: fun_rat_o,D3: rat,Q: fun_rat_o] :
      ( ! [X2: rat,K2: rat] :
          ( aa_rat_o(P,X2)
        <=> aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
     => ( ! [X2: rat,K2: rat] :
            ( aa_rat_o(Q,X2)
          <=> aa_rat_o(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
       => ! [X3: rat,K3: rat] :
            ( ( aa_rat_o(P,X3)
              & aa_rat_o(Q,X3) )
          <=> ( aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3)))
              & aa_rat_o(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_2106_inf__period_I1_J,axiom,
    ! [P: fun_int_o,D3: int,Q: fun_int_o] :
      ( ! [X2: int,K2: int] :
          ( aa_int_o(P,X2)
        <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(Q,X2)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
       => ! [X3: int,K3: int] :
            ( ( aa_int_o(P,X3)
              & aa_int_o(Q,X3) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))
              & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_2107_inf__period_I2_J,axiom,
    ! [P: fun_Code_integer_o,D3: code_integer,Q: fun_Code_integer_o] :
      ( ! [X2: code_integer,K2: code_integer] :
          ( aa_Code_integer_o(P,X2)
        <=> aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D3))) )
     => ( ! [X2: code_integer,K2: code_integer] :
            ( aa_Code_integer_o(Q,X2)
          <=> aa_Code_integer_o(Q,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D3))) )
       => ! [X3: code_integer,K3: code_integer] :
            ( ( aa_Code_integer_o(P,X3)
              | aa_Code_integer_o(Q,X3) )
          <=> ( aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3)))
              | aa_Code_integer_o(Q,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_2108_inf__period_I2_J,axiom,
    ! [P: fun_rat_o,D3: rat,Q: fun_rat_o] :
      ( ! [X2: rat,K2: rat] :
          ( aa_rat_o(P,X2)
        <=> aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
     => ( ! [X2: rat,K2: rat] :
            ( aa_rat_o(Q,X2)
          <=> aa_rat_o(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
       => ! [X3: rat,K3: rat] :
            ( ( aa_rat_o(P,X3)
              | aa_rat_o(Q,X3) )
          <=> ( aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3)))
              | aa_rat_o(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_2109_inf__period_I2_J,axiom,
    ! [P: fun_int_o,D3: int,Q: fun_int_o] :
      ( ! [X2: int,K2: int] :
          ( aa_int_o(P,X2)
        <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(Q,X2)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
       => ! [X3: int,K3: int] :
            ( ( aa_int_o(P,X3)
              | aa_int_o(Q,X3) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))
              | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_2110_diff__numeral__special_I5_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(N))) ) ).

% diff_numeral_special(5)
tff(fact_2111_diff__numeral__special_I5_J,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,inc(N))) ) ).

% diff_numeral_special(5)
tff(fact_2112_diff__numeral__special_I5_J,axiom,
    ! [N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),numeral_numeral_rat(N)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(inc(N))) ) ).

% diff_numeral_special(5)
tff(fact_2113_diff__numeral__special_I6_J,axiom,
    ! [M: num] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,inc(M)) ) ).

% diff_numeral_special(6)
tff(fact_2114_diff__numeral__special_I6_J,axiom,
    ! [M: num] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,inc(M)) ) ).

% diff_numeral_special(6)
tff(fact_2115_diff__numeral__special_I6_J,axiom,
    ! [M: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,numeral_numeral_rat(M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = numeral_numeral_rat(inc(M)) ) ).

% diff_numeral_special(6)
tff(fact_2116_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_2117_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_2118_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: int,A: int,B: int] :
      ( aa_int_o(aa_nat_fun_int_o(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_2119_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_2120_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_2121_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_2122_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(nO_MAT8252062027627875367nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_2123_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,C: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MAT7795273704451493282er_rat(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),C)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_2124_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,C: int,A: int,B: int] :
      ( aa_int_o(nO_MAT8427913294028938742er_int(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_2125_left__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT243659986023775650nteger(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_2126_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_2127_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
      ( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_2128_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,C: int] :
      ( aa_int_o(aa_nat_fun_int_o(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_2129_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_2130_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: rat,B: rat,C: rat] :
      ( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_2131_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_2132_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(nO_MAT8252062027627875367nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_2133_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: rat,B: rat,C: rat] :
      ( aa_rat_o(nO_MAT7795273704451493282er_rat(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_2134_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: int,B: int,C: int] :
      ( aa_int_o(nO_MAT8427913294028938742er_int(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_2135_right__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT243659986023775650nteger(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_2136_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_2137_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_2138_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_2139_dual__order_Orefl,axiom,
    ! [A: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),A) ).

% dual_order.refl
tff(fact_2140_dual__order_Orefl,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),A) ).

% dual_order.refl
tff(fact_2141_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_2142_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_2143_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_2144_order__refl,axiom,
    ! [X: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),X) ).

% order_refl
tff(fact_2145_order__refl,axiom,
    ! [X: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),X) ).

% order_refl
tff(fact_2146_order__refl,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),X) ).

% order_refl
tff(fact_2147_order__refl,axiom,
    ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X) ).

% order_refl
tff(fact_2148_order__refl,axiom,
    ! [X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X) ).

% order_refl
tff(fact_2149_empty__subsetI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,bot_bo2099793752762293965at_nat),A2) ).

% empty_subsetI
tff(fact_2150_empty__subsetI,axiom,
    ! [A2: set_o] : aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,bot_bot_set_o),A2) ).

% empty_subsetI
tff(fact_2151_empty__subsetI,axiom,
    ! [A2: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,bot_bot_set_int),A2) ).

% empty_subsetI
tff(fact_2152_empty__subsetI,axiom,
    ! [A2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A2) ).

% empty_subsetI
tff(fact_2153_subset__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),bot_bo2099793752762293965at_nat)
    <=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
tff(fact_2154_subset__empty,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),bot_bot_set_o)
    <=> ( A2 = bot_bot_set_o ) ) ).

% subset_empty
tff(fact_2155_subset__empty,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),bot_bot_set_int)
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_2156_subset__empty,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),bot_bot_set_nat)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_2157_Int__subset__iff,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),A2)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),B2) ) ) ).

% Int_subset_iff
tff(fact_2158_Int__subset__iff,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A2)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),B2) ) ) ).

% Int_subset_iff
tff(fact_2159_Un__subset__iff,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2),C2)
    <=> ( ord_le8081472938463900775at_nat(A2,C2)
        & ord_le8081472938463900775at_nat(B2,C2) ) ) ).

% Un_subset_iff
tff(fact_2160_Un__subset__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)),C2)
    <=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),C2)
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B2),C2) ) ) ).

% Un_subset_iff
tff(fact_2161_Un__subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),C2)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C2) ) ) ).

% Un_subset_iff
tff(fact_2162_Compl__subset__Compl__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B2))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2) ) ).

% Compl_subset_Compl_iff
tff(fact_2163_Compl__anti__mono,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,B2)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)) ) ).

% Compl_anti_mono
tff(fact_2164_le__zero__eq,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),zero_zero_nat)
    <=> ( N = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_2165_not__gr__zero,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
    <=> ( N = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_2166_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_2167_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_2168_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_2169_compl__le__compl__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y))
    <=> aa_Product_unit_o(ord_le3221252021190050221t_unit(Y),X) ) ).

% compl_le_compl_iff
tff(fact_2170_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X) ) ).

% compl_le_compl_iff
tff(fact_2171_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_2172_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_2173_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_2174_compl__less__compl__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( aa_Product_unit_o(ord_le361264281704409273t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y))
    <=> aa_Product_unit_o(ord_le361264281704409273t_unit(Y),X) ) ).

% compl_less_compl_iff
tff(fact_2175_le__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Z) ) ) ).

% le_inf_iff
tff(fact_2176_le__inf__iff,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z))
    <=> ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Y)
        & aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Z) ) ) ).

% le_inf_iff
tff(fact_2177_le__inf__iff,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),Z))
    <=> ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),Y)
        & aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),Z) ) ) ).

% le_inf_iff
tff(fact_2178_le__inf__iff,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Y),Z))
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
        & aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Z) ) ) ).

% le_inf_iff
tff(fact_2179_le__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z) ) ) ).

% le_inf_iff
tff(fact_2180_le__inf__iff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z) ) ) ).

% le_inf_iff
tff(fact_2181_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z) ) ) ).

% le_inf_iff
tff(fact_2182_le__inf__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z) ) ) ).

% le_inf_iff
tff(fact_2183_inf_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C) ) ) ).

% inf.bounded_iff
tff(fact_2184_inf_Obounded__iff,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C))
    <=> ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
        & aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C) ) ) ).

% inf.bounded_iff
tff(fact_2185_inf_Obounded__iff,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o,C: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,B),C))
    <=> ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),B)
        & aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),C) ) ) ).

% inf.bounded_iff
tff(fact_2186_inf_Obounded__iff,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,B),C))
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
        & aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),C) ) ) ).

% inf.bounded_iff
tff(fact_2187_inf_Obounded__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).

% inf.bounded_iff
tff(fact_2188_inf_Obounded__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% inf.bounded_iff
tff(fact_2189_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
    <=> ( 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),C) ) ) ).

% inf.bounded_iff
tff(fact_2190_inf_Obounded__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% inf.bounded_iff
tff(fact_2191_le__sup__iff,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y),Z)
    <=> ( ord_le8081472938463900775at_nat(X,Z)
        & ord_le8081472938463900775at_nat(Y,Z) ) ) ).

% le_sup_iff
tff(fact_2192_le__sup__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),Z)
    <=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),Z)
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),Z) ) ) ).

% le_sup_iff
tff(fact_2193_le__sup__iff,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y)),Z)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Z)
        & aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),Z) ) ) ).

% le_sup_iff
tff(fact_2194_le__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Z) ) ) ).

% le_sup_iff
tff(fact_2195_le__sup__iff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)),Z)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% le_sup_iff
tff(fact_2196_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) ) ) ).

% le_sup_iff
tff(fact_2197_le__sup__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) ) ) ).

% le_sup_iff
tff(fact_2198_sup_Obounded__iff,axiom,
    ! [B: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),C),A)
    <=> ( ord_le8081472938463900775at_nat(B,A)
        & ord_le8081472938463900775at_nat(C,A) ) ) ).

% sup.bounded_iff
tff(fact_2199_sup_Obounded__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)),A)
    <=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A) ) ) ).

% sup.bounded_iff
tff(fact_2200_sup_Obounded__iff,axiom,
    ! [B: filter_nat,C: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,B),C)),A)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
        & aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),A) ) ) ).

% sup.bounded_iff
tff(fact_2201_sup_Obounded__iff,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A) ) ) ).

% sup.bounded_iff
tff(fact_2202_sup_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),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,C),A) ) ) ).

% sup.bounded_iff
tff(fact_2203_sup_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),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,C),A) ) ) ).

% sup.bounded_iff
tff(fact_2204_sup_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),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,C),A) ) ) ).

% sup.bounded_iff
tff(fact_2205_Diff__eq__empty__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_2206_Diff__eq__empty__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( minus_minus_set_o(A2,B2) = bot_bot_set_o )
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_2207_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( minus_minus_set_int(A2,B2) = bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_2208_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_2209_diff__ge__0__iff__ge,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_2210_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(aa_rat_fun_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_2211_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_2212_diff__gt__0__iff__gt,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_2213_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(aa_rat_fun_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_2214_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_2215_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_2216_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_2217_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_2218_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_2219_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_2220_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_2221_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_2222_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_2223_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_2224_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_2225_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_2226_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_2227_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_2228_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_2229_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_2230_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_2231_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_2232_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_2233_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_2234_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_2235_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_2236_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_2237_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_2238_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_2239_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)))
    <=> ord_less_eq_num(N,M) ) ).

% neg_numeral_le_iff
tff(fact_2240_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)))
    <=> ord_less_eq_num(N,M) ) ).

% neg_numeral_le_iff
tff(fact_2241_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))
    <=> ord_less_eq_num(N,M) ) ).

% neg_numeral_le_iff
tff(fact_2242_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))
    <=> ord_less_num(N,M) ) ).

% neg_numeral_less_iff
tff(fact_2243_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)))
    <=> ord_less_num(N,M) ) ).

% neg_numeral_less_iff
tff(fact_2244_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)))
    <=> ord_less_num(N,M) ) ).

% neg_numeral_less_iff
tff(fact_2245_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_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_2246_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(aa_rat_fun_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_2247_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_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_2248_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(aa_rat_fun_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_2249_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(aa_rat_fun_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_2250_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(aa_rat_fun_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_2251_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(aa_rat_fun_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_2252_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(aa_rat_fun_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_2253_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),numeral_numeral_rat(W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2254_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),numeral_numeral_rat(W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2255_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),numeral_numeral_rat(W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2256_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),numeral_numeral_rat(W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2257_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(aa_rat_fun_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_2258_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(aa_rat_fun_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_2259_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(aa_rat_fun_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_2260_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(aa_rat_fun_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_2261_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_2262_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_2263_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_2264_minus__set__def,axiom,
    ! [A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] : ( minus_2612819937483484256nt_int(A2,B2) = collec5210948495886036740nt_int(minus_3219807633925007451_int_o(aTP_Lamm_ag(A2),aTP_Lamm_ag(B2))) ) ).

% minus_set_def
tff(fact_2265_minus__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : ( minus_2163939370556025621et_nat(A2,B2) = collect_set_nat(minus_6087510627544308050_nat_o(aTP_Lamm_ah(A2),aTP_Lamm_ah(B2))) ) ).

% minus_set_def
tff(fact_2266_minus__set__def,axiom,
    ! [A2: set_int,B2: set_int] : ( minus_minus_set_int(A2,B2) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_aj(A2),aTP_Lamm_aj(B2))) ) ).

% minus_set_def
tff(fact_2267_minus__set__def,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] : ( minus_6456595905716665654_nat_o(A2,B2) = collec6407654713642994138_nat_o(minus_8982612185847598341at_o_o(aTP_Lamm_ak(A2),aTP_Lamm_ak(B2))) ) ).

% minus_set_def
tff(fact_2268_minus__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = collect_nat(minus_8972599623528345224_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ai,A2),aa_set_nat_fun_nat_o(aTP_Lamm_ai,B2))) ) ).

% minus_set_def
tff(fact_2269_set__diff__eq,axiom,
    ! [A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] : ( minus_2612819937483484256nt_int(A2,B2) = collec5210948495886036740nt_int(aa_set4062574822058255039_int_o(aTP_Lamm_fm(A2),B2)) ) ).

% set_diff_eq
tff(fact_2270_set__diff__eq,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : ( minus_2163939370556025621et_nat(A2,B2) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_fn(A2),B2)) ) ).

% set_diff_eq
tff(fact_2271_set__diff__eq,axiom,
    ! [A2: set_int,B2: set_int] : ( minus_minus_set_int(A2,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_fo(A2),B2)) ) ).

% set_diff_eq
tff(fact_2272_set__diff__eq,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] : ( minus_6456595905716665654_nat_o(A2,B2) = collec6407654713642994138_nat_o(aa_set7516337885093543743at_o_o(aTP_Lamm_fp(A2),B2)) ) ).

% set_diff_eq
tff(fact_2273_set__diff__eq,axiom,
    ! [A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_fq(A2),B2)) ) ).

% set_diff_eq
tff(fact_2274_order__less__imp__not__less,axiom,
    ! [X: literal,Y: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
     => ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Y),X) ) ).

% order_less_imp_not_less
tff(fact_2275_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% order_less_imp_not_less
tff(fact_2276_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% order_less_imp_not_less
tff(fact_2277_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% order_less_imp_not_less
tff(fact_2278_order__le__imp__less__or__eq,axiom,
    ! [X: literal,Y: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,X),Y)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_2279_order__le__imp__less__or__eq,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_2280_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_2281_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_2282_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_2283_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_2284_linorder__le__less__linear,axiom,
    ! [X: literal,Y: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,X),Y)
      | aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Y),X) ) ).

% linorder_le_less_linear
tff(fact_2285_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% linorder_le_less_linear
tff(fact_2286_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% linorder_le_less_linear
tff(fact_2287_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% linorder_le_less_linear
tff(fact_2288_order__less__imp__not__eq2,axiom,
    ! [X: literal,Y: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_2289_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_2290_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_2291_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_2292_order__less__imp__not__eq,axiom,
    ! [X: literal,Y: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_2293_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_2294_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_2295_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_2296_order__less__le__subst2,axiom,
    ! [A: literal,B: literal,F: fun_literal_literal,C: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,aa_literal_literal(F,B)),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_literal_literal(F,X2)),aa_literal_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_literal_literal(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_2297_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_literal,C: literal] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,aa_rat_literal(F,B)),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_rat_literal(F,X2)),aa_rat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_rat_literal(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_2298_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_literal,C: literal] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,aa_nat_literal(F,B)),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,X2)),aa_nat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_2299_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_literal,C: literal] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,aa_int_literal(F,B)),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_int_literal(F,X2)),aa_int_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_int_literal(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_2300_order__less__le__subst2,axiom,
    ! [A: literal,B: literal,F: fun_literal_rat,C: rat] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_literal_rat(F,B)),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_literal_rat(F,X2)),aa_literal_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_literal_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_2301_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_2302_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: 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)),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_2303_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C: 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)),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_2304_order__less__le__subst2,axiom,
    ! [A: literal,B: literal,F: fun_literal_nat,C: nat] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_literal_nat(F,B)),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_literal_nat(F,X2)),aa_literal_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_literal_nat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_2305_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,B)),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_2306_order__less__le__subst1,axiom,
    ! [A: literal,F: fun_rat_literal,B: rat,C: rat] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_rat_literal(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,aa_rat_literal(F,X2)),aa_rat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_rat_literal(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_2307_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: 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),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_2308_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: 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),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_2309_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: 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),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_2310_order__less__le__subst1,axiom,
    ! [A: literal,F: fun_nat_literal,B: nat,C: nat] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_nat_literal(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,aa_nat_literal(F,X2)),aa_nat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_nat_literal(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_2311_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: 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_eq_nat,B),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_2312_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_2313_order__less__le__subst1,axiom,
    ! [A: int,F: fun_nat_int,B: nat,C: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_2314_order__less__le__subst1,axiom,
    ! [A: literal,F: fun_int_literal,B: int,C: int] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_int_literal(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,aa_int_literal(F,X2)),aa_int_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_int_literal(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_2315_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: 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_eq_int,B),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_2316_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_literal,C: literal] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_rat_literal(F,B)),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,aa_rat_literal(F,X2)),aa_rat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_rat_literal(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2317_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2318_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: 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)),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2319_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: 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)),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2320_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_literal,C: literal] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,B)),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,aa_nat_literal(F,X2)),aa_nat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2321_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: 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_rat,aa_nat_rat(F,B)),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2322_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,B)),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2323_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,B)),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2324_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: fun_int_literal,C: literal] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_int_literal(F,B)),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,aa_int_literal(F,X2)),aa_int_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_int_literal(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2325_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,B)),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2326_order__le__less__subst1,axiom,
    ! [A: literal,F: fun_literal_literal,B: literal,C: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,A),aa_literal_literal(F,B))
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_literal_literal(F,X2)),aa_literal_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_literal_literal(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2327_order__le__less__subst1,axiom,
    ! [A: literal,F: fun_rat_literal,B: rat,C: rat] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,A),aa_rat_literal(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_rat_literal(F,X2)),aa_rat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_rat_literal(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2328_order__le__less__subst1,axiom,
    ! [A: literal,F: fun_nat_literal,B: nat,C: nat] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,A),aa_nat_literal(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,X2)),aa_nat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_nat_literal(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2329_order__le__less__subst1,axiom,
    ! [A: literal,F: fun_int_literal,B: int,C: int] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,A),aa_int_literal(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_int_literal(F,X2)),aa_int_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_int_literal(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2330_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_literal_rat,B: literal,C: literal] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_literal_rat(F,B))
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_literal_rat(F,X2)),aa_literal_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_literal_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2331_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: 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),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2332_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: 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),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2333_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: 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),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2334_order__le__less__subst1,axiom,
    ! [A: nat,F: fun_literal_nat,B: literal,C: literal] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_literal_nat(F,B))
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_literal_nat(F,X2)),aa_literal_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_literal_nat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2335_order__le__less__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: 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_rat,B),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2336_linorder__less__linear,axiom,
    ! [X: literal,Y: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
      | ( X = Y )
      | aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Y),X) ) ).

% linorder_less_linear
tff(fact_2337_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
      | ( X = Y )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% linorder_less_linear
tff(fact_2338_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
      | ( X = Y )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% linorder_less_linear
tff(fact_2339_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
      | ( X = Y )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% linorder_less_linear
tff(fact_2340_order__less__le__trans,axiom,
    ! [X: literal,Y: literal,Z: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,Y),Z)
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Z) ) ) ).

% order_less_le_trans
tff(fact_2341_order__less__le__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),Z)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Z) ) ) ).

% order_less_le_trans
tff(fact_2342_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Z) ) ) ).

% order_less_le_trans
tff(fact_2343_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z) ) ) ).

% order_less_le_trans
tff(fact_2344_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z) ) ) ).

% order_less_le_trans
tff(fact_2345_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z) ) ) ).

% order_less_le_trans
tff(fact_2346_order__less__imp__triv,axiom,
    ! [X: literal,Y: literal,P: $o] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Y),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_2347_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_2348_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_2349_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_2350_order__le__less__trans,axiom,
    ! [X: literal,Y: literal,Z: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,X),Y)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Y),Z)
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Z) ) ) ).

% order_le_less_trans
tff(fact_2351_order__le__less__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,Y),Z)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Z) ) ) ).

% order_le_less_trans
tff(fact_2352_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Y),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Z) ) ) ).

% order_le_less_trans
tff(fact_2353_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z) ) ) ).

% order_le_less_trans
tff(fact_2354_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z) ) ) ).

% order_le_less_trans
tff(fact_2355_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z) ) ) ).

% order_le_less_trans
tff(fact_2356_order__neq__le__trans,axiom,
    ! [A: literal,B: literal] :
      ( ( A != B )
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,A),B)
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B) ) ) ).

% order_neq_le_trans
tff(fact_2357_order__neq__le__trans,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( A != B )
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_2358_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_2359_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_2360_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_2361_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_2362_order__less__not__sym,axiom,
    ! [X: literal,Y: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
     => ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Y),X) ) ).

% order_less_not_sym
tff(fact_2363_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% order_less_not_sym
tff(fact_2364_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% order_less_not_sym
tff(fact_2365_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% order_less_not_sym
tff(fact_2366_order__le__neq__trans,axiom,
    ! [A: literal,B: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,A),B)
     => ( ( A != B )
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B) ) ) ).

% order_le_neq_trans
tff(fact_2367_order__le__neq__trans,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
     => ( ( A != B )
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_2368_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( ( A != B )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_2369_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_2370_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_2371_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_2372_order__antisym__conv,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),X)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_2373_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_2374_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_2375_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_2376_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_2377_order__less__subst2,axiom,
    ! [A: literal,B: literal,F: fun_literal_literal,C: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_literal_literal(F,B)),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_literal_literal(F,X2)),aa_literal_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_literal_literal(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2378_order__less__subst2,axiom,
    ! [A: literal,B: literal,F: fun_literal_rat,C: rat] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_literal_rat(F,B)),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_literal_rat(F,X2)),aa_literal_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_literal_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2379_order__less__subst2,axiom,
    ! [A: literal,B: literal,F: fun_literal_nat,C: nat] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_literal_nat(F,B)),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_literal_nat(F,X2)),aa_literal_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_literal_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2380_order__less__subst2,axiom,
    ! [A: literal,B: literal,F: fun_literal_int,C: int] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_literal_int(F,B)),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_literal_int(F,X2)),aa_literal_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_literal_int(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2381_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_literal,C: literal] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_rat_literal(F,B)),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_rat_literal(F,X2)),aa_rat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_rat_literal(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2382_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2383_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: 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)),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2384_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: 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)),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2385_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_literal,C: literal] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,B)),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,X2)),aa_nat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2386_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,B)),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2387_order__less__subst1,axiom,
    ! [A: literal,F: fun_literal_literal,B: literal,C: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_literal_literal(F,B))
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_literal_literal(F,X2)),aa_literal_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_literal_literal(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2388_order__less__subst1,axiom,
    ! [A: literal,F: fun_rat_literal,B: rat,C: rat] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_rat_literal(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_rat_literal(F,X2)),aa_rat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_rat_literal(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2389_order__less__subst1,axiom,
    ! [A: literal,F: fun_nat_literal,B: nat,C: nat] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_nat_literal(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,X2)),aa_nat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_nat_literal(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2390_order__less__subst1,axiom,
    ! [A: literal,F: fun_int_literal,B: int,C: int] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_int_literal(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_int_literal(F,X2)),aa_int_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_int_literal(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2391_order__less__subst1,axiom,
    ! [A: rat,F: fun_literal_rat,B: literal,C: literal] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_literal_rat(F,B))
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_literal_rat(F,X2)),aa_literal_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_literal_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2392_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: 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),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2393_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: 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),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2394_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: 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),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2395_order__less__subst1,axiom,
    ! [A: nat,F: fun_literal_nat,B: literal,C: literal] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_literal_nat(F,B))
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_literal_nat(F,X2)),aa_literal_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_literal_nat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2396_order__less__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: 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_rat,B),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2397_order__less__irrefl,axiom,
    ! [X: literal] : ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),X) ).

% order_less_irrefl
tff(fact_2398_order__less__irrefl,axiom,
    ! [X: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X) ).

% order_less_irrefl
tff(fact_2399_order__less__irrefl,axiom,
    ! [X: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X) ).

% order_less_irrefl
tff(fact_2400_order__less__irrefl,axiom,
    ! [X: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),X) ).

% order_less_irrefl
tff(fact_2401_order__less__imp__le,axiom,
    ! [X: literal,Y: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
     => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,X),Y) ) ).

% order_less_imp_le
tff(fact_2402_order__less__imp__le,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y) ) ).

% order_less_imp_le
tff(fact_2403_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y) ) ).

% order_less_imp_le
tff(fact_2404_order__less__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% order_less_imp_le
tff(fact_2405_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% order_less_imp_le
tff(fact_2406_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% order_less_imp_le
tff(fact_2407_ord__less__eq__subst,axiom,
    ! [A: literal,B: literal,F: fun_literal_literal,C: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( ( aa_literal_literal(F,B) = C )
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_literal_literal(F,X2)),aa_literal_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_literal_literal(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2408_ord__less__eq__subst,axiom,
    ! [A: literal,B: literal,F: fun_literal_rat,C: rat] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( ( aa_literal_rat(F,B) = C )
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_literal_rat(F,X2)),aa_literal_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_literal_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2409_ord__less__eq__subst,axiom,
    ! [A: literal,B: literal,F: fun_literal_nat,C: nat] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( ( aa_literal_nat(F,B) = C )
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_literal_nat(F,X2)),aa_literal_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_literal_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2410_ord__less__eq__subst,axiom,
    ! [A: literal,B: literal,F: fun_literal_int,C: int] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( ( aa_literal_int(F,B) = C )
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_literal_int(F,X2)),aa_literal_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_literal_int(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2411_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_literal,C: literal] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_literal(F,B) = C )
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_rat_literal(F,X2)),aa_rat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_rat_literal(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2412_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2413_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2414_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2415_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_literal,C: literal] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( aa_nat_literal(F,B) = C )
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,X2)),aa_nat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2416_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2417_ord__eq__less__subst,axiom,
    ! [A: literal,F: fun_literal_literal,B: literal,C: literal] :
      ( ( A = aa_literal_literal(F,B) )
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_literal_literal(F,X2)),aa_literal_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_literal_literal(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2418_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_literal_rat,B: literal,C: literal] :
      ( ( A = aa_literal_rat(F,B) )
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_literal_rat(F,X2)),aa_literal_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_literal_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2419_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_literal_nat,B: literal,C: literal] :
      ( ( A = aa_literal_nat(F,B) )
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_literal_nat(F,X2)),aa_literal_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_literal_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2420_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_literal_int,B: literal,C: literal] :
      ( ( A = aa_literal_int(F,B) )
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),C)
       => ( ! [X2: literal,Y2: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_literal_int(F,X2)),aa_literal_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_literal_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2421_ord__eq__less__subst,axiom,
    ! [A: literal,F: fun_rat_literal,B: rat,C: rat] :
      ( ( A = aa_rat_literal(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_rat_literal(F,X2)),aa_rat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_rat_literal(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2422_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2423_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2424_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2425_ord__eq__less__subst,axiom,
    ! [A: literal,F: fun_nat_literal,B: nat,C: nat] :
      ( ( A = aa_nat_literal(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,X2)),aa_nat_literal(F,Y2)) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),aa_nat_literal(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2426_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2427_linorder__not__less,axiom,
    ! [X: literal,Y: literal] :
      ( ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
    <=> aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,Y),X) ) ).

% linorder_not_less
tff(fact_2428_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ).

% linorder_not_less
tff(fact_2429_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% linorder_not_less
tff(fact_2430_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% linorder_not_less
tff(fact_2431_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ).

% linorder_le_cases
tff(fact_2432_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% linorder_le_cases
tff(fact_2433_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% linorder_le_cases
tff(fact_2434_order__less__trans,axiom,
    ! [X: literal,Y: literal,Z: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Y),Z)
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Z) ) ) ).

% order_less_trans
tff(fact_2435_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z) ) ) ).

% order_less_trans
tff(fact_2436_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z) ) ) ).

% order_less_trans
tff(fact_2437_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z) ) ) ).

% order_less_trans
tff(fact_2438_order__less__asym_H,axiom,
    ! [A: literal,B: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),A) ) ).

% order_less_asym'
tff(fact_2439_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_2440_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_2441_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_2442_linorder__neq__iff,axiom,
    ! [X: literal,Y: literal] :
      ( ( X != Y )
    <=> ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
        | aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_2443_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_2444_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_2445_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_2446_order__less__asym,axiom,
    ! [X: literal,Y: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
     => ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Y),X) ) ).

% order_less_asym
tff(fact_2447_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% order_less_asym
tff(fact_2448_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% order_less_asym
tff(fact_2449_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% order_less_asym
tff(fact_2450_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2451_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2452_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2453_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2454_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_nat(F,B) = C )
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2455_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_int(F,B) = C )
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2456_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: fun_int_rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( aa_int_rat(F,B) = C )
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2457_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: fun_int_nat,C: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( aa_int_nat(F,B) = C )
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2458_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: fun_int_int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( aa_int_int(F,B) = C )
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2459_ord__le__eq__subst,axiom,
    ! [A: filter_nat,B: filter_nat,F: fun_filter_nat_rat,C: rat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
     => ( ( aa_filter_nat_rat(F,B) = C )
       => ( ! [X2: filter_nat,Y2: filter_nat] :
              ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_filter_nat_rat(F,X2)),aa_filter_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_filter_nat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2460_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2461_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2462_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2463_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2464_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C: nat] :
      ( ( A = aa_nat_nat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2465_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_nat_int,B: nat,C: nat] :
      ( ( A = aa_nat_int(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2466_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( ( A = aa_int_rat(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2467_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_int_nat,B: int,C: int] :
      ( ( A = aa_int_nat(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_int_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2468_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_int_int,B: int,C: int] :
      ( ( A = aa_int_int(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2469_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_filter_nat_rat,B: filter_nat,C: filter_nat] :
      ( ( A = aa_filter_nat_rat(F,B) )
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),C)
       => ( ! [X2: filter_nat,Y2: filter_nat] :
              ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_filter_nat_rat(F,X2)),aa_filter_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_filter_nat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2470_linorder__not__le,axiom,
    ! [X: literal,Y: literal] :
      ( ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,X),Y)
    <=> aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Y),X) ) ).

% linorder_not_le
tff(fact_2471_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% linorder_not_le
tff(fact_2472_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% linorder_not_le
tff(fact_2473_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% linorder_not_le
tff(fact_2474_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ).

% linorder_linear
tff(fact_2475_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% linorder_linear
tff(fact_2476_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% linorder_linear
tff(fact_2477_order__less__le,axiom,
    ! [X: literal,Y: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
    <=> ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2478_order__less__le,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2479_order__less__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2480_order__less__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2481_order__less__le,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2482_order__less__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2483_order__le__less,axiom,
    ! [X: literal,Y: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,X),Y)
    <=> ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2484_order__le__less,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2485_order__le__less,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2486_order__le__less,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2487_order__le__less,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2488_order__le__less,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2489_order__eq__refl,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( X = Y )
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y) ) ).

% order_eq_refl
tff(fact_2490_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y) ) ).

% order_eq_refl
tff(fact_2491_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% order_eq_refl
tff(fact_2492_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% order_eq_refl
tff(fact_2493_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% order_eq_refl
tff(fact_2494_linorder__neqE,axiom,
    ! [X: literal,Y: literal] :
      ( ( X != Y )
     => ( ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Y),X) ) ) ).

% linorder_neqE
tff(fact_2495_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ) ).

% linorder_neqE
tff(fact_2496_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_neqE
tff(fact_2497_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_neqE
tff(fact_2498_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2499_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: 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)),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2500_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: 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)),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2501_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: 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)),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2502_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,B)),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2503_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_int,C: int] :
      ( 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(F,B)),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2504_order__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,B)),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2505_order__subst2,axiom,
    ! [A: int,B: int,F: fun_int_nat,C: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,B)),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2506_order__subst2,axiom,
    ! [A: int,B: int,F: fun_int_int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,B)),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2507_order__subst2,axiom,
    ! [A: filter_nat,B: filter_nat,F: fun_filter_nat_rat,C: rat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_filter_nat_rat(F,B)),C)
       => ( ! [X2: filter_nat,Y2: filter_nat] :
              ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_filter_nat_rat(F,X2)),aa_filter_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_filter_nat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2508_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: 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),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2509_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: 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),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2510_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: 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),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2511_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: 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),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2512_order__subst1,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2513_order__subst1,axiom,
    ! [A: nat,F: fun_int_nat,B: int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_int_nat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_int_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2514_order__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_rat_int(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_rat_int(F,C)) ) ) ) ).

% order_subst1
tff(fact_2515_order__subst1,axiom,
    ! [A: int,F: fun_nat_int,B: nat,C: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(F,C)) ) ) ) ).

% order_subst1
tff(fact_2516_order__subst1,axiom,
    ! [A: int,F: fun_int_int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(F,C)) ) ) ) ).

% order_subst1
tff(fact_2517_order__subst1,axiom,
    ! [A: filter_nat,F: fun_rat_filter_nat,B: rat,C: rat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),aa_rat_filter_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_rat_filter_nat(F,X2)),aa_rat_filter_nat(F,Y2)) )
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),aa_rat_filter_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2518_Orderings_Oorder__eq__iff,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( A = B )
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
        & aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_2519_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_2520_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_2521_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_2522_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_2523_antisym,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_2524_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_2525_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_2526_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_2527_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_2528_dual__order_Ostrict__implies__order,axiom,
    ! [B: literal,A: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),A)
     => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_2529_dual__order_Ostrict__implies__order,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,B),A)
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_2530_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_2531_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_2532_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_2533_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_2534_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: literal,A: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_2535_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_2536_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_2537_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_2538_order_Ostrict__implies__order,axiom,
    ! [A: literal,B: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,A),B) ) ).

% order.strict_implies_order
tff(fact_2539_order_Ostrict__implies__order,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A),B)
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B) ) ).

% order.strict_implies_order
tff(fact_2540_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).

% order.strict_implies_order
tff(fact_2541_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_2542_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_2543_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_2544_dual__order_Ostrict__iff__not,axiom,
    ! [B: literal,A: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),A)
    <=> ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,B),A)
        & ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_2545_dual__order_Ostrict__iff__not,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,B),A)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
        & ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_2546_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
        & ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_2547_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_2548_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_2549_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_2550_dual__order_Ostrict__trans2,axiom,
    ! [B: literal,A: literal,C: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),A)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,C),B)
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2551_dual__order_Ostrict__trans2,axiom,
    ! [B: filter_nat,A: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,B),A)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),B)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2552_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2553_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2554_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: 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,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2555_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2556_dual__order_Ostrict__trans1,axiom,
    ! [B: literal,A: literal,C: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,B),A)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,C),B)
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2557_dual__order_Ostrict__trans1,axiom,
    ! [B: filter_nat,A: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,C),B)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2558_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2559_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2560_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: 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,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2561_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2562_order_Ostrict__implies__not__eq,axiom,
    ! [A: literal,B: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_2563_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_2564_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_2565_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_2566_dual__order_Ostrict__iff__order,axiom,
    ! [B: literal,A: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),A)
    <=> ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2567_dual__order_Ostrict__iff__order,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,B),A)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2568_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2569_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_2570_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_2571_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_2572_dual__order_Oorder__iff__strict,axiom,
    ! [B: literal,A: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,B),A)
    <=> ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2573_dual__order_Oorder__iff__strict,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2574_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2575_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_2576_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_2577_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_2578_dual__order_Ostrict__trans,axiom,
    ! [B: literal,A: literal,C: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),A)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,C),B)
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_2579_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_2580_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: 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,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_2581_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_2582_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Y)
             => 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,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_2583_dense__ge__bounded,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),X)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),W2) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_2584_not__less__iff__gr__or__eq,axiom,
    ! [X: literal,Y: literal] :
      ( ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
    <=> ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_2585_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_2586_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_2587_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_2588_order_Ostrict__iff__not,axiom,
    ! [A: literal,B: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
    <=> ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,A),B)
        & ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,B),A) ) ) ).

% order.strict_iff_not
tff(fact_2589_order_Ostrict__iff__not,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A),B)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
        & ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_2590_order_Ostrict__iff__not,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        & ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_2591_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_2592_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_2593_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_2594_order_Ostrict__trans2,axiom,
    ! [A: literal,B: literal,C: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,B),C)
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),C) ) ) ).

% order.strict_trans2
tff(fact_2595_order_Ostrict__trans2,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A),B)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),C)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A),C) ) ) ).

% order.strict_trans2
tff(fact_2596_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C) ) ) ).

% order.strict_trans2
tff(fact_2597_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans2
tff(fact_2598_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans2
tff(fact_2599_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans2
tff(fact_2600_order_Ostrict__trans1,axiom,
    ! [A: literal,B: literal,C: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,A),B)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),C)
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),C) ) ) ).

% order.strict_trans1
tff(fact_2601_order_Ostrict__trans1,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,B),C)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A),C) ) ) ).

% order.strict_trans1
tff(fact_2602_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C) ) ) ).

% order.strict_trans1
tff(fact_2603_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans1
tff(fact_2604_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans1
tff(fact_2605_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans1
tff(fact_2606_order_Ostrict__iff__order,axiom,
    ! [A: literal,B: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
    <=> ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2607_order_Ostrict__iff__order,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A),B)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2608_order_Ostrict__iff__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2609_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_2610_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_2611_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_2612_order_Oorder__iff__strict,axiom,
    ! [A: literal,B: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,A),B)
    <=> ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2613_order_Oorder__iff__strict,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2614_order_Oorder__iff__strict,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2615_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_2616_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_2617_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_2618_order_Ostrict__trans,axiom,
    ! [A: literal,B: literal,C: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),C)
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),C) ) ) ).

% order.strict_trans
tff(fact_2619_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans
tff(fact_2620_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans
tff(fact_2621_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans
tff(fact_2622_linorder__less__wlog,axiom,
    ! [P: fun_li4177005666941401710eral_o,A: literal,B: literal] :
      ( ! [A4: literal,B4: literal] :
          ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A4),B4)
         => aa_literal_o(aa_lit5986169069484365087eral_o(P,A4),B4) )
     => ( ! [A4: literal] : aa_literal_o(aa_lit5986169069484365087eral_o(P,A4),A4)
       => ( ! [A4: literal,B4: literal] :
              ( aa_literal_o(aa_lit5986169069484365087eral_o(P,B4),A4)
             => aa_literal_o(aa_lit5986169069484365087eral_o(P,A4),B4) )
         => aa_literal_o(aa_lit5986169069484365087eral_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_2623_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A4: rat,B4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A4),B4)
         => aa_rat_o(aa_rat_fun_rat_o(P,A4),B4) )
     => ( ! [A4: rat] : aa_rat_o(aa_rat_fun_rat_o(P,A4),A4)
       => ( ! [A4: rat,B4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(P,B4),A4)
             => aa_rat_o(aa_rat_fun_rat_o(P,A4),B4) )
         => aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_2624_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A4: nat,B4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4)
         => aa_nat_o(aa_nat_fun_nat_o(P,A4),B4) )
     => ( ! [A4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A4),A4)
       => ( ! [A4: nat,B4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,B4),A4)
             => aa_nat_o(aa_nat_fun_nat_o(P,A4),B4) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_2625_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A4: int,B4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4)
         => aa_int_o(aa_int_fun_int_o(P,A4),B4) )
     => ( ! [A4: int] : aa_int_o(aa_int_fun_int_o(P,A4),A4)
       => ( ! [A4: int,B4: int] :
              ( aa_int_o(aa_int_fun_int_o(P,B4),A4)
             => aa_int_o(aa_int_fun_int_o(P,A4),B4) )
         => aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_2626_exists__least__iff,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [N2: nat] :
          ( aa_nat_o(P,N2)
          & ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N2)
             => ~ aa_nat_o(P,M2) ) ) ) ).

% exists_least_iff
tff(fact_2627_dual__order_Oirrefl,axiom,
    ! [A: literal] : ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),A) ).

% dual_order.irrefl
tff(fact_2628_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% dual_order.irrefl
tff(fact_2629_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_2630_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_2631_dual__order_Otrans,axiom,
    ! [B: filter_nat,A: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),B)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),A) ) ) ).

% dual_order.trans
tff(fact_2632_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A) ) ) ).

% dual_order.trans
tff(fact_2633_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% dual_order.trans
tff(fact_2634_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: 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,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% dual_order.trans
tff(fact_2635_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% dual_order.trans
tff(fact_2636_dual__order_Oasym,axiom,
    ! [B: literal,A: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),A)
     => ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B) ) ).

% dual_order.asym
tff(fact_2637_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_2638_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_2639_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_2640_dual__order_Oantisym,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_2641_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_2642_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_2643_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_2644_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_2645_not__le__imp__less,axiom,
    ! [Y: literal,X: literal] :
      ( ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,Y),X)
     => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y) ) ).

% not_le_imp_less
tff(fact_2646_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% not_le_imp_less
tff(fact_2647_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).

% not_le_imp_less
tff(fact_2648_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ).

% not_le_imp_less
tff(fact_2649_less__le__not__le,axiom,
    ! [X: literal,Y: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
    <=> ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,X),Y)
        & ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,Y),X) ) ) ).

% less_le_not_le
tff(fact_2650_less__le__not__le,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
        & ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),X) ) ) ).

% less_le_not_le
tff(fact_2651_less__le__not__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
        & ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X) ) ) ).

% less_le_not_le
tff(fact_2652_less__le__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ) ).

% less_le_not_le
tff(fact_2653_less__le__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ) ).

% less_le_not_le
tff(fact_2654_less__le__not__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ) ).

% less_le_not_le
tff(fact_2655_dual__order_Oeq__iff,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( A = B )
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
        & aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_2656_dual__order_Oeq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_2657_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_2658_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_2659_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_2660_linorder__cases,axiom,
    ! [X: literal,Y: literal] :
      ( ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
     => ( ( X != Y )
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Y),X) ) ) ).

% linorder_cases
tff(fact_2661_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( ( X != Y )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ) ).

% linorder_cases
tff(fact_2662_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( ( X != Y )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_cases
tff(fact_2663_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( ( X != Y )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_cases
tff(fact_2664_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_le
tff(fact_2665_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X2) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_ge
tff(fact_2666_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A4: rat,B4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A4),B4)
         => aa_rat_o(aa_rat_fun_rat_o(P,A4),B4) )
     => ( ! [A4: rat,B4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(P,B4),A4)
           => aa_rat_o(aa_rat_fun_rat_o(P,A4),B4) )
       => aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_2667_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A4: nat,B4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A4),B4)
         => aa_nat_o(aa_nat_fun_nat_o(P,A4),B4) )
     => ( ! [A4: nat,B4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,B4),A4)
           => aa_nat_o(aa_nat_fun_nat_o(P,A4),B4) )
       => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_2668_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A4: int,B4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A4),B4)
         => aa_int_o(aa_int_fun_int_o(P,A4),B4) )
     => ( ! [A4: int,B4: int] :
            ( aa_int_o(aa_int_fun_int_o(P,B4),A4)
           => aa_int_o(aa_int_fun_int_o(P,A4),B4) )
       => aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_2669_antisym__conv3,axiom,
    ! [Y: literal,X: literal] :
      ( ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Y),X)
     => ( ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_2670_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_2671_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_2672_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_2673_less__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [X2: nat] :
          ( ! [Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),X2)
             => aa_nat_o(P,Y4) )
         => aa_nat_o(P,X2) )
     => aa_nat_o(P,A) ) ).

% less_induct
tff(fact_2674_ord__less__eq__trans,axiom,
    ! [A: literal,B: literal,C: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ( ( B = C )
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_2675_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_2676_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_2677_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_2678_ord__eq__less__trans,axiom,
    ! [A: literal,B: literal,C: literal] :
      ( ( A = B )
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),C)
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_2679_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_2680_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_2681_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_2682_order__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),Z)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Z) ) ) ).

% order_trans
tff(fact_2683_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z) ) ) ).

% order_trans
tff(fact_2684_order__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z) ) ) ).

% order_trans
tff(fact_2685_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z) ) ) ).

% order_trans
tff(fact_2686_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z) ) ) ).

% order_trans
tff(fact_2687_order_Otrans,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),C)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),C) ) ) ).

% order.trans
tff(fact_2688_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).

% order.trans
tff(fact_2689_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% order.trans
tff(fact_2690_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% order.trans
tff(fact_2691_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% order.trans
tff(fact_2692_order_Oasym,axiom,
    ! [A: literal,B: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
     => ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),A) ) ).

% order.asym
tff(fact_2693_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_2694_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_2695_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_2696_order__antisym,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_2697_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_2698_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_2699_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_2700_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_2701_antisym__conv2,axiom,
    ! [X: literal,Y: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,X),Y)
     => ( ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2702_antisym__conv2,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
     => ( ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2703_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2704_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2705_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2706_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2707_antisym__conv1,axiom,
    ! [X: literal,Y: literal] :
      ( ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2708_antisym__conv1,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2709_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2710_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2711_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2712_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2713_ord__le__eq__trans,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
     => ( ( B = C )
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_2714_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( ( B = C )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_2715_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_2716_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_2717_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_2718_ord__eq__le__trans,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( ( A = B )
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),C)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_2719_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( A = B )
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_2720_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_2721_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_2722_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_2723_order__class_Oorder__eq__iff,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( X = Y )
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
        & aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_2724_order__class_Oorder__eq__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_2725_order__class_Oorder__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_2726_order__class_Oorder__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_2727_order__class_Oorder__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_2728_less__imp__neq,axiom,
    ! [X: literal,Y: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_2729_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_2730_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_2731_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_2732_le__cases3,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
       => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) )
     => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z)
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y) )
         => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y)
             => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) )
           => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
               => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),X) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),X)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_2733_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
       => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
               => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_2734_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
       => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_2735_dense,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ? [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),Y) ) ) ).

% dense
tff(fact_2736_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_2737_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_2738_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_2739_nless__le,axiom,
    ! [A: literal,B: literal] :
      ( ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B)
    <=> ( ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2740_nless__le,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A),B)
    <=> ( ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2741_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
    <=> ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2742_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_2743_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_2744_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_2745_gt__ex,axiom,
    ! [X: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X_1) ).

% gt_ex
tff(fact_2746_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X_1) ).

% gt_ex
tff(fact_2747_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,X),X_1) ).

% gt_ex
tff(fact_2748_lt__ex,axiom,
    ! [X: rat] :
    ? [Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X) ).

% lt_ex
tff(fact_2749_lt__ex,axiom,
    ! [X: int] :
    ? [Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X) ).

% lt_ex
tff(fact_2750_leI,axiom,
    ! [X: literal,Y: literal] :
      ( ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y)
     => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,Y),X) ) ).

% leI
tff(fact_2751_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ).

% leI
tff(fact_2752_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% leI
tff(fact_2753_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% leI
tff(fact_2754_leD,axiom,
    ! [Y: literal,X: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,Y),X)
     => ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Y) ) ).

% leD
tff(fact_2755_leD,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),X)
     => ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y) ) ).

% leD
tff(fact_2756_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y) ) ).

% leD
tff(fact_2757_leD,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% leD
tff(fact_2758_leD,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).

% leD
tff(fact_2759_leD,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ).

% leD
tff(fact_2760_pred__subset__eq2,axiom,
    ! [R: set_Pr3486312109254157132nteger,S: set_Pr3486312109254157132nteger] :
      ( ord_le4825817444062779351eger_o(aTP_Lamm_dk(R),aTP_Lamm_dk(S))
    <=> ord_le2932375828722672556nteger(R,S) ) ).

% pred_subset_eq2
tff(fact_2761_pred__subset__eq2,axiom,
    ! [R: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat] :
      ( ord_le3580378406483493347_nat_o(aTP_Lamm_dl(R),aTP_Lamm_dl(S))
    <=> ord_le3919382008627836342et_nat(R,S) ) ).

% pred_subset_eq2
tff(fact_2762_pred__subset__eq2,axiom,
    ! [R: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat] :
      ( ord_le2834457679758710341_nat_o(aTP_Lamm_dm(R),aTP_Lamm_dm(S))
    <=> ord_le5189643068899110356et_nat(R,S) ) ).

% pred_subset_eq2
tff(fact_2763_pred__subset__eq2,axiom,
    ! [R: set_Pr6700895561065273913nt_int,S: set_Pr6700895561065273913nt_int] :
      ( ord_le4806730579060245290_int_o(aTP_Lamm_dn(R),aTP_Lamm_dn(S))
    <=> ord_le3990897393281686169nt_int(R,S) ) ).

% pred_subset_eq2
tff(fact_2764_pred__subset__eq2,axiom,
    ! [R: set_Pr4166680992137330533nt_int,S: set_Pr4166680992137330533nt_int] :
      ( ord_le2876595303158420734_int_o(aTP_Lamm_do(R),aTP_Lamm_do(S))
    <=> ord_le3197784583289274309nt_int(R,S) ) ).

% pred_subset_eq2
tff(fact_2765_mult__less__le__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2766_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2767_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2768_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2769_mult__le__less__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2770_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2771_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2772_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2773_mult__right__le__imp__le,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2774_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2775_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2776_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2777_mult__left__le__imp__le,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2778_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2779_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2780_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2781_mult__le__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2782_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2783_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2784_mult__le__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2785_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2786_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2787_mult__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2788_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2789_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2790_mult__strict__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2791_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2792_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2793_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2794_mult__right__less__imp__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2795_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2796_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2797_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2798_mult__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2799_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2800_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2801_mult__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2802_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2803_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2804_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2805_mult__left__less__imp__less,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2806_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2807_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2808_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2809_mult__le__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2810_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2811_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2812_mult__le__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2813_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2814_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2815_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
    <=> ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_2816_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_2817_not__less__zero,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),zero_zero_nat) ).

% not_less_zero
tff(fact_2818_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% gr_zeroI
tff(fact_2819_zero__le,axiom,
    ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X) ).

% zero_le
tff(fact_2820_less__numeral__extra_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),one_one_Code_integer) ).

% less_numeral_extra(4)
tff(fact_2821_less__numeral__extra_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),one_one_rat) ).

% less_numeral_extra(4)
tff(fact_2822_less__numeral__extra_I4_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),one_one_nat) ).

% less_numeral_extra(4)
tff(fact_2823_less__numeral__extra_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),one_one_int) ).

% less_numeral_extra(4)
tff(fact_2824_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) ) ).

% diff_strict_right_mono
tff(fact_2825_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_strict_right_mono
tff(fact_2826_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B)) ) ).

% diff_strict_left_mono
tff(fact_2827_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_strict_left_mono
tff(fact_2828_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D2) ) ) ).

% diff_eq_diff_less
tff(fact_2829_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2) ) ) ).

% diff_eq_diff_less
tff(fact_2830_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D2)) ) ) ).

% diff_strict_mono
tff(fact_2831_diff__strict__mono,axiom,
    ! [A: int,B: int,D2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)) ) ) ).

% diff_strict_mono
tff(fact_2832_bot_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),bot_bo2099793752762293965at_nat) ).

% bot.extremum_strict
tff(fact_2833_bot_Oextremum__strict,axiom,
    ! [A: set_o] : ~ aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,A),bot_bot_set_o) ).

% bot.extremum_strict
tff(fact_2834_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_2835_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_2836_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_2837_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,bot_bo2099793752762293965at_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_2838_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,bot_bot_set_o),A) ) ).

% bot.not_eq_extremum
tff(fact_2839_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_2840_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,bot_bot_set_int),A) ) ).

% bot.not_eq_extremum
tff(fact_2841_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_2842_not__psubset__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A2),bot_bo2099793752762293965at_nat) ).

% not_psubset_empty
tff(fact_2843_not__psubset__empty,axiom,
    ! [A2: set_o] : ~ aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,A2),bot_bot_set_o) ).

% not_psubset_empty
tff(fact_2844_not__psubset__empty,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_2845_not__psubset__empty,axiom,
    ! [A2: set_int] : ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A2),bot_bot_set_int) ).

% not_psubset_empty
tff(fact_2846_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_2847_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_2848_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_2849_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_2850_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_2851_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_2852_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_2853_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_2854_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_2855_compl__less__swap2,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( aa_Product_unit_o(ord_le361264281704409273t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)),X)
     => aa_Product_unit_o(ord_le361264281704409273t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),Y) ) ).

% compl_less_swap2
tff(fact_2856_compl__less__swap1,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( aa_Product_unit_o(ord_le361264281704409273t_unit(Y),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X))
     => aa_Product_unit_o(ord_le361264281704409273t_unit(X),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)) ) ).

% compl_less_swap1
tff(fact_2857_le__numeral__extra_I4_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),one_one_Code_integer) ).

% le_numeral_extra(4)
tff(fact_2858_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_2859_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_2860_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_2861_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2862_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2863_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) ) ).

% diff_right_mono
tff(fact_2864_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_right_mono
tff(fact_2865_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B)) ) ).

% diff_left_mono
tff(fact_2866_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_left_mono
tff(fact_2867_diff__mono,axiom,
    ! [A: rat,B: rat,D2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D2)) ) ) ).

% diff_mono
tff(fact_2868_diff__mono,axiom,
    ! [A: int,B: int,D2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)) ) ) ).

% diff_mono
tff(fact_2869_subrelI,axiom,
    ! [R2: set_Pr3486312109254157132nteger,S2: set_Pr3486312109254157132nteger] :
      ( ! [X2: fun_Pr5098669352780829690e_term,Y2: produc8923325533196201883nteger] :
          ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X2),Y2),R2)
         => member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X2),Y2),S2) )
     => ord_le2932375828722672556nteger(R2,S2) ) ).

% subrelI
tff(fact_2870_subrelI,axiom,
    ! [R2: set_Pr1312406793078187350et_nat,S2: set_Pr1312406793078187350et_nat] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] :
          ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X2),Y2),R2)
         => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X2),Y2),S2) )
     => ord_le3919382008627836342et_nat(R2,S2) ) ).

% subrelI
tff(fact_2871_subrelI,axiom,
    ! [R2: set_Pr4040394263813967220et_nat,S2: set_Pr4040394263813967220et_nat] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat] :
          ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X2),Y2),R2)
         => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X2),Y2),S2) )
     => ord_le5189643068899110356et_nat(R2,S2) ) ).

% subrelI
tff(fact_2872_subrelI,axiom,
    ! [R2: set_Pr6700895561065273913nt_int,S2: set_Pr6700895561065273913nt_int] :
      ( ! [X2: fun_Pr7763680713840462443e_term,Y2: product_prod_int_int] :
          ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X2),Y2),R2)
         => member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X2),Y2),S2) )
     => ord_le3990897393281686169nt_int(R2,S2) ) ).

% subrelI
tff(fact_2873_subrelI,axiom,
    ! [R2: set_Pr4166680992137330533nt_int,S2: set_Pr4166680992137330533nt_int] :
      ( ! [X2: fun_in5730379987311621695e_term,Y2: product_prod_int_int] :
          ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X2),Y2),R2)
         => member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X2),Y2),S2) )
     => ord_le3197784583289274309nt_int(R2,S2) ) ).

% subrelI
tff(fact_2874_bot_Oextremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,bot_bo2099793752762293965at_nat),A) ).

% bot.extremum
tff(fact_2875_bot_Oextremum,axiom,
    ! [A: set_o] : aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,bot_bot_set_o),A) ).

% bot.extremum
tff(fact_2876_bot_Oextremum,axiom,
    ! [A: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,bot_bot_set_int),A) ).

% bot.extremum
tff(fact_2877_bot_Oextremum,axiom,
    ! [A: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,bot_bot_filter_nat),A) ).

% bot.extremum
tff(fact_2878_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_2879_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_2880_bot_Oextremum__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),bot_bo2099793752762293965at_nat)
    <=> ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
tff(fact_2881_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),bot_bot_set_o)
    <=> ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
tff(fact_2882_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),bot_bot_set_int)
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_2883_bot_Oextremum__unique,axiom,
    ! [A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),bot_bot_filter_nat)
    <=> ( A = bot_bot_filter_nat ) ) ).

% bot.extremum_unique
tff(fact_2884_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_2885_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_2886_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),bot_bo2099793752762293965at_nat)
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_2887_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),bot_bot_set_o)
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
tff(fact_2888_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),bot_bot_set_int)
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_2889_bot_Oextremum__uniqueI,axiom,
    ! [A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),bot_bot_filter_nat)
     => ( A = bot_bot_filter_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_2890_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_2891_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_2892_top_Onot__eq__extremum,axiom,
    ! [A: set_list_nat] :
      ( ( A != top_top_set_list_nat )
    <=> aa_set_list_nat_o(aa_set8027019173009287559_nat_o(ord_le1190675801316882794st_nat,A),top_top_set_list_nat) ) ).

% top.not_eq_extremum
tff(fact_2893_top_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != top_top_set_o )
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,A),top_top_set_o) ) ).

% top.not_eq_extremum
tff(fact_2894_top_Onot__eq__extremum,axiom,
    ! [A: set_rat] :
      ( ( A != top_top_set_rat )
    <=> aa_set_rat_o(aa_set229218963967607191_rat_o(ord_less_set_rat,A),top_top_set_rat) ) ).

% top.not_eq_extremum
tff(fact_2895_top_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != top_top_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),top_top_set_nat) ) ).

% top.not_eq_extremum
tff(fact_2896_top_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != top_top_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),top_top_set_int) ) ).

% top.not_eq_extremum
tff(fact_2897_top_Oextremum__strict,axiom,
    ! [A: set_list_nat] : ~ aa_set_list_nat_o(aa_set8027019173009287559_nat_o(ord_le1190675801316882794st_nat,top_top_set_list_nat),A) ).

% top.extremum_strict
tff(fact_2898_top_Oextremum__strict,axiom,
    ! [A: set_o] : ~ aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,top_top_set_o),A) ).

% top.extremum_strict
tff(fact_2899_top_Oextremum__strict,axiom,
    ! [A: set_rat] : ~ aa_set_rat_o(aa_set229218963967607191_rat_o(ord_less_set_rat,top_top_set_rat),A) ).

% top.extremum_strict
tff(fact_2900_top_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,top_top_set_nat),A) ).

% top.extremum_strict
tff(fact_2901_top_Oextremum__strict,axiom,
    ! [A: set_int] : ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,top_top_set_int),A) ).

% top.extremum_strict
tff(fact_2902_less__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% less_infI1
tff(fact_2903_less__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% less_infI1
tff(fact_2904_less__infI1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),X)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),X) ) ).

% less_infI1
tff(fact_2905_less__infI1,axiom,
    ! [A: fun_list_char_o,X: fun_list_char_o,B: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le8755527016964500988char_o,A),X)
     => aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le8755527016964500988char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)),X) ) ).

% less_infI1
tff(fact_2906_less__infI1,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X) ) ).

% less_infI1
tff(fact_2907_less__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% less_infI1
tff(fact_2908_less__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% less_infI1
tff(fact_2909_less__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% less_infI2
tff(fact_2910_less__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% less_infI2
tff(fact_2911_less__infI2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,B),X)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),X) ) ).

% less_infI2
tff(fact_2912_less__infI2,axiom,
    ! [B: fun_list_char_o,X: fun_list_char_o,A: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le8755527016964500988char_o,B),X)
     => aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le8755527016964500988char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)),X) ) ).

% less_infI2
tff(fact_2913_less__infI2,axiom,
    ! [B: rat,X: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X) ) ).

% less_infI2
tff(fact_2914_less__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% less_infI2
tff(fact_2915_less__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% less_infI2
tff(fact_2916_inf_Oabsorb3,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),B)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_2917_inf_Oabsorb3,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_2918_inf_Oabsorb3,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),B)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = A ) ) ).

% inf.absorb3
tff(fact_2919_inf_Oabsorb3,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le8755527016964500988char_o,A),B)
     => ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) = A ) ) ).

% inf.absorb3
tff(fact_2920_inf_Oabsorb3,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_2921_inf_Oabsorb3,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(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_2922_inf_Oabsorb3,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(inf_inf_int,A),B) = A ) ) ).

% inf.absorb3
tff(fact_2923_inf_Oabsorb4,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B),A)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_2924_inf_Oabsorb4,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_2925_inf_Oabsorb4,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,B),A)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = B ) ) ).

% inf.absorb4
tff(fact_2926_inf_Oabsorb4,axiom,
    ! [B: fun_list_char_o,A: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le8755527016964500988char_o,B),A)
     => ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) = B ) ) ).

% inf.absorb4
tff(fact_2927_inf_Oabsorb4,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_2928_inf_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_2929_inf_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).

% inf.absorb4
tff(fact_2930_inf_Ostrict__boundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C))
     => ~ ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),B)
         => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_2931_inf_Ostrict__boundedE,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_2932_inf_Ostrict__boundedE,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C))
     => ~ ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),B)
         => ~ aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_2933_inf_Ostrict__boundedE,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o,C: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le8755527016964500988char_o,A),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,B),C))
     => ~ ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le8755527016964500988char_o,A),B)
         => ~ aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le8755527016964500988char_o,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_2934_inf_Ostrict__boundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C))
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_2935_inf_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_2936_inf_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_2937_inf_Ostrict__order__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),B)
    <=> ( ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_2938_inf_Ostrict__order__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
    <=> ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_2939_inf_Ostrict__order__iff,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),B)
    <=> ( ( A = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_2940_inf_Ostrict__order__iff,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le8755527016964500988char_o,A),B)
    <=> ( ( A = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_2941_inf_Ostrict__order__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_2942_inf_Ostrict__order__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_2943_inf_Ostrict__order__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_2944_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),C)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_2945_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_2946_inf_Ostrict__coboundedI1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),C)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_2947_inf_Ostrict__coboundedI1,axiom,
    ! [A: fun_list_char_o,C: fun_list_char_o,B: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le8755527016964500988char_o,A),C)
     => aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le8755527016964500988char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_2948_inf_Ostrict__coboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_2949_inf_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_2950_inf_Ostrict__coboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_2951_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B),C)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_2952_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),C)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_2953_inf_Ostrict__coboundedI2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,B),C)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_2954_inf_Ostrict__coboundedI2,axiom,
    ! [B: fun_list_char_o,C: fun_list_char_o,A: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le8755527016964500988char_o,B),C)
     => aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le8755527016964500988char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_2955_inf_Ostrict__coboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_2956_inf_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_2957_inf_Ostrict__coboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_2958_less__supI1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(X,A)
     => ord_le7642048601412989811at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% less_supI1
tff(fact_2959_less__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(X),A)
     => aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% less_supI1
tff(fact_2960_less__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% less_supI1
tff(fact_2961_less__supI1,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% less_supI1
tff(fact_2962_less__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% less_supI1
tff(fact_2963_less__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% less_supI1
tff(fact_2964_less__supI2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(X,B)
     => ord_le7642048601412989811at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% less_supI2
tff(fact_2965_less__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(X),B)
     => aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% less_supI2
tff(fact_2966_less__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% less_supI2
tff(fact_2967_less__supI2,axiom,
    ! [X: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% less_supI2
tff(fact_2968_less__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% less_supI2
tff(fact_2969_less__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% less_supI2
tff(fact_2970_sup_Oabsorb3,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(B,A)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_2971_sup_Oabsorb3,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(B),A)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_2972_sup_Oabsorb3,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_2973_sup_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_2974_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_2975_sup_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb3
tff(fact_2976_sup_Oabsorb4,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(A,B)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_2977_sup_Oabsorb4,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(A),B)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_2978_sup_Oabsorb4,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_2979_sup_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_2980_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_2981_sup_Oabsorb4,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(sup_sup_int,A),B) = B ) ) ).

% sup.absorb4
tff(fact_2982_sup_Ostrict__boundedE,axiom,
    ! [B: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),C),A)
     => ~ ( ord_le7642048601412989811at_nat(B,A)
         => ~ ord_le7642048601412989811at_nat(C,A) ) ) ).

% sup.strict_boundedE
tff(fact_2983_sup_Ostrict__boundedE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)),A)
     => ~ ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(B),A)
         => ~ aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C),A) ) ) ).

% sup.strict_boundedE
tff(fact_2984_sup_Ostrict__boundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_2985_sup_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_2986_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_2987_sup_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_2988_sup_Ostrict__order__iff,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(B,A)
    <=> ( ( A = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_2989_sup_Ostrict__order__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(B),A)
    <=> ( ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_2990_sup_Ostrict__order__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
    <=> ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_2991_sup_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_2992_sup_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_2993_sup_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_2994_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(C,A)
     => ord_le7642048601412989811at_nat(C,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_2995_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C),A)
     => aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_2996_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_2997_sup_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_2998_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_2999_sup_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_3000_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(C,B)
     => ord_le7642048601412989811at_nat(C,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_3001_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C),B)
     => aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_3002_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_3003_sup_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_3004_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_3005_sup_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_3006_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_3007_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_3008_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_3009_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_3010_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_3011_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_3012_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_3013_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_3014_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_3015_compl__le__swap2,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)),X)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),Y) ) ).

% compl_le_swap2
tff(fact_3016_compl__le__swap2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),Y) ) ).

% compl_le_swap2
tff(fact_3017_compl__le__swap1,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( aa_Product_unit_o(ord_le3221252021190050221t_unit(Y),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X))
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(X),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)) ) ).

% compl_le_swap1
tff(fact_3018_compl__le__swap1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(uminus5710092332889474511et_nat,X))
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ) ).

% compl_le_swap1
tff(fact_3019_compl__mono,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( aa_Product_unit_o(ord_le3221252021190050221t_unit(X),Y)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)) ) ).

% compl_mono
tff(fact_3020_compl__mono,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)) ) ).

% compl_mono
tff(fact_3021_subset__minus__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),B2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2) = bot_bo2099793752762293965at_nat ) ) ).

% subset_minus_empty
tff(fact_3022_subset__minus__empty,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),B2)
     => ( minus_minus_set_o(A2,B2) = bot_bot_set_o ) ) ).

% subset_minus_empty
tff(fact_3023_subset__minus__empty,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),B2)
     => ( minus_minus_set_int(A2,B2) = bot_bot_set_int ) ) ).

% subset_minus_empty
tff(fact_3024_subset__minus__empty,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = bot_bot_set_nat ) ) ).

% subset_minus_empty
tff(fact_3025_top_Oextremum__uniqueI,axiom,
    ! [A: set_list_nat] :
      ( aa_set_list_nat_o(aa_set8027019173009287559_nat_o(ord_le6045566169113846134st_nat,top_top_set_list_nat),A)
     => ( A = top_top_set_list_nat ) ) ).

% top.extremum_uniqueI
tff(fact_3026_top_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,top_top_set_o),A)
     => ( A = top_top_set_o ) ) ).

% top.extremum_uniqueI
tff(fact_3027_top_Oextremum__uniqueI,axiom,
    ! [A: set_rat] :
      ( aa_set_rat_o(aa_set229218963967607191_rat_o(ord_less_eq_set_rat,top_top_set_rat),A)
     => ( A = top_top_set_rat ) ) ).

% top.extremum_uniqueI
tff(fact_3028_top_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,top_top_set_int),A)
     => ( A = top_top_set_int ) ) ).

% top.extremum_uniqueI
tff(fact_3029_top_Oextremum__uniqueI,axiom,
    ! [A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,top_top_filter_nat),A)
     => ( A = top_top_filter_nat ) ) ).

% top.extremum_uniqueI
tff(fact_3030_top_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,top_top_set_nat),A)
     => ( A = top_top_set_nat ) ) ).

% top.extremum_uniqueI
tff(fact_3031_top_Oextremum__unique,axiom,
    ! [A: set_list_nat] :
      ( aa_set_list_nat_o(aa_set8027019173009287559_nat_o(ord_le6045566169113846134st_nat,top_top_set_list_nat),A)
    <=> ( A = top_top_set_list_nat ) ) ).

% top.extremum_unique
tff(fact_3032_top_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,top_top_set_o),A)
    <=> ( A = top_top_set_o ) ) ).

% top.extremum_unique
tff(fact_3033_top_Oextremum__unique,axiom,
    ! [A: set_rat] :
      ( aa_set_rat_o(aa_set229218963967607191_rat_o(ord_less_eq_set_rat,top_top_set_rat),A)
    <=> ( A = top_top_set_rat ) ) ).

% top.extremum_unique
tff(fact_3034_top_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,top_top_set_int),A)
    <=> ( A = top_top_set_int ) ) ).

% top.extremum_unique
tff(fact_3035_top_Oextremum__unique,axiom,
    ! [A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,top_top_filter_nat),A)
    <=> ( A = top_top_filter_nat ) ) ).

% top.extremum_unique
tff(fact_3036_top_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,top_top_set_nat),A)
    <=> ( A = top_top_set_nat ) ) ).

% top.extremum_unique
tff(fact_3037_top__greatest,axiom,
    ! [A: set_list_nat] : aa_set_list_nat_o(aa_set8027019173009287559_nat_o(ord_le6045566169113846134st_nat,A),top_top_set_list_nat) ).

% top_greatest
tff(fact_3038_top__greatest,axiom,
    ! [A: set_o] : aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),top_top_set_o) ).

% top_greatest
tff(fact_3039_top__greatest,axiom,
    ! [A: set_rat] : aa_set_rat_o(aa_set229218963967607191_rat_o(ord_less_eq_set_rat,A),top_top_set_rat) ).

% top_greatest
tff(fact_3040_top__greatest,axiom,
    ! [A: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),top_top_set_int) ).

% top_greatest
tff(fact_3041_top__greatest,axiom,
    ! [A: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),top_top_filter_nat) ).

% top_greatest
tff(fact_3042_top__greatest,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),top_top_set_nat) ).

% top_greatest
tff(fact_3043_inf__sup__ord_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_3044_inf__sup__ord_I2_J,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_3045_inf__sup__ord_I2_J,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_3046_inf__sup__ord_I2_J,axiom,
    ! [X: filter_nat,Y: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_3047_inf__sup__ord_I2_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_3048_inf__sup__ord_I2_J,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_3049_inf__sup__ord_I2_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_3050_inf__sup__ord_I2_J,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_3051_inf__sup__ord_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_3052_inf__sup__ord_I1_J,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_3053_inf__sup__ord_I1_J,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_3054_inf__sup__ord_I1_J,axiom,
    ! [X: filter_nat,Y: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_3055_inf__sup__ord_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_3056_inf__sup__ord_I1_J,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_3057_inf__sup__ord_I1_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_3058_inf__sup__ord_I1_J,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_3059_inf__le1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),X) ).

% inf_le1
tff(fact_3060_inf__le1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),X) ).

% inf_le1
tff(fact_3061_inf__le1,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)),X) ).

% inf_le1
tff(fact_3062_inf__le1,axiom,
    ! [X: filter_nat,Y: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y)),X) ).

% inf_le1
tff(fact_3063_inf__le1,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),X) ).

% inf_le1
tff(fact_3064_inf__le1,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),X) ).

% inf_le1
tff(fact_3065_inf__le1,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),X) ).

% inf_le1
tff(fact_3066_inf__le1,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),X) ).

% inf_le1
tff(fact_3067_inf__le2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y) ).

% inf_le2
tff(fact_3068_inf__le2,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Y) ).

% inf_le2
tff(fact_3069_inf__le2,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] : aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)),Y) ).

% inf_le2
tff(fact_3070_inf__le2,axiom,
    ! [X: filter_nat,Y: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y)),Y) ).

% inf_le2
tff(fact_3071_inf__le2,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y) ).

% inf_le2
tff(fact_3072_inf__le2,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),Y) ).

% inf_le2
tff(fact_3073_inf__le2,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y) ).

% inf_le2
tff(fact_3074_inf__le2,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),Y) ).

% inf_le2
tff(fact_3075_le__infE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B))
     => ~ ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),A)
         => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),B) ) ) ).

% le_infE
tff(fact_3076_le__infE,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B))
     => ~ ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),A)
         => ~ aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),B) ) ) ).

% le_infE
tff(fact_3077_le__infE,axiom,
    ! [X: fun_list_char_o,A: fun_list_char_o,B: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B))
     => ~ ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),A)
         => ~ aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),B) ) ) ).

% le_infE
tff(fact_3078_le__infE,axiom,
    ! [X: filter_nat,A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B))
     => ~ ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),A)
         => ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),B) ) ) ).

% le_infE
tff(fact_3079_le__infE,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B))
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B) ) ) ).

% le_infE
tff(fact_3080_le__infE,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B))
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),B) ) ) ).

% le_infE
tff(fact_3081_le__infE,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B) ) ) ).

% le_infE
tff(fact_3082_le__infE,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B) ) ) ).

% le_infE
tff(fact_3083_le__infI,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),A)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),B)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ) ) ).

% le_infI
tff(fact_3084_le__infI,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),A)
     => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),B)
       => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)) ) ) ).

% le_infI
tff(fact_3085_le__infI,axiom,
    ! [X: fun_list_char_o,A: fun_list_char_o,B: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),A)
     => ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),B)
       => aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)) ) ) ).

% le_infI
tff(fact_3086_le__infI,axiom,
    ! [X: filter_nat,A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),A)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),B)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B)) ) ) ).

% le_infI
tff(fact_3087_le__infI,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% le_infI
tff(fact_3088_le__infI,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)) ) ) ).

% le_infI
tff(fact_3089_le__infI,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).

% le_infI
tff(fact_3090_le__infI,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)) ) ) ).

% le_infI
tff(fact_3091_inf__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),D2)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C),D2)) ) ) ).

% inf_mono
tff(fact_3092_inf__mono,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,D2: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C)
     => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),D2)
       => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,C),D2)) ) ) ).

% inf_mono
tff(fact_3093_inf__mono,axiom,
    ! [A: fun_list_char_o,C: fun_list_char_o,B: fun_list_char_o,D2: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),C)
     => ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,B),D2)
       => aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,C),D2)) ) ) ).

% inf_mono
tff(fact_3094_inf__mono,axiom,
    ! [A: filter_nat,C: filter_nat,B: filter_nat,D2: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),C)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),D2)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,C),D2)) ) ) ).

% inf_mono
tff(fact_3095_inf__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),D2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C),D2)) ) ) ).

% inf_mono
tff(fact_3096_inf__mono,axiom,
    ! [A: rat,C: rat,B: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,C),D2)) ) ) ).

% inf_mono
tff(fact_3097_inf__mono,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,C),D2)) ) ) ).

% inf_mono
tff(fact_3098_inf__mono,axiom,
    ! [A: int,C: int,B: int,D2: 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,B),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),aa_int_int(aa_int_fun_int_int(inf_inf_int,C),D2)) ) ) ).

% inf_mono
tff(fact_3099_le__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% le_infI1
tff(fact_3100_le__infI1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),X)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),X) ) ).

% le_infI1
tff(fact_3101_le__infI1,axiom,
    ! [A: fun_list_char_o,X: fun_list_char_o,B: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),X)
     => aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)),X) ) ).

% le_infI1
tff(fact_3102_le__infI1,axiom,
    ! [A: filter_nat,X: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),X)
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B)),X) ) ).

% le_infI1
tff(fact_3103_le__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% le_infI1
tff(fact_3104_le__infI1,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X) ) ).

% le_infI1
tff(fact_3105_le__infI1,axiom,
    ! [A: nat,X: nat,B: 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_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% le_infI1
tff(fact_3106_le__infI1,axiom,
    ! [A: int,X: int,B: 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_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% le_infI1
tff(fact_3107_le__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% le_infI2
tff(fact_3108_le__infI2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),X)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),X) ) ).

% le_infI2
tff(fact_3109_le__infI2,axiom,
    ! [B: fun_list_char_o,X: fun_list_char_o,A: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,B),X)
     => aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)),X) ) ).

% le_infI2
tff(fact_3110_le__infI2,axiom,
    ! [B: filter_nat,X: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),X)
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B)),X) ) ).

% le_infI2
tff(fact_3111_le__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% le_infI2
tff(fact_3112_le__infI2,axiom,
    ! [B: rat,X: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X) ) ).

% le_infI2
tff(fact_3113_le__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% le_infI2
tff(fact_3114_le__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% le_infI2
tff(fact_3115_inf_OorderE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ).

% inf.orderE
tff(fact_3116_inf_OorderE,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
     => ( A = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) ) ) ).

% inf.orderE
tff(fact_3117_inf_OorderE,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),B)
     => ( A = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) ) ) ).

% inf.orderE
tff(fact_3118_inf_OorderE,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
     => ( A = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B) ) ) ).

% inf.orderE
tff(fact_3119_inf_OorderE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% inf.orderE
tff(fact_3120_inf_OorderE,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) ) ) ).

% inf.orderE
tff(fact_3121_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).

% inf.orderE
tff(fact_3122_inf_OorderE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).

% inf.orderE
tff(fact_3123_inf_OorderI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B) ) ).

% inf.orderI
tff(fact_3124_inf_OorderI,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( ( A = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) )
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B) ) ).

% inf.orderI
tff(fact_3125_inf_OorderI,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o] :
      ( ( A = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) )
     => aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),B) ) ).

% inf.orderI
tff(fact_3126_inf_OorderI,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( A = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B) )
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B) ) ).

% inf.orderI
tff(fact_3127_inf_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).

% inf.orderI
tff(fact_3128_inf_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% inf.orderI
tff(fact_3129_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% inf.orderI
tff(fact_3130_inf_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% inf.orderI
tff(fact_3131_inf__unique,axiom,
    ! [F: fun_se3560235952666129655at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X2),Y2)),X2)
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X2),Y2)),Y2)
       => ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X2),Y2)
             => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X2),Z2)
               => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,Y2),Z2)) ) )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3132_inf__unique,axiom,
    ! [F: fun_fu959379683396315552_int_o,X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: fun_Pr3238374608701176162_int_o,Y2: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(F,X2),Y2)),X2)
     => ( ! [X2: fun_Pr3238374608701176162_int_o,Y2: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(F,X2),Y2)),Y2)
       => ( ! [X2: fun_Pr3238374608701176162_int_o,Y2: fun_Pr3238374608701176162_int_o,Z2: fun_Pr3238374608701176162_int_o] :
              ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X2),Y2)
             => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X2),Z2)
               => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X2),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(F,Y2),Z2)) ) )
         => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3133_inf__unique,axiom,
    ! [F: fun_fu8416120441428932390char_o,X: fun_list_char_o,Y: fun_list_char_o] :
      ( ! [X2: fun_list_char_o,Y2: fun_list_char_o] : aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(F,X2),Y2)),X2)
     => ( ! [X2: fun_list_char_o,Y2: fun_list_char_o] : aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(F,X2),Y2)),Y2)
       => ( ! [X2: fun_list_char_o,Y2: fun_list_char_o,Z2: fun_list_char_o] :
              ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X2),Y2)
             => ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X2),Z2)
               => aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X2),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(F,Y2),Z2)) ) )
         => ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3134_inf__unique,axiom,
    ! [F: fun_fi7557781610047541018er_nat,X: filter_nat,Y: filter_nat] :
      ( ! [X2: filter_nat,Y2: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F,X2),Y2)),X2)
     => ( ! [X2: filter_nat,Y2: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F,X2),Y2)),Y2)
       => ( ! [X2: filter_nat,Y2: filter_nat,Z2: filter_nat] :
              ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X2),Y2)
             => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X2),Z2)
               => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X2),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F,Y2),Z2)) ) )
         => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3135_inf__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2)),X2)
     => ( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2)),Y2)
       => ( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),Y2)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),Z2)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y2),Z2)) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3136_inf__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,X: rat,Y: rat] :
      ( ! [X2: rat,Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,X2),Y2)),X2)
     => ( ! [X2: rat,Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,X2),Y2)),Y2)
       => ( ! [X2: rat,Y2: rat,Z2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y2)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Z2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),aa_rat_rat(aa_rat_fun_rat_rat(F,Y2),Z2)) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3137_inf__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2)),X2)
     => ( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2)),Y2)
       => ( ! [X2: nat,Y2: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(F,Y2),Z2)) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3138_inf__unique,axiom,
    ! [F: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X2),Y2)),X2)
     => ( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X2),Y2)),Y2)
       => ( ! [X2: int,Y2: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Z2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),aa_int_int(aa_int_fun_int_int(F,Y2),Z2)) ) )
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = aa_int_int(aa_int_fun_int_int(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3139_le__iff__inf,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3140_le__iff__inf,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Y)
    <=> ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3141_le__iff__inf,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),Y)
    <=> ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3142_le__iff__inf,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3143_le__iff__inf,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3144_le__iff__inf,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3145_le__iff__inf,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3146_le__iff__inf,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3147_inf_Oabsorb1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3148_inf_Oabsorb1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3149_inf_Oabsorb1,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),B)
     => ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3150_inf_Oabsorb1,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3151_inf_Oabsorb1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3152_inf_Oabsorb1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3153_inf_Oabsorb1,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(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3154_inf_Oabsorb1,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(inf_inf_int,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3155_inf_Oabsorb2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),A)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3156_inf_Oabsorb2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),A)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3157_inf_Oabsorb2,axiom,
    ! [B: fun_list_char_o,A: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,B),A)
     => ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3158_inf_Oabsorb2,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3159_inf_Oabsorb2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3160_inf_Oabsorb2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3161_inf_Oabsorb2,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(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3162_inf_Oabsorb2,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(inf_inf_int,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3163_inf__absorb1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3164_inf__absorb1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Y)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3165_inf__absorb1,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),Y)
     => ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3166_inf__absorb1,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3167_inf__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3168_inf__absorb1,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3169_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3170_inf__absorb1,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3171_inf__absorb2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Y),X)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3172_inf__absorb2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,Y),X)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3173_inf__absorb2,axiom,
    ! [Y: fun_list_char_o,X: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,Y),X)
     => ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3174_inf__absorb2,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),X)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3175_inf__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3176_inf__absorb2,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3177_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3178_inf__absorb2,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3179_inf_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C))
     => ~ ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
         => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C) ) ) ).

% inf.boundedE
tff(fact_3180_inf_OboundedE,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C))
     => ~ ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
         => ~ aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C) ) ) ).

% inf.boundedE
tff(fact_3181_inf_OboundedE,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o,C: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,B),C))
     => ~ ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),B)
         => ~ aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),C) ) ) ).

% inf.boundedE
tff(fact_3182_inf_OboundedE,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,B),C))
     => ~ ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
         => ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),C) ) ) ).

% inf.boundedE
tff(fact_3183_inf_OboundedE,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).

% inf.boundedE
tff(fact_3184_inf_OboundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C))
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% inf.boundedE
tff(fact_3185_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
     => ~ ( 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),C) ) ) ).

% inf.boundedE
tff(fact_3186_inf_OboundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% inf.boundedE
tff(fact_3187_inf_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) ) ) ).

% inf.boundedI
tff(fact_3188_inf_OboundedI,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
     => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C)
       => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C)) ) ) ).

% inf.boundedI
tff(fact_3189_inf_OboundedI,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o,C: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),B)
     => ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),C)
       => aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,B),C)) ) ) ).

% inf.boundedI
tff(fact_3190_inf_OboundedI,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),C)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,B),C)) ) ) ).

% inf.boundedI
tff(fact_3191_inf_OboundedI,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ) ) ).

% inf.boundedI
tff(fact_3192_inf_OboundedI,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C)) ) ) ).

% inf.boundedI
tff(fact_3193_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C)) ) ) ).

% inf.boundedI
tff(fact_3194_inf_OboundedI,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C)) ) ) ).

% inf.boundedI
tff(fact_3195_inf__greatest,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Z)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_3196_inf__greatest,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Y)
     => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Z)
       => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) ) ) ).

% inf_greatest
tff(fact_3197_inf__greatest,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),Y)
     => ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),Z)
       => aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),Z)) ) ) ).

% inf_greatest
tff(fact_3198_inf__greatest,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Z)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_3199_inf__greatest,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_3200_inf__greatest,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_3201_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_3202_inf__greatest,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)) ) ) ).

% inf_greatest
tff(fact_3203_inf_Oorder__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
    <=> ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_3204_inf_Oorder__iff,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
    <=> ( A = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) ) ) ).

% inf.order_iff
tff(fact_3205_inf_Oorder__iff,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),B)
    <=> ( A = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) ) ) ).

% inf.order_iff
tff(fact_3206_inf_Oorder__iff,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
    <=> ( A = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_3207_inf_Oorder__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_3208_inf_Oorder__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) ) ) ).

% inf.order_iff
tff(fact_3209_inf_Oorder__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_3210_inf_Oorder__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).

% inf.order_iff
tff(fact_3211_inf_Ocobounded1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_3212_inf_Ocobounded1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),A) ).

% inf.cobounded1
tff(fact_3213_inf_Ocobounded1,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o] : aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)),A) ).

% inf.cobounded1
tff(fact_3214_inf_Ocobounded1,axiom,
    ! [A: filter_nat,B: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_3215_inf_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_3216_inf_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),A) ).

% inf.cobounded1
tff(fact_3217_inf_Ocobounded1,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(inf_inf_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_3218_inf_Ocobounded1,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(inf_inf_int,A),B)),A) ).

% inf.cobounded1
tff(fact_3219_inf_Ocobounded2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_3220_inf_Ocobounded2,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),B) ).

% inf.cobounded2
tff(fact_3221_inf_Ocobounded2,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o] : aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)),B) ).

% inf.cobounded2
tff(fact_3222_inf_Ocobounded2,axiom,
    ! [A: filter_nat,B: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_3223_inf_Ocobounded2,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_3224_inf_Ocobounded2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),B) ).

% inf.cobounded2
tff(fact_3225_inf_Ocobounded2,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(inf_inf_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_3226_inf_Ocobounded2,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(inf_inf_int,A),B)),B) ).

% inf.cobounded2
tff(fact_3227_inf_Oabsorb__iff1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3228_inf_Oabsorb__iff1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
    <=> ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3229_inf_Oabsorb__iff1,axiom,
    ! [A: fun_list_char_o,B: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),B)
    <=> ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3230_inf_Oabsorb__iff1,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3231_inf_Oabsorb__iff1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3232_inf_Oabsorb__iff1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3233_inf_Oabsorb__iff1,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(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3234_inf_Oabsorb__iff1,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(inf_inf_int,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3235_inf_Oabsorb__iff2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),A)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3236_inf_Oabsorb__iff2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),A)
    <=> ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3237_inf_Oabsorb__iff2,axiom,
    ! [B: fun_list_char_o,A: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,B),A)
    <=> ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3238_inf_Oabsorb__iff2,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3239_inf_Oabsorb__iff2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3240_inf_Oabsorb__iff2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3241_inf_Oabsorb__iff2,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(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3242_inf_Oabsorb__iff2,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(inf_inf_int,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3243_inf_OcoboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_3244_inf_OcoboundedI1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_3245_inf_OcoboundedI1,axiom,
    ! [A: fun_list_char_o,C: fun_list_char_o,B: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,A),C)
     => aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_3246_inf_OcoboundedI1,axiom,
    ! [A: filter_nat,C: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),C)
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_3247_inf_OcoboundedI1,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_3248_inf_OcoboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_3249_inf_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_3250_inf_OcoboundedI1,axiom,
    ! [A: int,C: int,B: 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,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_3251_inf_OcoboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),C)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_3252_inf_OcoboundedI2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),C)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_3253_inf_OcoboundedI2,axiom,
    ! [B: fun_list_char_o,C: fun_list_char_o,A: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,B),C)
     => aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_3254_inf_OcoboundedI2,axiom,
    ! [B: filter_nat,C: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),C)
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_3255_inf_OcoboundedI2,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_3256_inf_OcoboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_3257_inf_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_3258_inf_OcoboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_3259_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(Y,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_3260_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_3261_inf__sup__ord_I4_J,axiom,
    ! [Y: filter_nat,X: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_3262_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_3263_inf__sup__ord_I4_J,axiom,
    ! [Y: rat,X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_3264_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_3265_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% inf_sup_ord(4)
tff(fact_3266_inf__sup__ord_I3_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_3267_inf__sup__ord_I3_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_3268_inf__sup__ord_I3_J,axiom,
    ! [X: filter_nat,Y: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_3269_inf__sup__ord_I3_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_3270_inf__sup__ord_I3_J,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_3271_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_3272_inf__sup__ord_I3_J,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% inf_sup_ord(3)
tff(fact_3273_le__supE,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B),X)
     => ~ ( ord_le8081472938463900775at_nat(A,X)
         => ~ ord_le8081472938463900775at_nat(B,X) ) ) ).

% le_supE
tff(fact_3274_le__supE,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),X)
     => ~ ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),X)
         => ~ aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),X) ) ) ).

% le_supE
tff(fact_3275_le__supE,axiom,
    ! [A: filter_nat,B: filter_nat,X: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B)),X)
     => ~ ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),X)
         => ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),X) ) ) ).

% le_supE
tff(fact_3276_le__supE,axiom,
    ! [A: set_nat,B: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X) ) ) ).

% le_supE
tff(fact_3277_le__supE,axiom,
    ! [A: rat,B: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),X)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),X) ) ) ).

% le_supE
tff(fact_3278_le__supE,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X) ) ) ).

% le_supE
tff(fact_3279_le__supE,axiom,
    ! [A: int,B: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X) ) ) ).

% le_supE
tff(fact_3280_le__supI,axiom,
    ! [A: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A,X)
     => ( ord_le8081472938463900775at_nat(B,X)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B),X) ) ) ).

% le_supI
tff(fact_3281_le__supI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),X)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),X)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),X) ) ) ).

% le_supI
tff(fact_3282_le__supI,axiom,
    ! [A: filter_nat,X: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),X)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),X)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B)),X) ) ) ).

% le_supI
tff(fact_3283_le__supI,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X) ) ) ).

% le_supI
tff(fact_3284_le__supI,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),X)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),X) ) ) ).

% le_supI
tff(fact_3285_le__supI,axiom,
    ! [A: nat,X: nat,B: 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_eq_nat,B),X)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X) ) ) ).

% le_supI
tff(fact_3286_le__supI,axiom,
    ! [A: int,X: int,B: 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_eq_int,B),X)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X) ) ) ).

% le_supI
tff(fact_3287_sup__ge1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) ).

% sup_ge1
tff(fact_3288_sup__ge1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% sup_ge1
tff(fact_3289_sup__ge1,axiom,
    ! [X: filter_nat,Y: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y)) ).

% sup_ge1
tff(fact_3290_sup__ge1,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% sup_ge1
tff(fact_3291_sup__ge1,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)) ).

% sup_ge1
tff(fact_3292_sup__ge1,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% sup_ge1
tff(fact_3293_sup__ge1,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% sup_ge1
tff(fact_3294_sup__ge2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(Y,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) ).

% sup_ge2
tff(fact_3295_sup__ge2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% sup_ge2
tff(fact_3296_sup__ge2,axiom,
    ! [Y: filter_nat,X: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y)) ).

% sup_ge2
tff(fact_3297_sup__ge2,axiom,
    ! [Y: set_nat,X: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% sup_ge2
tff(fact_3298_sup__ge2,axiom,
    ! [Y: rat,X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)) ).

% sup_ge2
tff(fact_3299_sup__ge2,axiom,
    ! [Y: nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% sup_ge2
tff(fact_3300_sup__ge2,axiom,
    ! [Y: int,X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% sup_ge2
tff(fact_3301_le__supI1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(X,A)
     => ord_le8081472938463900775at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% le_supI1
tff(fact_3302_le__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),A)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% le_supI1
tff(fact_3303_le__supI1,axiom,
    ! [X: filter_nat,A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),A)
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B)) ) ).

% le_supI1
tff(fact_3304_le__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% le_supI1
tff(fact_3305_le__supI1,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% le_supI1
tff(fact_3306_le__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% le_supI1
tff(fact_3307_le__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% le_supI1
tff(fact_3308_le__supI2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(X,B)
     => ord_le8081472938463900775at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% le_supI2
tff(fact_3309_le__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),B)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% le_supI2
tff(fact_3310_le__supI2,axiom,
    ! [X: filter_nat,B: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),B)
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B)) ) ).

% le_supI2
tff(fact_3311_le__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% le_supI2
tff(fact_3312_le__supI2,axiom,
    ! [X: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% le_supI2
tff(fact_3313_le__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% le_supI2
tff(fact_3314_le__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% le_supI2
tff(fact_3315_sup_Omono,axiom,
    ! [C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,D2: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(C,A)
     => ( ord_le8081472938463900775at_nat(D2,B)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,C),D2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ) ).

% sup.mono
tff(fact_3316_sup_Omono,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(D2),B)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C),D2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ) ).

% sup.mono
tff(fact_3317_sup_Omono,axiom,
    ! [C: filter_nat,A: filter_nat,D2: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),A)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,D2),B)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,C),D2)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B)) ) ) ).

% sup.mono
tff(fact_3318_sup_Omono,axiom,
    ! [C: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,D2),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% sup.mono
tff(fact_3319_sup_Omono,axiom,
    ! [C: rat,A: rat,D2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C),D2)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ) ).

% sup.mono
tff(fact_3320_sup_Omono,axiom,
    ! [C: nat,A: nat,D2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D2)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ) ).

% sup.mono
tff(fact_3321_sup_Omono,axiom,
    ! [C: int,A: int,D2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D2)),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ) ).

% sup.mono
tff(fact_3322_sup__mono,axiom,
    ! [A: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,D2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A,C)
     => ( ord_le8081472938463900775at_nat(B,D2)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,C),D2)) ) ) ).

% sup_mono
tff(fact_3323_sup__mono,axiom,
    ! [A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),C)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),D2)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C),D2)) ) ) ).

% sup_mono
tff(fact_3324_sup__mono,axiom,
    ! [A: filter_nat,C: filter_nat,B: filter_nat,D2: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),C)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),D2)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,C),D2)) ) ) ).

% sup_mono
tff(fact_3325_sup__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),D2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D2)) ) ) ).

% sup_mono
tff(fact_3326_sup__mono,axiom,
    ! [A: rat,C: rat,B: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C),D2)) ) ) ).

% sup_mono
tff(fact_3327_sup__mono,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D2)) ) ) ).

% sup_mono
tff(fact_3328_sup__mono,axiom,
    ! [A: int,C: int,B: int,D2: 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,B),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D2)) ) ) ).

% sup_mono
tff(fact_3329_sup__least,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(Y,X)
     => ( ord_le8081472938463900775at_nat(Z,X)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z),X) ) ) ).

% sup_least
tff(fact_3330_sup__least,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),X)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Z),X)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)),X) ) ) ).

% sup_least
tff(fact_3331_sup__least,axiom,
    ! [Y: filter_nat,X: filter_nat,Z: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),X)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Z),X)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Y),Z)),X) ) ) ).

% sup_least
tff(fact_3332_sup__least,axiom,
    ! [Y: set_nat,X: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z),X)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X) ) ) ).

% sup_least
tff(fact_3333_sup__least,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),X)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z)),X) ) ) ).

% sup_least
tff(fact_3334_sup__least,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)),X) ) ) ).

% sup_least
tff(fact_3335_sup__least,axiom,
    ! [Y: int,X: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)),X) ) ) ).

% sup_least
tff(fact_3336_le__iff__sup,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(X,Y)
    <=> ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3337_le__iff__sup,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),Y)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3338_le__iff__sup,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3339_le__iff__sup,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3340_le__iff__sup,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3341_le__iff__sup,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3342_le__iff__sup,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3343_sup_OorderE,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(B,A)
     => ( A = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) ) ) ).

% sup.orderE
tff(fact_3344_sup_OorderE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
     => ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ) ) ).

% sup.orderE
tff(fact_3345_sup_OorderE,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
     => ( A = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B) ) ) ).

% sup.orderE
tff(fact_3346_sup_OorderE,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.orderE
tff(fact_3347_sup_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) ) ) ).

% sup.orderE
tff(fact_3348_sup_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(sup_sup_nat,A),B) ) ) ).

% sup.orderE
tff(fact_3349_sup_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(sup_sup_int,A),B) ) ) ).

% sup.orderE
tff(fact_3350_sup_OorderI,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( A = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) )
     => ord_le8081472938463900775at_nat(B,A) ) ).

% sup.orderI
tff(fact_3351_sup_OorderI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) )
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A) ) ).

% sup.orderI
tff(fact_3352_sup_OorderI,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( A = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B) )
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A) ) ).

% sup.orderI
tff(fact_3353_sup_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ).

% sup.orderI
tff(fact_3354_sup_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% sup.orderI
tff(fact_3355_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% sup.orderI
tff(fact_3356_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% sup.orderI
tff(fact_3357_sup__unique,axiom,
    ! [F: fun_se6682931024248017501at_nat,X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ! [X2: set_Pr8551490117392284871at_nat,Y2: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(X2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(F,X2),Y2))
     => ( ! [X2: set_Pr8551490117392284871at_nat,Y2: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(Y2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(F,X2),Y2))
       => ( ! [X2: set_Pr8551490117392284871at_nat,Y2: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] :
              ( ord_le8081472938463900775at_nat(Y2,X2)
             => ( ord_le8081472938463900775at_nat(Z2,X2)
               => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(F,Y2),Z2),X2) ) )
         => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_3358_sup__unique,axiom,
    ! [F: fun_se177437144654672861at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X2),Y2))
     => ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X2),Y2))
       => ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y2),X2)
             => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Z2),X2)
               => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,Y2),Z2)),X2) ) )
         => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_3359_sup__unique,axiom,
    ! [F: fun_fi7557781610047541018er_nat,X: filter_nat,Y: filter_nat] :
      ( ! [X2: filter_nat,Y2: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X2),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F,X2),Y2))
     => ( ! [X2: filter_nat,Y2: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y2),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F,X2),Y2))
       => ( ! [X2: filter_nat,Y2: filter_nat,Z2: filter_nat] :
              ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y2),X2)
             => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Z2),X2)
               => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F,Y2),Z2)),X2) ) )
         => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_3360_sup__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2))
     => ( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2))
       => ( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y2),X2)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z2),X2)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y2),Z2)),X2) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_3361_sup__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,X: rat,Y: rat] :
      ( ! [X2: rat,Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),aa_rat_rat(aa_rat_fun_rat_rat(F,X2),Y2))
     => ( ! [X2: rat,Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),aa_rat_rat(aa_rat_fun_rat_rat(F,X2),Y2))
       => ( ! [X2: rat,Y2: rat,Z2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X2)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z2),X2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,Y2),Z2)),X2) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_3362_sup__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2))
     => ( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2))
       => ( ! [X2: nat,Y2: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),X2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,Y2),Z2)),X2) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_3363_sup__unique,axiom,
    ! [F: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),aa_int_int(aa_int_fun_int_int(F,X2),Y2))
     => ( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),aa_int_int(aa_int_fun_int_int(F,X2),Y2))
       => ( ! [X2: int,Y2: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),X2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,Y2),Z2)),X2) ) )
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = aa_int_int(aa_int_fun_int_int(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_3364_sup_Oabsorb1,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(B,A)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_3365_sup_Oabsorb1,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_3366_sup_Oabsorb1,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_3367_sup_Oabsorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_3368_sup_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_3369_sup_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(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_3370_sup_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(sup_sup_int,A),B) = A ) ) ).

% sup.absorb1
tff(fact_3371_sup_Oabsorb2,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A,B)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_3372_sup_Oabsorb2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),B)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_3373_sup_Oabsorb2,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_3374_sup_Oabsorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_3375_sup_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_3376_sup_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(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_3377_sup_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(sup_sup_int,A),B) = B ) ) ).

% sup.absorb2
tff(fact_3378_sup__absorb1,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(Y,X)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_3379_sup__absorb1,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),X)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_3380_sup__absorb1,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),X)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_3381_sup__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_3382_sup__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_3383_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_3384_sup__absorb1,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_3385_sup__absorb2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(X,Y)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3386_sup__absorb2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),Y)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3387_sup__absorb2,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3388_sup__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3389_sup__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3390_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3391_sup__absorb2,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3392_sup_OboundedE,axiom,
    ! [B: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),C),A)
     => ~ ( ord_le8081472938463900775at_nat(B,A)
         => ~ ord_le8081472938463900775at_nat(C,A) ) ) ).

% sup.boundedE
tff(fact_3393_sup_OboundedE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)),A)
     => ~ ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
         => ~ aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A) ) ) ).

% sup.boundedE
tff(fact_3394_sup_OboundedE,axiom,
    ! [B: filter_nat,C: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,B),C)),A)
     => ~ ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
         => ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),A) ) ) ).

% sup.boundedE
tff(fact_3395_sup_OboundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A) ) ) ).

% sup.boundedE
tff(fact_3396_sup_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),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,C),A) ) ) ).

% sup.boundedE
tff(fact_3397_sup_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),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,C),A) ) ) ).

% sup.boundedE
tff(fact_3398_sup_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),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,C),A) ) ) ).

% sup.boundedE
tff(fact_3399_sup_OboundedI,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(B,A)
     => ( ord_le8081472938463900775at_nat(C,A)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),C),A) ) ) ).

% sup.boundedI
tff(fact_3400_sup_OboundedI,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_3401_sup_OboundedI,axiom,
    ! [B: filter_nat,A: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),A)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_3402_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_3403_sup_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_3404_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C: 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,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_3405_sup_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_3406_sup_Oorder__iff,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(B,A)
    <=> ( A = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_3407_sup_Oorder__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
    <=> ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_3408_sup_Oorder__iff,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
    <=> ( A = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_3409_sup_Oorder__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_3410_sup_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(aa_rat_fun_rat_rat(sup_sup_rat,A),B) ) ) ).

% sup.order_iff
tff(fact_3411_sup_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(sup_sup_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_3412_sup_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(sup_sup_int,A),B) ) ) ).

% sup.order_iff
tff(fact_3413_sup_Ocobounded1,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(A,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ).

% sup.cobounded1
tff(fact_3414_sup_Ocobounded1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ).

% sup.cobounded1
tff(fact_3415_sup_Ocobounded1,axiom,
    ! [A: filter_nat,B: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B)) ).

% sup.cobounded1
tff(fact_3416_sup_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% sup.cobounded1
tff(fact_3417_sup_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ).

% sup.cobounded1
tff(fact_3418_sup_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(sup_sup_nat,A),B)) ).

% sup.cobounded1
tff(fact_3419_sup_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(sup_sup_int,A),B)) ).

% sup.cobounded1
tff(fact_3420_sup_Ocobounded2,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(B,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ).

% sup.cobounded2
tff(fact_3421_sup_Ocobounded2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ).

% sup.cobounded2
tff(fact_3422_sup_Ocobounded2,axiom,
    ! [B: filter_nat,A: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B)) ).

% sup.cobounded2
tff(fact_3423_sup_Ocobounded2,axiom,
    ! [B: set_nat,A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% sup.cobounded2
tff(fact_3424_sup_Ocobounded2,axiom,
    ! [B: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ).

% sup.cobounded2
tff(fact_3425_sup_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(sup_sup_nat,A),B)) ).

% sup.cobounded2
tff(fact_3426_sup_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(sup_sup_int,A),B)) ).

% sup.cobounded2
tff(fact_3427_sup_Oabsorb__iff1,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(B,A)
    <=> ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_3428_sup_Oabsorb__iff1,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_3429_sup_Oabsorb__iff1,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_3430_sup_Oabsorb__iff1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_3431_sup_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_3432_sup_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(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_3433_sup_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(sup_sup_int,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_3434_sup_Oabsorb__iff2,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A,B)
    <=> ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_3435_sup_Oabsorb__iff2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),B)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_3436_sup_Oabsorb__iff2,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_3437_sup_Oabsorb__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_3438_sup_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_3439_sup_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(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_3440_sup_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(sup_sup_int,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_3441_sup_OcoboundedI1,axiom,
    ! [C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(C,A)
     => ord_le8081472938463900775at_nat(C,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_3442_sup_OcoboundedI1,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_3443_sup_OcoboundedI1,axiom,
    ! [C: filter_nat,A: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),A)
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_3444_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_3445_sup_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.coboundedI1
tff(fact_3446_sup_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_3447_sup_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.coboundedI1
tff(fact_3448_sup_OcoboundedI2,axiom,
    ! [C: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(C,B)
     => ord_le8081472938463900775at_nat(C,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_3449_sup_OcoboundedI2,axiom,
    ! [C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),B)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_3450_sup_OcoboundedI2,axiom,
    ! [C: filter_nat,B: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),B)
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_3451_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_3452_sup_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.coboundedI2
tff(fact_3453_sup_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_3454_sup_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.coboundedI2
tff(fact_3455_subset__UNIV,axiom,
    ! [A2: set_list_nat] : aa_set_list_nat_o(aa_set8027019173009287559_nat_o(ord_le6045566169113846134st_nat,A2),top_top_set_list_nat) ).

% subset_UNIV
tff(fact_3456_subset__UNIV,axiom,
    ! [A2: set_o] : aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),top_top_set_o) ).

% subset_UNIV
tff(fact_3457_subset__UNIV,axiom,
    ! [A2: set_rat] : aa_set_rat_o(aa_set229218963967607191_rat_o(ord_less_eq_set_rat,A2),top_top_set_rat) ).

% subset_UNIV
tff(fact_3458_subset__UNIV,axiom,
    ! [A2: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),top_top_set_int) ).

% subset_UNIV
tff(fact_3459_subset__UNIV,axiom,
    ! [A2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),top_top_set_nat) ).

% subset_UNIV
tff(fact_3460_inter__eq__subsetI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),S3)
     => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),S3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),S3) )
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),S) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_3461_inter__eq__subsetI,axiom,
    ! [S: set_nat,S3: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),S3)
     => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),S3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),S3) )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),S) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_3462_Int__Collect__mono,axiom,
    ! [A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int,P: fun_se7249947736802449410_int_o,Q: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A2),B2)
     => ( ! [X2: set_Pr958786334691620121nt_int] :
            ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),A2)
           => ( aa_set6170482117830564987_int_o(P,X2)
             => aa_set6170482117830564987_int_o(Q,X2) ) )
       => aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(inf_in8396524679539076455nt_int(A2,collec5210948495886036740nt_int(P))),inf_in8396524679539076455nt_int(B2,collec5210948495886036740nt_int(Q))) ) ) ).

% Int_Collect_mono
tff(fact_3463_Int__Collect__mono,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
     => ( ! [X2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X2),A2)
           => ( aa_set_nat_o(P,X2)
             => aa_set_nat_o(Q,X2) ) )
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(inf_inf_set_set_nat(A2,collect_set_nat(P))),inf_inf_set_set_nat(B2,collect_set_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_3464_Int__Collect__mono,axiom,
    ! [A2: set_int,B2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),B2)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
           => ( aa_int_o(P,X2)
             => aa_int_o(Q,X2) ) )
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),collect_int(Q))) ) ) ).

% Int_Collect_mono
tff(fact_3465_Int__Collect__mono,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o,P: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),B2)
     => ( ! [X2: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A2)
           => ( aa_fun1675921005893230629at_o_o(P,X2)
             => aa_fun1675921005893230629at_o_o(Q,X2) ) )
       => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(inf_in3376754634429691197_nat_o(A2,collec6407654713642994138_nat_o(P))),inf_in3376754634429691197_nat_o(B2,collec6407654713642994138_nat_o(Q))) ) ) ).

% Int_Collect_mono
tff(fact_3466_Int__Collect__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),B2)
     => ( ! [X2: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),A2)
           => ( aa_Pro6150637604445964947_nat_o(P,X2)
             => aa_Pro6150637604445964947_nat_o(Q,X2) ) )
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),collec3392354462482085612at_nat(P))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),collec3392354462482085612at_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_3467_Int__Collect__mono,axiom,
    ! [A2: set_nat,B2: set_nat,P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
           => ( aa_nat_o(P,X2)
             => aa_nat_o(Q,X2) ) )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),collect_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_3468_Int__greatest,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),A2)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),B2)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ) ) ).

% Int_greatest
tff(fact_3469_Int__greatest,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),B2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ) ).

% Int_greatest
tff(fact_3470_Int__absorb2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),B2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = A2 ) ) ).

% Int_absorb2
tff(fact_3471_Int__absorb2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = A2 ) ) ).

% Int_absorb2
tff(fact_3472_Int__absorb1,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B2),A2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = B2 ) ) ).

% Int_absorb1
tff(fact_3473_Int__absorb1,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = B2 ) ) ).

% Int_absorb1
tff(fact_3474_Int__lower2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),B2) ).

% Int_lower2
tff(fact_3475_Int__lower2,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),B2) ).

% Int_lower2
tff(fact_3476_Int__lower1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),A2) ).

% Int_lower1
tff(fact_3477_Int__lower1,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),A2) ).

% Int_lower1
tff(fact_3478_Int__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,D3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),C2)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B2),D3)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),D3)) ) ) ).

% Int_mono
tff(fact_3479_Int__mono,axiom,
    ! [A2: set_nat,C2: set_nat,B2: set_nat,D3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),D3)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),D3)) ) ) ).

% Int_mono
tff(fact_3480_subset__Un__eq,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A2,B2)
    <=> ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2) = B2 ) ) ).

% subset_Un_eq
tff(fact_3481_subset__Un__eq,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),B2)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = B2 ) ) ).

% subset_Un_eq
tff(fact_3482_subset__Un__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = B2 ) ) ).

% subset_Un_eq
tff(fact_3483_subset__UnE,axiom,
    ! [C2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(C2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2))
     => ~ ! [A5: set_Pr8551490117392284871at_nat] :
            ( ord_le8081472938463900775at_nat(A5,A2)
           => ! [B6: set_Pr8551490117392284871at_nat] :
                ( ord_le8081472938463900775at_nat(B6,B2)
               => ( C2 != aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A5),B6) ) ) ) ) ).

% subset_UnE
tff(fact_3484_subset__UnE,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2))
     => ~ ! [A5: set_Pr4329608150637261639at_nat] :
            ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A5),A2)
           => ! [B6: set_Pr4329608150637261639at_nat] :
                ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B6),B2)
               => ( C2 != aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A5),B6) ) ) ) ) ).

% subset_UnE
tff(fact_3485_subset__UnE,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
     => ~ ! [A5: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A5),A2)
           => ! [B6: set_nat] :
                ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),B2)
               => ( C2 != aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A5),B6) ) ) ) ) ).

% subset_UnE
tff(fact_3486_Un__absorb2,axiom,
    ! [B2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(B2,A2)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2) = A2 ) ) ).

% Un_absorb2
tff(fact_3487_Un__absorb2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B2),A2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = A2 ) ) ).

% Un_absorb2
tff(fact_3488_Un__absorb2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = A2 ) ) ).

% Un_absorb2
tff(fact_3489_Un__absorb1,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A2,B2)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2) = B2 ) ) ).

% Un_absorb1
tff(fact_3490_Un__absorb1,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),B2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = B2 ) ) ).

% Un_absorb1
tff(fact_3491_Un__absorb1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = B2 ) ) ).

% Un_absorb1
tff(fact_3492_Un__upper2,axiom,
    ! [B2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(B2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) ).

% Un_upper2
tff(fact_3493_Un__upper2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ).

% Un_upper2
tff(fact_3494_Un__upper2,axiom,
    ! [B2: set_nat,A2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ).

% Un_upper2
tff(fact_3495_Un__upper1,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(A2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) ).

% Un_upper1
tff(fact_3496_Un__upper1,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ).

% Un_upper1
tff(fact_3497_Un__upper1,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ).

% Un_upper1
tff(fact_3498_Un__least,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A2,C2)
     => ( ord_le8081472938463900775at_nat(B2,C2)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2),C2) ) ) ).

% Un_least
tff(fact_3499_Un__least,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),C2)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B2),C2)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)),C2) ) ) ).

% Un_least
tff(fact_3500_Un__least,axiom,
    ! [A2: set_nat,C2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),C2) ) ) ).

% Un_least
tff(fact_3501_Un__mono,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,D3: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A2,C2)
     => ( ord_le8081472938463900775at_nat(B2,D3)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,C2),D3)) ) ) ).

% Un_mono
tff(fact_3502_Un__mono,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,D3: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),C2)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B2),D3)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),D3)) ) ) ).

% Un_mono
tff(fact_3503_Un__mono,axiom,
    ! [A2: set_nat,C2: set_nat,B2: set_nat,D3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),D3)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D3)) ) ) ).

% Un_mono
tff(fact_3504_Diff__partition,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A2,B2)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,B2),A2)) = B2 ) ) ).

% Diff_partition
tff(fact_3505_Diff__partition,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),B2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B2),A2)) = B2 ) ) ).

% Diff_partition
tff(fact_3506_Diff__partition,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)) = B2 ) ) ).

% Diff_partition
tff(fact_3507_Diff__subset__conv,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),B2),C2)
    <=> ord_le8081472938463900775at_nat(A2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B2),C2)) ) ).

% Diff_subset_conv
tff(fact_3508_Diff__subset__conv,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),B2)),C2)
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) ) ).

% Diff_subset_conv
tff(fact_3509_Diff__subset__conv,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),C2)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) ) ).

% Diff_subset_conv
tff(fact_3510_relH__subset,axiom,
    ! [Bs: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,As: set_nat] :
      ( relH(Bs,H2,H3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,As),Bs)
       => relH(As,H2,H3) ) ) ).

% relH_subset
tff(fact_3511_mult__le__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3512_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3513_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3514_mult__le__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3515_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3516_mult__le__cancel__left2,axiom,
    ! [C: 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,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3517_mult__le__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3518_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3519_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3520_mult__le__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3521_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3522_mult__le__cancel__right2,axiom,
    ! [A: int,C: 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),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3523_mult__less__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3524_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3525_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3526_mult__less__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3527_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3528_mult__less__cancel__left2,axiom,
    ! [C: 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,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3529_mult__less__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3530_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3531_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3532_mult__less__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3533_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3534_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3535_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z2),X)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% field_le_mult_one_interval
tff(fact_3536_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3537_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),X)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3538_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3539_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3540_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3541_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3542_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3543_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B)) ) ) ) ).

% divide_left_mono
tff(fact_3544_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_3545_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_3546_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(aa_rat_fun_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_3547_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_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_3548_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(W)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3549_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3550_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_3551_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_3552_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_3553_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_3554_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),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_3555_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_3556_less__eq__assn__def,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( A = inf_inf_assn(A,B) ) ) ).

% less_eq_assn_def
tff(fact_3557_mult__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).

% mult_neg_neg
tff(fact_3558_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_neg_neg
tff(fact_3559_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_3560_not__square__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A)),zero_z3403309356797280102nteger) ).

% not_square_less_zero
tff(fact_3561_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_3562_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_3563_mult__less__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ) ).

% mult_less_0_iff
tff(fact_3564_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_3565_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_3566_mult__neg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_neg_pos
tff(fact_3567_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_3568_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_3569_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_3570_mult__pos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_pos_neg
tff(fact_3571_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_3572_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_3573_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_3574_mult__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).

% mult_pos_pos
tff(fact_3575_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_3576_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_3577_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_3578_mult__pos__neg2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),zero_z3403309356797280102nteger) ) ) ).

% mult_pos_neg2
tff(fact_3579_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_3580_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_3581_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_3582_zero__less__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger) ) ) ) ).

% zero_less_mult_iff
tff(fact_3583_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_3584_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_3585_zero__less__mult__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ).

% zero_less_mult_pos
tff(fact_3586_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_3587_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_3588_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_3589_zero__less__mult__pos2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3590_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3591_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_3592_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_3593_mult__less__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3594_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3595_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3596_mult__less__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3597_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3598_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3599_mult__strict__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3600_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3601_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3602_mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3603_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3604_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3605_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3606_mult__less__cancel__left__disj,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3607_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3608_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3609_mult__strict__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3610_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3611_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3612_mult__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_3613_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_3614_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_3615_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_3616_mult__less__cancel__right__disj,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3617_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3618_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3619_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3620_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3621_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3622_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3623_zero__less__one,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% zero_less_one
tff(fact_3624_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_3625_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_3626_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_3627_not__one__less__zero,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),zero_z3403309356797280102nteger) ).

% not_one_less_zero
tff(fact_3628_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_3629_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_3630_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_3631_less__numeral__extra_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% less_numeral_extra(1)
tff(fact_3632_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_3633_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_3634_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_3635_less__iff__diff__less__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),zero_z3403309356797280102nteger) ) ).

% less_iff_diff_less_0
tff(fact_3636_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(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_3637_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_3638_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(N)),one_one_nat) ).

% not_numeral_less_one
tff(fact_3639_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N)),one_one_int) ).

% not_numeral_less_one
tff(fact_3640_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),one_one_Code_integer) ).

% not_numeral_less_one
tff(fact_3641_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(N)),one_one_rat) ).

% not_numeral_less_one
tff(fact_3642_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_Code_natural_o(ord_le5570908160329646204atural(numera5444537566228673987atural(N)),one_one_Code_natural) ).

% not_numeral_less_one
tff(fact_3643_less__1__mult,axiom,
    ! [M: code_integer,N: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),N)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,M),N)) ) ) ).

% less_1_mult
tff(fact_3644_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),N)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),N)) ) ) ).

% less_1_mult
tff(fact_3645_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ) ) ).

% less_1_mult
tff(fact_3646_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),N)) ) ) ).

% less_1_mult
tff(fact_3647_mult__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_3648_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_3649_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_3650_mult__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_3651_mult__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_3652_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_3653_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_3654_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_3655_zero__le__square,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A)) ).

% zero_le_square
tff(fact_3656_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)) ).

% zero_le_square
tff(fact_3657_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_3658_split__mult__pos__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ).

% split_mult_pos_le
tff(fact_3659_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ).

% split_mult_pos_le
tff(fact_3660_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_3661_mult__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3662_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3663_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3664_mult__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3665_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3666_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_3667_mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_left_mono
tff(fact_3668_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_left_mono
tff(fact_3669_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_left_mono
tff(fact_3670_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono
tff(fact_3671_mult__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3672_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3673_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3674_mult__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_right_mono
tff(fact_3675_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_right_mono
tff(fact_3676_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_right_mono
tff(fact_3677_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono
tff(fact_3678_mult__le__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3679_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3680_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_3681_split__mult__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ).

% split_mult_neg_le
tff(fact_3682_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_3683_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_3684_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_3685_mult__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3686_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3687_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_3688_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_3689_mult__nonneg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonneg_nonpos
tff(fact_3690_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_3691_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_3692_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_3693_mult__nonpos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonpos_nonneg
tff(fact_3694_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_3695_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_3696_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_3697_mult__nonneg__nonpos2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3698_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3699_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_3700_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_3701_zero__le__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) ) ) ).

% zero_le_mult_iff
tff(fact_3702_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_3703_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_3704_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3705_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3706_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3707_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3708_zero__less__one__class_Ozero__le__one,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% zero_less_one_class.zero_le_one
tff(fact_3709_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_3710_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_3711_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_3712_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_3713_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_3714_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_3715_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_3716_not__one__le__zero,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),zero_z3403309356797280102nteger) ).

% not_one_le_zero
tff(fact_3717_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_3718_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_3719_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_3720_le__iff__diff__le__0,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(minus_8373710615458151222nteger(A),B)),zero_z3403309356797280102nteger) ) ).

% le_iff_diff_le_0
tff(fact_3721_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(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_3722_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_3723_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) ).

% not_numeral_less_neg_numeral
tff(fact_3724_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% not_numeral_less_neg_numeral
tff(fact_3725_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(M)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) ).

% not_numeral_less_neg_numeral
tff(fact_3726_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) ).

% neg_numeral_less_numeral
tff(fact_3727_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% neg_numeral_less_numeral
tff(fact_3728_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),numeral_numeral_rat(N)) ).

% neg_numeral_less_numeral
tff(fact_3729_less__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(4)
tff(fact_3730_less__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(4)
tff(fact_3731_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_3732_less__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% less_minus_one_simps(2)
tff(fact_3733_less__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% less_minus_one_simps(2)
tff(fact_3734_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_3735_one__le__numeral,axiom,
    ! [N: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% one_le_numeral
tff(fact_3736_one__le__numeral,axiom,
    ! [N: num] : aa_Code_natural_o(ord_le1926595141338095240atural(one_one_Code_natural),numera5444537566228673987atural(N)) ).

% one_le_numeral
tff(fact_3737_one__le__numeral,axiom,
    ! [N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),numeral_numeral_rat(N)) ).

% one_le_numeral
tff(fact_3738_one__le__numeral,axiom,
    ! [N: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),numeral_numeral_nat(N)) ).

% one_le_numeral
tff(fact_3739_one__le__numeral,axiom,
    ! [N: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,N)) ).

% one_le_numeral
tff(fact_3740_diff__shunt__var,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,X),Y) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y) ) ).

% diff_shunt_var
tff(fact_3741_diff__shunt__var,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( minus_minus_set_o(X,Y) = bot_bot_set_o )
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,X),Y) ) ).

% diff_shunt_var
tff(fact_3742_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( minus_minus_set_int(X,Y) = bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y) ) ).

% diff_shunt_var
tff(fact_3743_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X),Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y) ) ).

% diff_shunt_var
tff(fact_3744_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% not_numeral_le_neg_numeral
tff(fact_3745_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(M)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) ).

% not_numeral_le_neg_numeral
tff(fact_3746_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) ).

% not_numeral_le_neg_numeral
tff(fact_3747_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% neg_numeral_le_numeral
tff(fact_3748_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),numeral_numeral_rat(N)) ).

% neg_numeral_le_numeral
tff(fact_3749_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) ).

% neg_numeral_le_numeral
tff(fact_3750_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_3751_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_3752_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_3753_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_3754_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_3755_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_3756_disjoint__alt__simp3,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),A2)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp3
tff(fact_3757_disjoint__alt__simp3,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,minus_minus_set_o(A2,B2)),A2)
    <=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) != bot_bot_set_o ) ) ).

% disjoint_alt_simp3
tff(fact_3758_disjoint__alt__simp3,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,minus_minus_set_int(A2,B2)),A2)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) != bot_bot_set_int ) ) ).

% disjoint_alt_simp3
tff(fact_3759_disjoint__alt__simp3,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),A2)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp3
tff(fact_3760_distrib__inf__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_3761_distrib__inf__le,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z))),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z))) ).

% distrib_inf_le
tff(fact_3762_distrib__inf__le,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] : aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Z))),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,Y),Z))) ).

% distrib_inf_le
tff(fact_3763_distrib__inf__le,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Z)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_3764_distrib__inf__le,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_3765_distrib__inf__le,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Z))),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_3766_distrib__inf__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_3767_distrib__inf__le,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Z))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z))) ).

% distrib_inf_le
tff(fact_3768_distrib__inf__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_3769_distrib__inf__le,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z))) ).

% distrib_inf_le
tff(fact_3770_distrib__sup__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z))) ).

% distrib_sup_le
tff(fact_3771_distrib__sup__le,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z))),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Z))) ).

% distrib_sup_le
tff(fact_3772_distrib__sup__le,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] : aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,Y),Z))),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),Y)),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,X),Z))) ).

% distrib_sup_le
tff(fact_3773_distrib__sup__le,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),Z)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Z))) ).

% distrib_sup_le
tff(fact_3774_distrib__sup__le,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z))) ).

% distrib_sup_le
tff(fact_3775_distrib__sup__le,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Y),Z))),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Z))) ).

% distrib_sup_le
tff(fact_3776_distrib__sup__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z))) ).

% distrib_sup_le
tff(fact_3777_distrib__sup__le,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Z))) ).

% distrib_sup_le
tff(fact_3778_distrib__sup__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z))) ).

% distrib_sup_le
tff(fact_3779_distrib__sup__le,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z))) ).

% distrib_sup_le
tff(fact_3780_disjoint__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),A3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),B3)
       => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B3) = bot_bo2099793752762293965at_nat )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ) ) ).

% disjoint_mono
tff(fact_3781_disjoint__mono,axiom,
    ! [A: set_o,A3: set_o,B: set_o,B3: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),A3)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B),B3)
       => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A3),B3) = bot_bot_set_o )
         => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A),B) = bot_bot_set_o ) ) ) ) ).

% disjoint_mono
tff(fact_3782_disjoint__mono,axiom,
    ! [A: set_int,A3: set_int,B: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),A3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),B3)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B3) = bot_bot_set_int )
         => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = bot_bot_set_int ) ) ) ) ).

% disjoint_mono
tff(fact_3783_disjoint__mono,axiom,
    ! [A: set_nat,A3: set_nat,B: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),A3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),B3)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B3) = bot_bot_set_nat )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ) ) ).

% disjoint_mono
tff(fact_3784_Un__Int__assoc__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2)) )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),A2) ) ).

% Un_Int_assoc_eq
tff(fact_3785_Un__Int__assoc__eq,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
      ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),B2)),C2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B2),C2)) )
    <=> ord_le8081472938463900775at_nat(C2,A2) ) ).

% Un_Int_assoc_eq
tff(fact_3786_Un__Int__assoc__eq,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) )
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C2),A2) ) ).

% Un_Int_assoc_eq
tff(fact_3787_Un__Int__assoc__eq,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A2) ) ).

% Un_Int_assoc_eq
tff(fact_3788_subset__Compl__self__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,A2))
    <=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% subset_Compl_self_eq
tff(fact_3789_subset__Compl__self__eq,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),aa_set_o_set_o(uminus_uminus_set_o,A2))
    <=> ( A2 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
tff(fact_3790_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),aa_set_int_set_int(uminus1532241313380277803et_int,A2))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_3791_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_3792_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_3793_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_3794_in__range__subset,axiom,
    ! [As: set_nat,As3: set_nat,H2: heap_e7401611519738050253t_unit] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,As),As3)
     => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As3))
       => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As)) ) ) ).

% in_range_subset
tff(fact_3795_not__zero__less__neg__numeral,axiom,
    ! [N: 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,N))) ).

% not_zero_less_neg_numeral
tff(fact_3796_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% not_zero_less_neg_numeral
tff(fact_3797_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) ).

% not_zero_less_neg_numeral
tff(fact_3798_neg__numeral__less__zero,axiom,
    ! [N: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_3799_neg__numeral__less__zero,axiom,
    ! [N: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_3800_neg__numeral__less__zero,axiom,
    ! [N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_3801_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_3802_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_3803_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_3804_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_3805_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_3806_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_3807_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_3808_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_3809_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_3810_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_3811_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_3812_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_3813_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_3814_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),X)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_3815_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_3816_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3817_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_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_3818_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(aa_rat_fun_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_3819_neg__numeral__less__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) ).

% neg_numeral_less_one
tff(fact_3820_neg__numeral__less__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_3821_neg__numeral__less__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_3822_neg__one__less__numeral,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M)) ).

% neg_one_less_numeral
tff(fact_3823_neg__one__less__numeral,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,M)) ).

% neg_one_less_numeral
tff(fact_3824_neg__one__less__numeral,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),numeral_numeral_rat(M)) ).

% neg_one_less_numeral
tff(fact_3825_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_3826_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_3827_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_3828_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_one_less_neg_numeral
tff(fact_3829_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% not_one_less_neg_numeral
tff(fact_3830_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))) ).

% not_one_less_neg_numeral
tff(fact_3831_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_3832_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_3833_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))) ).

% not_neg_one_less_neg_numeral
tff(fact_3834_mult__left__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_3835_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_3836_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_3837_mult__right__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_3838_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_3839_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_3840_mult__le__one,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ) ).

% mult_le_one
tff(fact_3841_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_3842_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_3843_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_3844_mult__left__le,axiom,
    ! [C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),A) ) ) ).

% mult_left_le
tff(fact_3845_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_3846_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_3847_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A) ) ) ).

% mult_left_le
tff(fact_3848_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% not_zero_le_neg_numeral
tff(fact_3849_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) ).

% not_zero_le_neg_numeral
tff(fact_3850_not__zero__le__neg__numeral,axiom,
    ! [N: 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,N))) ).

% not_zero_le_neg_numeral
tff(fact_3851_neg__numeral__le__zero,axiom,
    ! [N: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_3852_neg__numeral__le__zero,axiom,
    ! [N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_3853_neg__numeral__le__zero,axiom,
    ! [N: 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,N))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_3854_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_3855_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_3856_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_3857_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_3858_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_3859_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_3860_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3861_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3862_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3863_neg__numeral__le__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_3864_neg__numeral__le__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_3865_neg__numeral__le__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) ).

% neg_numeral_le_one
tff(fact_3866_neg__one__le__numeral,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,M)) ).

% neg_one_le_numeral
tff(fact_3867_neg__one__le__numeral,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),numeral_numeral_rat(M)) ).

% neg_one_le_numeral
tff(fact_3868_neg__one__le__numeral,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M)) ).

% neg_one_le_numeral
tff(fact_3869_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_3870_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_3871_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_3872_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_3873_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_3874_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_3875_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% not_one_le_neg_numeral
tff(fact_3876_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))) ).

% not_one_le_neg_numeral
tff(fact_3877_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_one_le_neg_numeral
tff(fact_3878_inf__shunt,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = bot_bo3516217578726220278_int_o )
    <=> aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,Y)) ) ).

% inf_shunt
tff(fact_3879_inf__shunt,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o] :
      ( ( aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y) = bot_bo8860746597864711612char_o )
    <=> aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,Y)) ) ).

% inf_shunt
tff(fact_3880_inf__shunt,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Y)) ) ).

% inf_shunt
tff(fact_3881_inf__shunt,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),Y) = bot_bot_set_o )
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,X),aa_set_o_set_o(uminus_uminus_set_o,Y)) ) ).

% inf_shunt
tff(fact_3882_inf__shunt,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(uminus1532241313380277803et_int,Y)) ) ).

% inf_shunt
tff(fact_3883_inf__shunt,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = bot_bot_Product_unit )
    <=> aa_Product_unit_o(ord_le3221252021190050221t_unit(X),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)) ) ).

% inf_shunt
tff(fact_3884_inf__shunt,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ) ).

% inf_shunt
tff(fact_3885_sup__shunt,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = top_to8764741339697478167at_nat )
    <=> ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,X),Y) ) ).

% sup_shunt
tff(fact_3886_sup__shunt,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = top_to6833984726390702231at_nat )
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(uminus935396558254630718at_nat(X)),Y) ) ).

% sup_shunt
tff(fact_3887_sup__shunt,axiom,
    ! [X: set_list_nat,Y: set_list_nat] :
      ( ( aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(sup_sup_set_list_nat,X),Y) = top_top_set_list_nat )
    <=> aa_set_list_nat_o(aa_set8027019173009287559_nat_o(ord_le6045566169113846134st_nat,aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,X)),Y) ) ).

% sup_shunt
tff(fact_3888_sup__shunt,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),Y) = top_top_set_o )
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),Y) ) ).

% sup_shunt
tff(fact_3889_sup__shunt,axiom,
    ! [X: set_rat,Y: set_rat] :
      ( ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,X),Y) = top_top_set_rat )
    <=> aa_set_rat_o(aa_set229218963967607191_rat_o(ord_less_eq_set_rat,aa_set_rat_set_rat(uminus2201863774496077783et_rat,X)),Y) ) ).

% sup_shunt
tff(fact_3890_sup__shunt,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = top_top_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),Y) ) ).

% sup_shunt
tff(fact_3891_sup__shunt,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y) = top_top_Product_unit )
    <=> aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),Y) ) ).

% sup_shunt
tff(fact_3892_sup__shunt,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = top_top_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),Y) ) ).

% sup_shunt
tff(fact_3893_shunt1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Z)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Y)),Z)) ) ).

% shunt1
tff(fact_3894_shunt1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Z)
    <=> aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,Y)),Z)) ) ).

% shunt1
tff(fact_3895_shunt1,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),Y)),Z)
    <=> aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,Y)),Z)) ) ).

% shunt1
tff(fact_3896_shunt1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y),Z)
    <=> ord_le8081472938463900775at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,Y)),Z)) ) ).

% shunt1
tff(fact_3897_shunt1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),Z)
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(Y)),Z)) ) ).

% shunt1
tff(fact_3898_shunt1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),Z)
    <=> aa_Product_unit_o(ord_le3221252021190050221t_unit(X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)),Z)) ) ).

% shunt1
tff(fact_3899_shunt1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Z)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)),Z)) ) ).

% shunt1
tff(fact_3900_shunt2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Y))),Z)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)) ) ).

% shunt2
tff(fact_3901_shunt2,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,Y))),Z)
    <=> aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z)) ) ).

% shunt2
tff(fact_3902_shunt2,axiom,
    ! [X: fun_list_char_o,Y: fun_list_char_o,Z: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,X),aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,Y))),Z)
    <=> aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,X),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,Y),Z)) ) ).

% shunt2
tff(fact_3903_shunt2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,Y)),Z)
    <=> ord_le8081472938463900775at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z)) ) ).

% shunt2
tff(fact_3904_shunt2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),uminus935396558254630718at_nat(Y))),Z)
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) ) ).

% shunt2
tff(fact_3905_shunt2,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y))),Z)
    <=> aa_Product_unit_o(ord_le3221252021190050221t_unit(X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),Z)) ) ).

% shunt2
tff(fact_3906_shunt2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y))),Z)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ) ).

% shunt2
tff(fact_3907_sup__neg__inf,axiom,
    ! [P3: set_Pr1261947904930325089at_nat,Q3: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,P3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Q3),R2))
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,P3),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Q3))),R2) ) ).

% sup_neg_inf
tff(fact_3908_sup__neg__inf,axiom,
    ! [P3: fun_Pr3238374608701176162_int_o,Q3: fun_Pr3238374608701176162_int_o,R2: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,P3),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Q3),R2))
    <=> aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,P3),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,Q3))),R2) ) ).

% sup_neg_inf
tff(fact_3909_sup__neg__inf,axiom,
    ! [P3: fun_list_char_o,Q3: fun_list_char_o,R2: fun_list_char_o] :
      ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,P3),aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(sup_su7322457434961926436char_o,Q3),R2))
    <=> aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(ord_le7536583267156233968char_o,aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,P3),aa_fun3665998754152086460char_o(uminus8184685084194811975char_o,Q3))),R2) ) ).

% sup_neg_inf
tff(fact_3910_sup__neg__inf,axiom,
    ! [P3: set_Pr8551490117392284871at_nat,Q3: set_Pr8551490117392284871at_nat,R2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(P3,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Q3),R2))
    <=> ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,P3),aa_set6734554978791863420at_nat(uminus2330091110623919550at_nat,Q3)),R2) ) ).

% sup_neg_inf
tff(fact_3911_sup__neg__inf,axiom,
    ! [P3: set_Pr4329608150637261639at_nat,Q3: set_Pr4329608150637261639at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(P3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Q3),R2))
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,P3),uminus935396558254630718at_nat(Q3))),R2) ) ).

% sup_neg_inf
tff(fact_3912_sup__neg__inf,axiom,
    ! [P3: product_unit,Q3: product_unit,R2: product_unit] :
      ( aa_Product_unit_o(ord_le3221252021190050221t_unit(P3),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Q3),R2))
    <=> aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,P3),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Q3))),R2) ) ).

% sup_neg_inf
tff(fact_3913_sup__neg__inf,axiom,
    ! [P3: set_nat,Q3: set_nat,R2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,P3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Q3),R2))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,P3),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Q3))),R2) ) ).

% sup_neg_inf
tff(fact_3914_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_3915_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o )
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),aa_set_o_set_o(uminus_uminus_set_o,B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_3916_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),aa_set_int_set_int(uminus1532241313380277803et_int,B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_3917_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_3918_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(W)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3919_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3920_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_3921_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_3922_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_3923_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_3924_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_3925_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),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_3926_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_3927_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_3928_mult__le__cancel__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Z))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3929_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3930_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3931_mult__le__cancel__iff2,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),Y))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3932_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3933_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3934_mult__less__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Z))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y) ) ) ).

% mult_less_iff1
tff(fact_3935_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ) ).

% mult_less_iff1
tff(fact_3936_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ) ).

% mult_less_iff1
tff(fact_3937_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila5207306265035627125at_nat(sup_su3035147773818789531at_nat,bot_bo8422036546324065075at_nat,aTP_Lamm_fr,aTP_Lamm_fs) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_3938_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila6534579987270727413at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat,aTP_Lamm_ft,aTP_Lamm_fu) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_3939_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila4918476307565957903at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat,aTP_Lamm_fv,aTP_Lamm_fw) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_3940_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila2554085542299052326_set_o(sup_sup_set_o,bot_bot_set_o,aTP_Lamm_fx,aTP_Lamm_fy) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_3941_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila6712789903965657268et_int(sup_sup_set_int,bot_bot_set_int,aTP_Lamm_fz,aTP_Lamm_ga) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_3942_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila5926144314033625522er_nat(sup_sup_filter_nat,bot_bot_filter_nat,aTP_Lamm_gb,aTP_Lamm_gc) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_3943_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila1667268886620078168et_nat(sup_sup_set_nat,bot_bot_set_nat,aTP_Lamm_gd,aTP_Lamm_ge) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_3944_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila4918476307565957903at_nat(inf_in2572325071724192079at_nat,top_to4669805908274784177at_nat,ord_le3146513528884898305at_nat,ord_le7866589430770878221at_nat) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_3945_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila4431159887699967284_int_o(inf_in1782598092406300404_int_o,top_to8487252020945707026_int_o,ord_le14110778710936002_int_o,ord_le7515035663206590902_int_o) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_3946_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila4133154140287329534char_o(inf_in94156054473216830char_o,top_to7404587909000090784char_o,ord_le7536583267156233968char_o,ord_le8755527016964500988char_o) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_3947_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila2613578647866687720st_nat(inf_inf_set_list_nat,top_top_set_list_nat,ord_le6045566169113846134st_nat,ord_le1190675801316882794st_nat) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_3948_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila2554085542299052326_set_o(inf_inf_set_o,top_top_set_o,ord_less_eq_set_o,ord_less_set_o) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_3949_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila7382412365081457248et_rat(inf_inf_set_rat,top_top_set_rat,ord_less_eq_set_rat,ord_less_set_rat) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_3950_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila6712789903965657268et_int(inf_inf_set_int,top_top_set_int,ord_less_eq_set_int,ord_less_set_int) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_3951_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila5926144314033625522er_nat(inf_inf_filter_nat,top_top_filter_nat,ord_le2510731241096832064er_nat,ord_less_filter_nat) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_3952_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila1667268886620078168et_nat(inf_inf_set_nat,top_top_set_nat,ord_less_eq_set_nat,ord_less_set_nat) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_3953_bot_Oordering__top__axioms,axiom,
    aa_set3702741264012733747_nat_o(orderi2172309028950807442at_nat(aTP_Lamm_fv,aTP_Lamm_fw),bot_bo2099793752762293965at_nat) ).

% bot.ordering_top_axioms
tff(fact_3954_bot_Oordering__top__axioms,axiom,
    aa_set_o_o(ordering_top_set_o(aTP_Lamm_fx,aTP_Lamm_fy),bot_bot_set_o) ).

% bot.ordering_top_axioms
tff(fact_3955_bot_Oordering__top__axioms,axiom,
    aa_set_int_o(ordering_top_set_int(aTP_Lamm_fz,aTP_Lamm_ga),bot_bot_set_int) ).

% bot.ordering_top_axioms
tff(fact_3956_bot_Oordering__top__axioms,axiom,
    orderi2807725203697798511er_nat(aTP_Lamm_gb,aTP_Lamm_gc,bot_bot_filter_nat) ).

% bot.ordering_top_axioms
tff(fact_3957_bot_Oordering__top__axioms,axiom,
    aa_set_nat_o(ordering_top_set_nat(aTP_Lamm_gd,aTP_Lamm_ge),bot_bot_set_nat) ).

% bot.ordering_top_axioms
tff(fact_3958_bot_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(aTP_Lamm_gf,aTP_Lamm_gg),bot_bot_nat) ).

% bot.ordering_top_axioms
tff(fact_3959_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3960_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3961_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3962_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3963_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: 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,numeral_numeral_rat(M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3964_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3965_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(M))) ) ).

% add_neg_numeral_special(6)
tff(fact_3966_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,inc(M))) ) ).

% add_neg_numeral_special(6)
tff(fact_3967_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(inc(M))) ) ).

% add_neg_numeral_special(6)
tff(fact_3968_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : ( 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,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(N))) ) ).

% add_neg_numeral_special(5)
tff(fact_3969_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,inc(N))) ) ).

% add_neg_numeral_special(5)
tff(fact_3970_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(inc(N))) ) ).

% add_neg_numeral_special(5)
tff(fact_3971_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_3972_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_3973_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_3974_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_3975_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_3976_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_3977_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_3978_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_3979_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_3980_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_3981_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_3982_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_3983_add_Oright__neutral,axiom,
    ! [A: literal] : ( aa_literal_literal(aa_lit1193766896341886811iteral(plus_plus_literal,A),zero_zero_literal) = A ) ).

% add.right_neutral
tff(fact_3984_add_Oright__neutral,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ) ).

% add.right_neutral
tff(fact_3985_add_Oright__neutral,axiom,
    ! [A: multis2468970476368604999at_nat] : ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),zero_z1048942125864253310at_nat) = A ) ).

% add.right_neutral
tff(fact_3986_add_Oright__neutral,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ) ).

% add.right_neutral
tff(fact_3987_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_3988_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_3989_double__zero__sym,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% double_zero_sym
tff(fact_3990_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_3991_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_3992_add__cancel__left__left,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) = A )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_left
tff(fact_3993_add__cancel__left__left,axiom,
    ! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B),A) = A )
    <=> ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_left_left
tff(fact_3994_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_3995_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_3996_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_3997_add__cancel__left__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = A )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_right
tff(fact_3998_add__cancel__left__right,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B) = A )
    <=> ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_left_right
tff(fact_3999_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_4000_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_4001_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_4002_add__cancel__right__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_left
tff(fact_4003_add__cancel__right__left,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( A = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B),A) )
    <=> ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_right_left
tff(fact_4004_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_4005_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_4006_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_4007_add__cancel__right__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_right
tff(fact_4008_add__cancel__right__right,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( A = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B) )
    <=> ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_right_right
tff(fact_4009_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_4010_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_4011_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_4012_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_4013_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_4014_add__0,axiom,
    ! [A: literal] : ( aa_literal_literal(aa_lit1193766896341886811iteral(plus_plus_literal,zero_zero_literal),A) = A ) ).

% add_0
tff(fact_4015_add__0,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ) ).

% add_0
tff(fact_4016_add__0,axiom,
    ! [A: multis2468970476368604999at_nat] : ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,zero_z1048942125864253310at_nat),A) = A ) ).

% add_0
tff(fact_4017_add__0,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ) ).

% add_0
tff(fact_4018_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_4019_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_4020_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_4021_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_4022_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_4023_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_4024_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_4025_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_4026_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ) ).

% add_diff_cancel
tff(fact_4027_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_4028_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),B) = A ) ).

% diff_add_cancel
tff(fact_4029_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_4030_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ) ).

% add_diff_cancel_left
tff(fact_4031_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ).

% add_diff_cancel_left
tff(fact_4032_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ).

% add_diff_cancel_left
tff(fact_4033_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),A) = B ) ).

% add_diff_cancel_left'
tff(fact_4034_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_4035_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_4036_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ) ).

% add_diff_cancel_right
tff(fact_4037_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ).

% add_diff_cancel_right
tff(fact_4038_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ).

% add_diff_cancel_right
tff(fact_4039_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ) ).

% add_diff_cancel_right'
tff(fact_4040_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_4041_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_4042_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_4043_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ) ).

% minus_add_distrib
tff(fact_4044_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ) ).

% minus_add_distrib
tff(fact_4045_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_4046_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = B ) ).

% minus_add_cancel
tff(fact_4047_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = B ) ).

% minus_add_cancel
tff(fact_4048_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_4049_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) = B ) ).

% add_minus_cancel
tff(fact_4050_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) = B ) ).

% add_minus_cancel
tff(fact_4051_add__le__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% add_le_same_cancel1
tff(fact_4052_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_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_4053_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_4054_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_4055_add__le__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% add_le_same_cancel2
tff(fact_4056_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(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_4057_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_4058_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_4059_le__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ).

% le_add_same_cancel1
tff(fact_4060_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(aa_rat_fun_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_4061_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_4062_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_4063_le__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ).

% le_add_same_cancel2
tff(fact_4064_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(aa_rat_fun_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_4065_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_4066_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_4067_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_4068_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(aa_rat_fun_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_4069_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_4070_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_4071_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(aa_rat_fun_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_4072_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_4073_add__less__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% add_less_same_cancel1
tff(fact_4074_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_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_4075_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_4076_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_4077_add__less__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% add_less_same_cancel2
tff(fact_4078_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_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_4079_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_4080_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_4081_less__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ).

% less_add_same_cancel1
tff(fact_4082_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_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_4083_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_4084_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_4085_less__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ).

% less_add_same_cancel2
tff(fact_4086_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_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_4087_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_4088_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_4089_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_4090_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(aa_rat_fun_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_4091_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_4092_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_4093_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(aa_rat_fun_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_4094_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_4095_diff__add__zero,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] : ( minus_4286766774447292334at_nat(A,aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B)) = zero_z1048942125864253310at_nat ) ).

% diff_add_zero
tff(fact_4096_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_4097_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),numeral_numeral_nat(V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),numeral_numeral_nat(V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),numeral_numeral_nat(V))) ) ).

% distrib_right_numeral
tff(fact_4098_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_4099_distrib__right__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),aa_num_Code_integer(numera6620942414471956472nteger,V)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_num_Code_integer(numera6620942414471956472nteger,V))) ) ).

% distrib_right_numeral
tff(fact_4100_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),numeral_numeral_rat(V)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),numeral_numeral_rat(V))) ) ).

% distrib_right_numeral
tff(fact_4101_distrib__right__numeral,axiom,
    ! [A: code_natural,B: code_natural,V: num] : ( times_2397367101498566445atural(plus_p4538020629002901425atural(A,B),numera5444537566228673987atural(V)) = plus_p4538020629002901425atural(times_2397367101498566445atural(A,numera5444537566228673987atural(V)),times_2397367101498566445atural(B,numera5444537566228673987atural(V))) ) ).

% distrib_right_numeral
tff(fact_4102_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),C)) ) ).

% distrib_left_numeral
tff(fact_4103_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: 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),C)) = 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)),C)) ) ).

% distrib_left_numeral
tff(fact_4104_distrib__left__numeral,axiom,
    ! [V: num,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),C)) ) ).

% distrib_left_numeral
tff(fact_4105_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),C)) ) ).

% distrib_left_numeral
tff(fact_4106_distrib__left__numeral,axiom,
    ! [V: num,B: code_natural,C: code_natural] : ( times_2397367101498566445atural(numera5444537566228673987atural(V),plus_p4538020629002901425atural(B,C)) = plus_p4538020629002901425atural(times_2397367101498566445atural(numera5444537566228673987atural(V),B),times_2397367101498566445atural(numera5444537566228673987atural(V),C)) ) ).

% distrib_left_numeral
tff(fact_4107_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_4108_ab__left__minus,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A) = zero_z3403309356797280102nteger ) ).

% ab_left_minus
tff(fact_4109_ab__left__minus,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ) ).

% ab_left_minus
tff(fact_4110_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_4111_add_Oright__inverse,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ) ).

% add.right_inverse
tff(fact_4112_add_Oright__inverse,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ) ).

% add.right_inverse
tff(fact_4113_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: 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,W))),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,plus_plus_num(V,W)))),Y) ) ).

% semiring_norm(168)
tff(fact_4114_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,plus_plus_num(V,W)))),Y) ) ).

% semiring_norm(168)
tff(fact_4115_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(plus_plus_num(V,W)))),Y) ) ).

% semiring_norm(168)
tff(fact_4116_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ) ).

% add_neg_numeral_simps(3)
tff(fact_4117_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N))) ) ).

% add_neg_numeral_simps(3)
tff(fact_4118_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(M)),numeral_numeral_rat(N))) ) ).

% add_neg_numeral_simps(3)
tff(fact_4119_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat(N) = one_one_nat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_4120_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_int(numeral_numeral_int,N) = one_one_int )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_4121_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_Code_integer(numera6620942414471956472nteger,N) = one_one_Code_integer )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_4122_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat(N) = one_one_rat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_4123_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera5444537566228673987atural(N) = one_one_Code_natural )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_4124_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat = numeral_numeral_nat(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_4125_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_4126_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_integer = aa_num_Code_integer(numera6620942414471956472nteger,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_4127_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat = numeral_numeral_rat(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_4128_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_natural = numera5444537566228673987atural(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_4129_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,plus_plus_num(M,N))) ) ).

% diff_numeral_simps(3)
tff(fact_4130_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,plus_plus_num(M,N))) ) ).

% diff_numeral_simps(3)
tff(fact_4131_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),numeral_numeral_rat(N)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(plus_plus_num(M,N))) ) ).

% diff_numeral_simps(3)
tff(fact_4132_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,plus_plus_num(M,N)) ) ).

% diff_numeral_simps(2)
tff(fact_4133_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,plus_plus_num(M,N)) ) ).

% diff_numeral_simps(2)
tff(fact_4134_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,numeral_numeral_rat(M)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = numeral_numeral_rat(plus_plus_num(M,N)) ) ).

% diff_numeral_simps(2)
tff(fact_4135_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_4136_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A) ) ).

% uminus_add_conv_diff
tff(fact_4137_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A) ) ).

% uminus_add_conv_diff
tff(fact_4138_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_4139_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) ) ).

% diff_minus_eq_add
tff(fact_4140_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) ) ).

% diff_minus_eq_add
tff(fact_4141_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_4142_add__neg__numeral__special_I7_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_4143_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_4144_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_4145_add__neg__numeral__special_I8_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_4146_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_4147_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_4148_div__mult__self4,axiom,
    ! [B: int,C: 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),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_4149_div__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).

% div_mult_self4
tff(fact_4150_div__mult__self4,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(plus_p4538020629002901425atural(times_2397367101498566445atural(B,C),A),B) = plus_p4538020629002901425atural(C,divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self4
tff(fact_4151_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_4152_div__mult__self3,axiom,
    ! [B: int,C: 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,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_4153_div__mult__self3,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).

% div_mult_self3
tff(fact_4154_div__mult__self3,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(plus_p4538020629002901425atural(times_2397367101498566445atural(C,B),A),B) = plus_p4538020629002901425atural(C,divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self3
tff(fact_4155_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_4156_div__mult__self2,axiom,
    ! [B: int,A: int,C: 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),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_4157_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).

% div_mult_self2
tff(fact_4158_div__mult__self2,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(plus_p4538020629002901425atural(A,times_2397367101498566445atural(B,C)),B) = plus_p4538020629002901425atural(C,divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self2
tff(fact_4159_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_4160_div__mult__self1,axiom,
    ! [B: int,A: int,C: 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,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_4161_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).

% div_mult_self1
tff(fact_4162_div__mult__self1,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(plus_p4538020629002901425atural(A,times_2397367101498566445atural(C,B)),B) = plus_p4538020629002901425atural(C,divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self1
tff(fact_4163_one__plus__numeral,axiom,
    ! [N: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),numeral_numeral_nat(N)) = numeral_numeral_nat(plus_plus_num(one,N)) ) ).

% one_plus_numeral
tff(fact_4164_one__plus__numeral,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,plus_plus_num(one,N)) ) ).

% one_plus_numeral
tff(fact_4165_one__plus__numeral,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_num_Code_integer(numera6620942414471956472nteger,plus_plus_num(one,N)) ) ).

% one_plus_numeral
tff(fact_4166_one__plus__numeral,axiom,
    ! [N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),numeral_numeral_rat(N)) = numeral_numeral_rat(plus_plus_num(one,N)) ) ).

% one_plus_numeral
tff(fact_4167_one__plus__numeral,axiom,
    ! [N: num] : ( plus_p4538020629002901425atural(one_one_Code_natural,numera5444537566228673987atural(N)) = numera5444537566228673987atural(plus_plus_num(one,N)) ) ).

% one_plus_numeral
tff(fact_4168_numeral__plus__one,axiom,
    ! [N: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(N)),one_one_nat) = numeral_numeral_nat(plus_plus_num(N,one)) ) ).

% numeral_plus_one
tff(fact_4169_numeral__plus__one,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),one_one_int) = aa_num_int(numeral_numeral_int,plus_plus_num(N,one)) ) ).

% numeral_plus_one
tff(fact_4170_numeral__plus__one,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,plus_plus_num(N,one)) ) ).

% numeral_plus_one
tff(fact_4171_numeral__plus__one,axiom,
    ! [N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(N)),one_one_rat) = numeral_numeral_rat(plus_plus_num(N,one)) ) ).

% numeral_plus_one
tff(fact_4172_numeral__plus__one,axiom,
    ! [N: num] : ( plus_p4538020629002901425atural(numera5444537566228673987atural(N),one_one_Code_natural) = numera5444537566228673987atural(plus_plus_num(N,one)) ) ).

% numeral_plus_one
tff(fact_4173_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4174_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4175_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4176_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4177_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4178_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4179_diff__numeral__special_I3_J,axiom,
    ! [N: 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,N))) = aa_num_int(numeral_numeral_int,plus_plus_num(one,N)) ) ).

% diff_numeral_special(3)
tff(fact_4180_diff__numeral__special_I3_J,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,plus_plus_num(one,N)) ) ).

% diff_numeral_special(3)
tff(fact_4181_diff__numeral__special_I3_J,axiom,
    ! [N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = numeral_numeral_rat(plus_plus_num(one,N)) ) ).

% diff_numeral_special(3)
tff(fact_4182_diff__numeral__special_I4_J,axiom,
    ! [M: num] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,plus_plus_num(M,one))) ) ).

% diff_numeral_special(4)
tff(fact_4183_diff__numeral__special_I4_J,axiom,
    ! [M: num] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,plus_plus_num(M,one))) ) ).

% diff_numeral_special(4)
tff(fact_4184_diff__numeral__special_I4_J,axiom,
    ! [M: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(plus_plus_num(M,one))) ) ).

% diff_numeral_special(4)
tff(fact_4185_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),one_one_Code_integer)
    <=> ord_less_eq_num(N,one) ) ).

% numeral_le_one_iff
tff(fact_4186_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(numera5444537566228673987atural(N)),one_one_Code_natural)
    <=> ord_less_eq_num(N,one) ) ).

% numeral_le_one_iff
tff(fact_4187_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(N)),one_one_rat)
    <=> ord_less_eq_num(N,one) ) ).

% numeral_le_one_iff
tff(fact_4188_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(N)),one_one_nat)
    <=> ord_less_eq_num(N,one) ) ).

% numeral_le_one_iff
tff(fact_4189_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),one_one_int)
    <=> ord_less_eq_num(N,one) ) ).

% numeral_le_one_iff
tff(fact_4190_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),numeral_numeral_nat(N))
    <=> ord_less_num(one,N) ) ).

% one_less_numeral_iff
tff(fact_4191_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,N))
    <=> ord_less_num(one,N) ) ).

% one_less_numeral_iff
tff(fact_4192_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N))
    <=> ord_less_num(one,N) ) ).

% one_less_numeral_iff
tff(fact_4193_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),numeral_numeral_rat(N))
    <=> ord_less_num(one,N) ) ).

% one_less_numeral_iff
tff(fact_4194_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(one_one_Code_natural),numera5444537566228673987atural(N))
    <=> ord_less_num(one,N) ) ).

% one_less_numeral_iff
tff(fact_4195_pred__subset__eq,axiom,
    ! [R: set_se6260736226359567993nt_int,S: set_se6260736226359567993nt_int] :
      ( ord_le2758740137681283170_int_o(aTP_Lamm_ag(R),aTP_Lamm_ag(S))
    <=> aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(R),S) ) ).

% pred_subset_eq
tff(fact_4196_pred__subset__eq,axiom,
    ! [R: set_set_nat,S: set_set_nat] :
      ( ord_le6468474429079055883_nat_o(aTP_Lamm_ah(R),aTP_Lamm_ah(S))
    <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(R),S) ) ).

% pred_subset_eq
tff(fact_4197_pred__subset__eq,axiom,
    ! [R: set_int,S: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_aj(R),aTP_Lamm_aj(S))
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,R),S) ) ).

% pred_subset_eq
tff(fact_4198_pred__subset__eq,axiom,
    ! [R: set_fu3555266933746557007_nat_o,S: set_fu3555266933746557007_nat_o] :
      ( ord_le2214281183343576076at_o_o(aTP_Lamm_ak(R),aTP_Lamm_ak(S))
    <=> aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(R),S) ) ).

% pred_subset_eq
tff(fact_4199_pred__subset__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ord_le7240017405068453697_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ai,R),aa_set_nat_fun_nat_o(aTP_Lamm_ai,S))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,R),S) ) ).

% pred_subset_eq
tff(fact_4200_less__eq__set__def,axiom,
    ! [A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A2),B2)
    <=> ord_le2758740137681283170_int_o(aTP_Lamm_ag(A2),aTP_Lamm_ag(B2)) ) ).

% less_eq_set_def
tff(fact_4201_less__eq__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
    <=> ord_le6468474429079055883_nat_o(aTP_Lamm_ah(A2),aTP_Lamm_ah(B2)) ) ).

% less_eq_set_def
tff(fact_4202_less__eq__set__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),B2)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_aj(A2),aTP_Lamm_aj(B2)) ) ).

% less_eq_set_def
tff(fact_4203_less__eq__set__def,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),B2)
    <=> ord_le2214281183343576076at_o_o(aTP_Lamm_ak(A2),aTP_Lamm_ak(B2)) ) ).

% less_eq_set_def
tff(fact_4204_less__eq__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
    <=> ord_le7240017405068453697_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ai,A2),aa_set_nat_fun_nat_o(aTP_Lamm_ai,B2)) ) ).

% less_eq_set_def
tff(fact_4205_less__set__def,axiom,
    ! [A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le1924305788584680229nt_int(A2),B2)
    <=> ord_le8970291702602496086_int_o(aTP_Lamm_ag(A2),aTP_Lamm_ag(B2)) ) ).

% less_set_def
tff(fact_4206_less__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A2),B2)
    <=> ord_le921202756137246487_nat_o(aTP_Lamm_ah(A2),aTP_Lamm_ah(B2)) ) ).

% less_set_def
tff(fact_4207_less__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
    <=> ord_less_fun_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ai,A2),aa_set_nat_fun_nat_o(aTP_Lamm_ai,B2)) ) ).

% less_set_def
tff(fact_4208_less__set__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A2),B2)
    <=> ord_less_fun_int_o(aTP_Lamm_aj(A2),aTP_Lamm_aj(B2)) ) ).

% less_set_def
tff(fact_4209_less__set__def,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(A2),B2)
    <=> ord_le5201174217191475200at_o_o(aTP_Lamm_ak(A2),aTP_Lamm_ak(B2)) ) ).

% less_set_def
tff(fact_4210_less__assn__def,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        & ( A != B ) ) ) ).

% less_assn_def
tff(fact_4211_Collect__subset,axiom,
    ! [A2: set_se6260736226359567993nt_int,P: fun_se7249947736802449410_int_o] : aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_gh(A2),P))),A2) ).

% Collect_subset
tff(fact_4212_Collect__subset,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_o] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_gi(A2),P))),A2) ).

% Collect_subset
tff(fact_4213_Collect__subset,axiom,
    ! [A2: set_int,P: fun_int_o] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gj(A2),P))),A2) ).

% Collect_subset
tff(fact_4214_Collect__subset,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,P: fun_fu5135969500543783852at_o_o] : aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_gk(A2),P))),A2) ).

% Collect_subset
tff(fact_4215_Collect__subset,axiom,
    ! [A2: set_nat,P: fun_nat_o] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gl(A2),P))),A2) ).

% Collect_subset
tff(fact_4216_conj__subset__def,axiom,
    ! [A2: set_se6260736226359567993nt_int,P: fun_se7249947736802449410_int_o,Q: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A2),collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_bu(P),Q)))
    <=> ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A2),collec5210948495886036740nt_int(P))
        & aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A2),collec5210948495886036740nt_int(Q)) ) ) ).

% conj_subset_def
tff(fact_4217_conj__subset__def,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bv(P),Q)))
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),collect_set_nat(P))
        & aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),collect_set_nat(Q)) ) ) ).

% conj_subset_def
tff(fact_4218_conj__subset__def,axiom,
    ! [A2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bw(P),Q)))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),collect_int(P))
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),collect_int(Q)) ) ) ).

% conj_subset_def
tff(fact_4219_conj__subset__def,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,P: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_bx(P),Q)))
    <=> ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),collec6407654713642994138_nat_o(P))
        & aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),collec6407654713642994138_nat_o(Q)) ) ) ).

% conj_subset_def
tff(fact_4220_conj__subset__def,axiom,
    ! [A2: set_nat,P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bz(P),Q)))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),collect_nat(P))
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),collect_nat(Q)) ) ) ).

% conj_subset_def
tff(fact_4221_ordering__top_Oextremum,axiom,
    ! [Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,Top: nat,A: nat] :
      ( aa_nat_o(ordering_top_nat(Less_eq,Less),Top)
     => aa_nat_o(aa_nat_fun_nat_o(Less_eq,A),Top) ) ).

% ordering_top.extremum
tff(fact_4222_ordering__top_Oextremum__strict,axiom,
    ! [Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,Top: nat,A: nat] :
      ( aa_nat_o(ordering_top_nat(Less_eq,Less),Top)
     => ~ aa_nat_o(aa_nat_fun_nat_o(Less,Top),A) ) ).

% ordering_top.extremum_strict
tff(fact_4223_ordering__top_Oextremum__unique,axiom,
    ! [Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,Top: nat,A: nat] :
      ( aa_nat_o(ordering_top_nat(Less_eq,Less),Top)
     => ( aa_nat_o(aa_nat_fun_nat_o(Less_eq,Top),A)
      <=> ( A = Top ) ) ) ).

% ordering_top.extremum_unique
tff(fact_4224_ordering__top_Onot__eq__extremum,axiom,
    ! [Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,Top: nat,A: nat] :
      ( aa_nat_o(ordering_top_nat(Less_eq,Less),Top)
     => ( ( A != Top )
      <=> aa_nat_o(aa_nat_fun_nat_o(Less,A),Top) ) ) ).

% ordering_top.not_eq_extremum
tff(fact_4225_ordering__top_Oextremum__uniqueI,axiom,
    ! [Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,Top: nat,A: nat] :
      ( aa_nat_o(ordering_top_nat(Less_eq,Less),Top)
     => ( aa_nat_o(aa_nat_fun_nat_o(Less_eq,Top),A)
       => ( A = Top ) ) ) ).

% ordering_top.extremum_uniqueI
tff(fact_4226_semilattice__neutr__order_Oeq__neutr__iff,axiom,
    ! [F: fun_nat_fun_nat_nat,Z: nat,Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( semila1623282765462674594er_nat(F,Z,Less_eq,Less)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(F,A),B) = Z )
      <=> ( ( A = Z )
          & ( B = Z ) ) ) ) ).

% semilattice_neutr_order.eq_neutr_iff
tff(fact_4227_semilattice__neutr__order_Oneutr__eq__iff,axiom,
    ! [F: fun_nat_fun_nat_nat,Z: nat,Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( semila1623282765462674594er_nat(F,Z,Less_eq,Less)
     => ( ( Z = aa_nat_nat(aa_nat_fun_nat_nat(F,A),B) )
      <=> ( ( A = Z )
          & ( B = Z ) ) ) ) ).

% semilattice_neutr_order.neutr_eq_iff
tff(fact_4228_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_4229_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_4230_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_4231_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_4232_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_4233_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_4234_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_4235_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_4236_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_4237_add_Oassoc,axiom,
    ! [A: literal,B: literal,C: literal] : ( aa_literal_literal(aa_lit1193766896341886811iteral(plus_plus_literal,aa_literal_literal(aa_lit1193766896341886811iteral(plus_plus_literal,A),B)),C) = aa_literal_literal(aa_lit1193766896341886811iteral(plus_plus_literal,A),aa_literal_literal(aa_lit1193766896341886811iteral(plus_plus_literal,B),C)) ) ).

% add.assoc
tff(fact_4238_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add.assoc
tff(fact_4239_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add.assoc
tff(fact_4240_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add.assoc
tff(fact_4241_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_4242_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_4243_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_4244_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_4245_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_4246_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_4247_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_4248_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_4249_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_4250_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_4251_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_4252_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_4253_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_4254_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_4255_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_4256_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_4257_add_Oright__assoc,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add.right_assoc
tff(fact_4258_add_Oright__assoc,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add.right_assoc
tff(fact_4259_add_Oright__assoc,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add.right_assoc
tff(fact_4260_add_Oright__commute,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ).

% add.right_commute
tff(fact_4261_add_Oright__commute,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ).

% add.right_commute
tff(fact_4262_add_Oright__commute,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ).

% add.right_commute
tff(fact_4263_numerals_I1_J,axiom,
    numeral_numeral_nat(one) = one_one_nat ).

% numerals(1)
tff(fact_4264_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_4265_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_4266_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_4267_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_4268_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_4269_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_4270_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_4271_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_4272_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_4273_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_mono
tff(fact_4274_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_mono
tff(fact_4275_add__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_mono
tff(fact_4276_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_left_mono
tff(fact_4277_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_left_mono
tff(fact_4278_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_left_mono
tff(fact_4279_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C3: nat] : ( B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ) ).

% less_eqE
tff(fact_4280_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_right_mono
tff(fact_4281_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_right_mono
tff(fact_4282_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_right_mono
tff(fact_4283_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C4: nat] : ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) ) ) ).

% le_iff_add
tff(fact_4284_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_4285_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_4286_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_4287_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_4288_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_4289_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_4290_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ) ).

% comm_monoid_add_class.add_0
tff(fact_4291_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: multis2468970476368604999at_nat] : ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,zero_z1048942125864253310at_nat),A) = A ) ).

% comm_monoid_add_class.add_0
tff(fact_4292_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ) ).

% comm_monoid_add_class.add_0
tff(fact_4293_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_4294_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_4295_add_Ocomm__neutral,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ) ).

% add.comm_neutral
tff(fact_4296_add_Ocomm__neutral,axiom,
    ! [A: multis2468970476368604999at_nat] : ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),zero_z1048942125864253310at_nat) = A ) ).

% add.comm_neutral
tff(fact_4297_add_Ocomm__neutral,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ) ).

% add.comm_neutral
tff(fact_4298_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_4299_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_4300_add_Ogroup__left__neutral,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ) ).

% add.group_left_neutral
tff(fact_4301_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ) ).

% add.group_left_neutral
tff(fact_4302_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_4303_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_4304_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_4305_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_4306_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_4307_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_4308_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_4309_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_4310_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_4311_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_4312_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_strict_mono
tff(fact_4313_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_strict_mono
tff(fact_4314_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_strict_mono
tff(fact_4315_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_strict_left_mono
tff(fact_4316_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_strict_left_mono
tff(fact_4317_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_strict_left_mono
tff(fact_4318_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_strict_right_mono
tff(fact_4319_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_strict_right_mono
tff(fact_4320_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_strict_right_mono
tff(fact_4321_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_4322_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_4323_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_4324_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_4325_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_4326_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_4327_combine__common__factor,axiom,
    ! [A: code_integer,E2: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),E2)),C) ) ).

% combine_common_factor
tff(fact_4328_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),E2)),C) ) ).

% combine_common_factor
tff(fact_4329_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E2)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E2)),C) ) ).

% combine_common_factor
tff(fact_4330_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E2)),C) ) ).

% combine_common_factor
tff(fact_4331_distrib__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% distrib_right
tff(fact_4332_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% distrib_right
tff(fact_4333_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% distrib_right
tff(fact_4334_distrib__right,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% distrib_right
tff(fact_4335_distrib__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ).

% distrib_left
tff(fact_4336_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ).

% distrib_left
tff(fact_4337_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ) ).

% distrib_left
tff(fact_4338_distrib__left,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ).

% distrib_left
tff(fact_4339_comm__semiring__class_Odistrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% comm_semiring_class.distrib
tff(fact_4340_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% comm_semiring_class.distrib
tff(fact_4341_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% comm_semiring_class.distrib
tff(fact_4342_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% comm_semiring_class.distrib
tff(fact_4343_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ).

% ring_class.ring_distribs(1)
tff(fact_4344_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ).

% ring_class.ring_distribs(1)
tff(fact_4345_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ).

% ring_class.ring_distribs(1)
tff(fact_4346_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% ring_class.ring_distribs(2)
tff(fact_4347_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% ring_class.ring_distribs(2)
tff(fact_4348_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% ring_class.ring_distribs(2)
tff(fact_4349_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4350_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4351_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = C )
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) ) ) ).

% diff_eq_eq
tff(fact_4352_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).

% diff_eq_eq
tff(fact_4353_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4354_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4355_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% add_diff_eq
tff(fact_4356_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% add_diff_eq
tff(fact_4357_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ).

% diff_diff_eq2
tff(fact_4358_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ).

% diff_diff_eq2
tff(fact_4359_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ).

% diff_add_eq
tff(fact_4360_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ).

% diff_add_eq
tff(fact_4361_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),B) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_4362_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_4363_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) = A )
     => ( C = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ) ) ).

% add_implies_diff
tff(fact_4364_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_4365_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = A )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_4366_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% diff_diff_eq
tff(fact_4367_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% diff_diff_eq
tff(fact_4368_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% diff_diff_eq
tff(fact_4369_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_4370_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% is_num_normalize(8)
tff(fact_4371_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% is_num_normalize(8)
tff(fact_4372_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_4373_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% add.inverse_distrib_swap
tff(fact_4374_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% add.inverse_distrib_swap
tff(fact_4375_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_4376_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_4377_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_4378_add_Osafe__commute,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(syntax3730441303064801268at_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ) ) ).

% add.safe_commute
tff(fact_4379_add_Osafe__commute,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(syntax4682126007086162916at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),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,B),A) ) ) ).

% add.safe_commute
tff(fact_4380_add_Osafe__commute,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(syntax5678989248478167196nt_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),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,B),A) ) ) ).

% add.safe_commute
tff(fact_4381_add__nonpos__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),zero_z3403309356797280102nteger)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y) = zero_z3403309356797280102nteger )
        <=> ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_4382_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_4383_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_4384_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_4385_add__nonneg__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y) = zero_z3403309356797280102nteger )
        <=> ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_4386_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_4387_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_4388_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_4389_add__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_nonpos_nonpos
tff(fact_4390_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_4391_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_4392_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_4393_add__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_4394_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_4395_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_4396_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_4397_add__increasing2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% add_increasing2
tff(fact_4398_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing2
tff(fact_4399_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing2
tff(fact_4400_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing2
tff(fact_4401_add__decreasing2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_4402_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_4403_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_4404_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_4405_add__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% add_increasing
tff(fact_4406_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing
tff(fact_4407_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing
tff(fact_4408_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing
tff(fact_4409_add__decreasing,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ) ) ).

% add_decreasing
tff(fact_4410_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_4411_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_4412_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing
tff(fact_4413_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_4414_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_4415_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_4416_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_4417_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_4418_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_4419_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_4420_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_4421_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_4422_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_4423_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_4424_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_4425_pos__add__strict,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% pos_add_strict
tff(fact_4426_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% pos_add_strict
tff(fact_4427_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% pos_add_strict
tff(fact_4428_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% pos_add_strict
tff(fact_4429_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C3: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_4430_add__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% add_pos_pos
tff(fact_4431_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_pos_pos
tff(fact_4432_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_4433_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_4434_add__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_neg_neg
tff(fact_4435_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_4436_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_4437_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_4438_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_le_eq
tff(fact_4439_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_le_eq
tff(fact_4440_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% le_diff_eq
tff(fact_4441_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),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)),C) ) ).

% le_diff_eq
tff(fact_4442_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_4443_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A)) ) ).

% le_add_diff
tff(fact_4444_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_4445_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_4446_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_4447_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_4448_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_4449_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_4450_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_4451_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_4452_add__mono1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),one_one_Code_integer)) ) ).

% add_mono1
tff(fact_4453_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(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),one_one_rat)) ) ).

% add_mono1
tff(fact_4454_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_4455_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_4456_less__add__one,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)) ).

% less_add_one
tff(fact_4457_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)) ).

% less_add_one
tff(fact_4458_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_4459_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_4460_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_less_eq
tff(fact_4461_diff__less__eq,axiom,
    ! [A: int,B: int,C: 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)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_less_eq
tff(fact_4462_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% less_diff_eq
tff(fact_4463_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),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)),C) ) ).

% less_diff_eq
tff(fact_4464_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_4465_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4466_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4467_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_4468_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4469_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4470_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_4471_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).

% add.inverse_unique
tff(fact_4472_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_4473_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_4474_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A) = zero_z3403309356797280102nteger ) ).

% ab_group_add_class.ab_left_minus
tff(fact_4475_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ) ).

% ab_group_add_class.ab_left_minus
tff(fact_4476_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_4477_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% add_eq_0_iff
tff(fact_4478_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_4479_one__plus__numeral__commute,axiom,
    ! [X: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),numeral_numeral_nat(X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(X)),one_one_nat) ) ).

% one_plus_numeral_commute
tff(fact_4480_one__plus__numeral__commute,axiom,
    ! [X: num] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ) ).

% one_plus_numeral_commute
tff(fact_4481_one__plus__numeral__commute,axiom,
    ! [X: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)),one_one_Code_integer) ) ).

% one_plus_numeral_commute
tff(fact_4482_one__plus__numeral__commute,axiom,
    ! [X: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),numeral_numeral_rat(X)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(X)),one_one_rat) ) ).

% one_plus_numeral_commute
tff(fact_4483_one__plus__numeral__commute,axiom,
    ! [X: num] : ( plus_p4538020629002901425atural(one_one_Code_natural,numera5444537566228673987atural(X)) = plus_p4538020629002901425atural(numera5444537566228673987atural(X),one_one_Code_natural) ) ).

% one_plus_numeral_commute
tff(fact_4484_square__diff__square__factored,axiom,
    ! [X: code_integer,Y: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),Y)) ) ).

% square_diff_square_factored
tff(fact_4485_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)) ) ).

% square_diff_square_factored
tff(fact_4486_square__diff__square__factored,axiom,
    ! [X: int,Y: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) ) ).

% square_diff_square_factored
tff(fact_4487_eq__add__iff2,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E2)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2) )
    <=> ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A)),E2)),D2) ) ) ).

% eq_add_iff2
tff(fact_4488_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2) )
    <=> ( C = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E2)),D2) ) ) ).

% eq_add_iff2
tff(fact_4489_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D2) ) ) ).

% eq_add_iff2
tff(fact_4490_eq__add__iff1,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E2)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),E2)),C) = D2 ) ) ).

% eq_add_iff1
tff(fact_4491_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E2)),C) = D2 ) ) ).

% eq_add_iff1
tff(fact_4492_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C) = D2 ) ) ).

% eq_add_iff1
tff(fact_4493_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_4494_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_4495_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_4496_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_4497_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ) ).

% diff_conv_add_uminus
tff(fact_4498_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ) ).

% diff_conv_add_uminus
tff(fact_4499_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4500_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),B) )
     => ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4501_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4502_eq__numeral__iff__iszero_I7_J,axiom,
    ! [X: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X)) = one_one_int )
    <=> aa_int_o(ring_1_iszero_int,aa_num_int(numeral_numeral_int,plus_plus_num(X,one))) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_4503_eq__numeral__iff__iszero_I7_J,axiom,
    ! [X: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)) = one_one_Code_integer )
    <=> aa_Code_integer_o(ring_16219924574208605041nteger,aa_num_Code_integer(numera6620942414471956472nteger,plus_plus_num(X,one))) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_4504_eq__numeral__iff__iszero_I7_J,axiom,
    ! [X: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(X)) = one_one_rat )
    <=> aa_rat_o(ring_1_iszero_rat,numeral_numeral_rat(plus_plus_num(X,one))) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_4505_eq__numeral__iff__iszero_I8_J,axiom,
    ! [Y: num] :
      ( ( one_one_int = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Y)) )
    <=> aa_int_o(ring_1_iszero_int,aa_num_int(numeral_numeral_int,plus_plus_num(one,Y))) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_4506_eq__numeral__iff__iszero_I8_J,axiom,
    ! [Y: num] :
      ( ( one_one_Code_integer = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Y)) )
    <=> aa_Code_integer_o(ring_16219924574208605041nteger,aa_num_Code_integer(numera6620942414471956472nteger,plus_plus_num(one,Y))) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_4507_eq__numeral__iff__iszero_I8_J,axiom,
    ! [Y: num] :
      ( ( one_one_rat = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(Y)) )
    <=> aa_rat_o(ring_1_iszero_rat,numeral_numeral_rat(plus_plus_num(one,Y))) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_4508_mult__numeral__1__right,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),numeral_numeral_nat(one)) = A ) ).

% mult_numeral_1_right
tff(fact_4509_mult__numeral__1__right,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,one)) = A ) ).

% mult_numeral_1_right
tff(fact_4510_mult__numeral__1__right,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,one)) = A ) ).

% mult_numeral_1_right
tff(fact_4511_mult__numeral__1__right,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(one)) = A ) ).

% mult_numeral_1_right
tff(fact_4512_mult__numeral__1__right,axiom,
    ! [A: code_natural] : ( times_2397367101498566445atural(A,numera5444537566228673987atural(one)) = A ) ).

% mult_numeral_1_right
tff(fact_4513_mult__numeral__1,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(one)),A) = A ) ).

% mult_numeral_1
tff(fact_4514_mult__numeral__1,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,one)),A) = A ) ).

% mult_numeral_1
tff(fact_4515_mult__numeral__1,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)),A) = A ) ).

% mult_numeral_1
tff(fact_4516_mult__numeral__1,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(one)),A) = A ) ).

% mult_numeral_1
tff(fact_4517_mult__numeral__1,axiom,
    ! [A: code_natural] : ( times_2397367101498566445atural(numera5444537566228673987atural(one),A) = A ) ).

% mult_numeral_1
tff(fact_4518_numeral__One,axiom,
    numeral_numeral_nat(one) = one_one_nat ).

% numeral_One
tff(fact_4519_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_4520_numeral__One,axiom,
    aa_num_Code_integer(numera6620942414471956472nteger,one) = one_one_Code_integer ).

% numeral_One
tff(fact_4521_numeral__One,axiom,
    numeral_numeral_rat(one) = one_one_rat ).

% numeral_One
tff(fact_4522_numeral__One,axiom,
    numera5444537566228673987atural(one) = one_one_Code_natural ).

% numeral_One
tff(fact_4523_add_Ocomm__monoid__axioms,axiom,
    aa_Code_integer_o(comm_m1654649897431166824nteger(plus_p5714425477246183910nteger),zero_z3403309356797280102nteger) ).

% add.comm_monoid_axioms
tff(fact_4524_add_Ocomm__monoid__axioms,axiom,
    comm_m1862543879040115196at_nat(plus_p7104986032573967614at_nat,zero_z1048942125864253310at_nat) ).

% add.comm_monoid_axioms
tff(fact_4525_add_Ocomm__monoid__axioms,axiom,
    aa_rat_o(comm_monoid_rat(plus_plus_rat),zero_zero_rat) ).

% add.comm_monoid_axioms
tff(fact_4526_add_Ocomm__monoid__axioms,axiom,
    aa_nat_o(comm_monoid_nat(plus_plus_nat),zero_zero_nat) ).

% add.comm_monoid_axioms
tff(fact_4527_add_Ocomm__monoid__axioms,axiom,
    aa_int_o(comm_monoid_int(plus_plus_int),zero_zero_int) ).

% add.comm_monoid_axioms
tff(fact_4528_add_Omonoid__axioms,axiom,
    aa_literal_o(monoid_literal(plus_plus_literal),zero_zero_literal) ).

% add.monoid_axioms
tff(fact_4529_add_Omonoid__axioms,axiom,
    aa_Code_integer_o(monoid_Code_integer(plus_p5714425477246183910nteger),zero_z3403309356797280102nteger) ).

% add.monoid_axioms
tff(fact_4530_add_Omonoid__axioms,axiom,
    monoid66256789251010760at_nat(plus_p7104986032573967614at_nat,zero_z1048942125864253310at_nat) ).

% add.monoid_axioms
tff(fact_4531_add_Omonoid__axioms,axiom,
    aa_rat_o(monoid_rat(plus_plus_rat),zero_zero_rat) ).

% add.monoid_axioms
tff(fact_4532_add_Omonoid__axioms,axiom,
    aa_nat_o(monoid_nat(plus_plus_nat),zero_zero_nat) ).

% add.monoid_axioms
tff(fact_4533_add_Omonoid__axioms,axiom,
    aa_int_o(monoid_int(plus_plus_int),zero_zero_int) ).

% add.monoid_axioms
tff(fact_4534_semilattice__neutr__order_Oaxioms_I1_J,axiom,
    ! [F: fun_nat_fun_nat_nat,Z: nat,Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o] :
      ( semila1623282765462674594er_nat(F,Z,Less_eq,Less)
     => aa_nat_o(semila9081495762789891438tr_nat(F),Z) ) ).

% semilattice_neutr_order.axioms(1)
tff(fact_4535_dbl__inc__def,axiom,
    ! [X: code_integer] : ( aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,X) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),X)),one_one_Code_integer) ) ).

% dbl_inc_def
tff(fact_4536_dbl__inc__def,axiom,
    ! [X: rat] : ( aa_rat_rat(neg_nu5219082963157363817nc_rat,X) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X)),one_one_rat) ) ).

% dbl_inc_def
tff(fact_4537_dbl__inc__def,axiom,
    ! [X: int] : ( aa_int_int(neg_nu5851722552734809277nc_int,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ) ).

% dbl_inc_def
tff(fact_4538_add__strict__increasing2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_4539_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_4540_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_4541_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_4542_add__strict__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% add_strict_increasing
tff(fact_4543_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_4544_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_4545_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing
tff(fact_4546_add__pos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_4547_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_4548_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_4549_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_4550_add__nonpos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_nonpos_neg
tff(fact_4551_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_4552_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_4553_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_4554_add__nonneg__pos,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_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_4555_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_4556_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_4557_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_4558_add__neg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_neg_nonpos
tff(fact_4559_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_4560_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_4561_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_4562_sum__squares__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_4563_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_4564_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_4565_not__sum__squares__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))),zero_z3403309356797280102nteger) ).

% not_sum_squares_lt_zero
tff(fact_4566_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_4567_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_4568_discrete,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),B) ) ).

% discrete
tff(fact_4569_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_4570_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_4571_zero__less__two,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),one_one_Code_integer)) ).

% zero_less_two
tff(fact_4572_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_4573_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_4574_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_4575_le__add__iff1,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E2)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),E2)),C)),D2) ) ).

% le_add_iff1
tff(fact_4576_le__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E2)),C)),D2) ) ).

% le_add_iff1
tff(fact_4577_le__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: 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)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2))
    <=> 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)),C)),D2) ) ).

% le_add_iff1
tff(fact_4578_le__add__iff2,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E2)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A)),E2)),D2)) ) ).

% le_add_iff2
tff(fact_4579_le__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E2)),D2)) ) ).

% le_add_iff2
tff(fact_4580_le__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: 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)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D2)) ) ).

% le_add_iff2
tff(fact_4581_less__add__iff1,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E2)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),E2)),C)),D2) ) ).

% less_add_iff1
tff(fact_4582_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E2)),C)),D2) ) ).

% less_add_iff1
tff(fact_4583_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: 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)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2))
    <=> 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)),C)),D2) ) ).

% less_add_iff1
tff(fact_4584_less__add__iff2,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E2)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A)),E2)),D2)) ) ).

% less_add_iff2
tff(fact_4585_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E2)),D2)) ) ).

% less_add_iff2
tff(fact_4586_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: 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)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D2)) ) ).

% less_add_iff2
tff(fact_4587_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_4588_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_4589_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_4590_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_4591_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_4592_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B)),Z)) ) ).

% add_divide_eq_if_simps(1)
tff(fact_4593_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z)),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z)) ) ).

% add_divide_eq_if_simps(2)
tff(fact_4594_square__diff__one__factored,axiom,
    ! [X: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),one_one_Code_integer)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),one_one_Code_integer)) ) ).

% square_diff_one_factored
tff(fact_4595_square__diff__one__factored,axiom,
    ! [X: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)) ) ).

% square_diff_one_factored
tff(fact_4596_square__diff__one__factored,axiom,
    ! [X: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),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,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ) ).

% square_diff_one_factored
tff(fact_4597_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_4598_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_4599_div__add__self2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),one_one_Code_integer) ) ) ).

% div_add_self2
tff(fact_4600_div__add__self2,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(plus_p4538020629002901425atural(A,B),B) = plus_p4538020629002901425atural(divide5121882707175180666atural(A,B),one_one_Code_natural) ) ) ).

% div_add_self2
tff(fact_4601_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_4602_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_4603_div__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),one_one_Code_integer) ) ) ).

% div_add_self1
tff(fact_4604_div__add__self1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(plus_p4538020629002901425atural(B,A),B) = plus_p4538020629002901425atural(divide5121882707175180666atural(A,B),one_one_Code_natural) ) ) ).

% div_add_self1
tff(fact_4605_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(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_4606_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(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_4607_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))),B) = aa_int_int(uminus_uminus_int,B) ) ).

% mult_1s_ring_1(1)
tff(fact_4608_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ).

% mult_1s_ring_1(1)
tff(fact_4609_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(one))),B) = aa_rat_rat(uminus_uminus_rat,B) ) ).

% mult_1s_ring_1(1)
tff(fact_4610_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) = aa_int_int(uminus_uminus_int,B) ) ).

% mult_1s_ring_1(2)
tff(fact_4611_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ).

% mult_1s_ring_1(2)
tff(fact_4612_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(one))) = aa_rat_rat(uminus_uminus_rat,B) ) ).

% mult_1s_ring_1(2)
tff(fact_4613_uminus__numeral__One,axiom,
    aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% uminus_numeral_One
tff(fact_4614_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_4615_uminus__numeral__One,axiom,
    aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(one)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% uminus_numeral_One
tff(fact_4616_numeral__inc,axiom,
    ! [X: num] : ( numeral_numeral_nat(inc(X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(X)),one_one_nat) ) ).

% numeral_inc
tff(fact_4617_numeral__inc,axiom,
    ! [X: num] : ( aa_num_int(numeral_numeral_int,inc(X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ) ).

% numeral_inc
tff(fact_4618_numeral__inc,axiom,
    ! [X: num] : ( aa_num_Code_integer(numera6620942414471956472nteger,inc(X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)),one_one_Code_integer) ) ).

% numeral_inc
tff(fact_4619_numeral__inc,axiom,
    ! [X: num] : ( numeral_numeral_rat(inc(X)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(X)),one_one_rat) ) ).

% numeral_inc
tff(fact_4620_numeral__inc,axiom,
    ! [X: num] : ( numera5444537566228673987atural(inc(X)) = plus_p4538020629002901425atural(numera5444537566228673987atural(X),one_one_Code_natural) ) ).

% numeral_inc
tff(fact_4621_eq__numeral__iff__iszero_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X)) = aa_num_int(numeral_numeral_int,Y) )
    <=> aa_int_o(ring_1_iszero_int,aa_num_int(numeral_numeral_int,plus_plus_num(X,Y))) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_4622_eq__numeral__iff__iszero_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)) = aa_num_Code_integer(numera6620942414471956472nteger,Y) )
    <=> aa_Code_integer_o(ring_16219924574208605041nteger,aa_num_Code_integer(numera6620942414471956472nteger,plus_plus_num(X,Y))) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_4623_eq__numeral__iff__iszero_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(X)) = numeral_numeral_rat(Y) )
    <=> aa_rat_o(ring_1_iszero_rat,numeral_numeral_rat(plus_plus_num(X,Y))) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_4624_eq__numeral__iff__iszero_I2_J,axiom,
    ! [X: num,Y: num] :
      ( ( aa_num_int(numeral_numeral_int,X) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Y)) )
    <=> aa_int_o(ring_1_iszero_int,aa_num_int(numeral_numeral_int,plus_plus_num(X,Y))) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_4625_eq__numeral__iff__iszero_I2_J,axiom,
    ! [X: num,Y: num] :
      ( ( aa_num_Code_integer(numera6620942414471956472nteger,X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Y)) )
    <=> aa_Code_integer_o(ring_16219924574208605041nteger,aa_num_Code_integer(numera6620942414471956472nteger,plus_plus_num(X,Y))) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_4626_eq__numeral__iff__iszero_I2_J,axiom,
    ! [X: num,Y: num] :
      ( ( numeral_numeral_rat(X) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(Y)) )
    <=> aa_rat_o(ring_1_iszero_rat,numeral_numeral_rat(plus_plus_num(X,Y))) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_4627_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4628_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
      ( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4629_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(nO_MATCH_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4630_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,C: int] :
      ( aa_int_o(aa_nat_fun_int_o(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4631_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4632_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: rat,B: rat,C: rat] :
      ( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4633_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(nO_MATCH_int_nat,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4634_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4635_distrib__left__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(nO_MAT8252062027627875367nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4636_distrib__left__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: rat,B: rat,C: rat] :
      ( aa_rat_o(nO_MAT7795273704451493282er_rat(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4637_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4638_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4639_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(nO_MATCH_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4640_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: int,A: int,B: int] :
      ( aa_int_o(aa_nat_fun_int_o(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4641_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4642_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4643_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(nO_MATCH_int_nat,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4644_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4645_distrib__right__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(nO_MAT8252062027627875367nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4646_distrib__right__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,C: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MAT7795273704451493282er_rat(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),C)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4647_dbl__dec__def,axiom,
    ! [X: code_integer] : ( aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,X) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),X)),one_one_Code_integer) ) ).

% dbl_dec_def
tff(fact_4648_dbl__dec__def,axiom,
    ! [X: rat] : ( aa_rat_rat(neg_nu3179335615603231917ec_rat,X) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X)),one_one_rat) ) ).

% dbl_dec_def
tff(fact_4649_dbl__dec__def,axiom,
    ! [X: int] : ( aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ) ).

% dbl_dec_def
tff(fact_4650_not__iszero__neg__Numeral1,axiom,
    ~ aa_int_o(ring_1_iszero_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) ).

% not_iszero_neg_Numeral1
tff(fact_4651_not__iszero__neg__Numeral1,axiom,
    ~ aa_Code_integer_o(ring_16219924574208605041nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))) ).

% not_iszero_neg_Numeral1
tff(fact_4652_not__iszero__neg__Numeral1,axiom,
    ~ aa_rat_o(ring_1_iszero_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(one))) ).

% not_iszero_neg_Numeral1
tff(fact_4653_convex__bound__le,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V)
           => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),V) = one_one_Code_integer )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_4654_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_4655_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(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),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_4656_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z))),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4657_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z))),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z)) ) ).

% add_divide_eq_if_simps(3)
tff(fact_4658_convex__bound__lt,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V)
           => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),V) = one_one_Code_integer )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_4659_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_4660_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(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),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_4661_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S2: 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),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,V),U))),S2))),V) ) ) ) ).

% scaling_mono
tff(fact_4662_top_Oordering__top__axioms,axiom,
    orderi7613023277850353061st_nat(ord_le6045566169113846134st_nat,ord_le1190675801316882794st_nat,top_top_set_list_nat) ).

% top.ordering_top_axioms
tff(fact_4663_top_Oordering__top__axioms,axiom,
    aa_set_o_o(ordering_top_set_o(ord_less_eq_set_o,ord_less_set_o),top_top_set_o) ).

% top.ordering_top_axioms
tff(fact_4664_top_Oordering__top__axioms,axiom,
    aa_set_rat_o(ordering_top_set_rat(ord_less_eq_set_rat,ord_less_set_rat),top_top_set_rat) ).

% top.ordering_top_axioms
tff(fact_4665_top_Oordering__top__axioms,axiom,
    aa_set_int_o(ordering_top_set_int(ord_less_eq_set_int,ord_less_set_int),top_top_set_int) ).

% top.ordering_top_axioms
tff(fact_4666_top_Oordering__top__axioms,axiom,
    orderi2807725203697798511er_nat(ord_le2510731241096832064er_nat,ord_less_filter_nat,top_top_filter_nat) ).

% top.ordering_top_axioms
tff(fact_4667_top_Oordering__top__axioms,axiom,
    aa_set_nat_o(ordering_top_set_nat(ord_less_eq_set_nat,ord_less_set_nat),top_top_set_nat) ).

% top.ordering_top_axioms
tff(fact_4668_sum__squares__gt__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)))
    <=> ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_4669_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_4670_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_4671_sum__squares__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))),zero_z3403309356797280102nteger)
    <=> ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_4672_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat)
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_4673_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_4674_add__scale__eq__noteq,axiom,
    ! [R2: code_integer,A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( R2 != zero_z3403309356797280102nteger )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),C)) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4675_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D2: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4676_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D2: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4677_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D2: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4678_sum__squares__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)) = zero_z3403309356797280102nteger )
    <=> ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_4679_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_4680_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_4681_less__by__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( A2 = bot_bo2099793752762293965at_nat )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),B2) ) ).

% less_by_empty
tff(fact_4682_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : ( unique5052692396658037445od_int(M,one) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,M)),zero_zero_int) ) ).

% divmod_algorithm_code(2)
tff(fact_4683_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : ( unique5055182867167087721od_nat(M,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(numeral_numeral_nat(M)),zero_zero_nat) ) ).

% divmod_algorithm_code(2)
tff(fact_4684_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : ( unique3479559517661332726nteger(M,one) = produc1086072967326762835nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),zero_z3403309356797280102nteger) ) ).

% divmod_algorithm_code(2)
tff(fact_4685_minus__sub__one__diff__one,axiom,
    ! [M: num] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,neg_numeral_sub_int(M,one))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) ) ).

% minus_sub_one_diff_one
tff(fact_4686_minus__sub__one__diff__one,axiom,
    ! [M: num] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5755505904847501662nteger(M,one))),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)) ) ).

% minus_sub_one_diff_one
tff(fact_4687_minus__sub__one__diff__one,axiom,
    ! [M: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,neg_numeral_sub_rat(M,one))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M)) ) ).

% minus_sub_one_diff_one
tff(fact_4688_semiring__norm_I166_J,axiom,
    ! [V: num,W: num,Y: 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_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,neg_numeral_sub_int(V,W)),Y) ) ).

% semiring_norm(166)
tff(fact_4689_semiring__norm_I166_J,axiom,
    ! [V: num,W: num,Y: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,neg_nu5755505904847501662nteger(V,W)),Y) ) ).

% semiring_norm(166)
tff(fact_4690_semiring__norm_I166_J,axiom,
    ! [V: num,W: num,Y: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,neg_numeral_sub_rat(V,W)),Y) ) ).

% semiring_norm(166)
tff(fact_4691_semiring__norm_I167_J,axiom,
    ! [V: num,W: num,Y: 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_num_int(numeral_numeral_int,W)),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,neg_numeral_sub_int(W,V)),Y) ) ).

% semiring_norm(167)
tff(fact_4692_semiring__norm_I167_J,axiom,
    ! [V: num,W: num,Y: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,neg_nu5755505904847501662nteger(W,V)),Y) ) ).

% semiring_norm(167)
tff(fact_4693_semiring__norm_I167_J,axiom,
    ! [V: num,W: num,Y: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(W)),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,neg_numeral_sub_rat(W,V)),Y) ) ).

% semiring_norm(167)
tff(fact_4694_add__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = neg_numeral_sub_int(M,N) ) ).

% add_neg_numeral_simps(1)
tff(fact_4695_add__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = neg_nu5755505904847501662nteger(M,N) ) ).

% add_neg_numeral_simps(1)
tff(fact_4696_add__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(M)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = neg_numeral_sub_rat(M,N) ) ).

% add_neg_numeral_simps(1)
tff(fact_4697_add__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = neg_numeral_sub_int(N,M) ) ).

% add_neg_numeral_simps(2)
tff(fact_4698_add__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N)) = neg_nu5755505904847501662nteger(N,M) ) ).

% add_neg_numeral_simps(2)
tff(fact_4699_add__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),numeral_numeral_rat(N)) = neg_numeral_sub_rat(N,M) ) ).

% add_neg_numeral_simps(2)
tff(fact_4700_diff__numeral__simps_I4_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = neg_numeral_sub_int(N,M) ) ).

% diff_numeral_simps(4)
tff(fact_4701_diff__numeral__simps_I4_J,axiom,
    ! [M: num,N: num] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = neg_nu5755505904847501662nteger(N,M) ) ).

% diff_numeral_simps(4)
tff(fact_4702_diff__numeral__simps_I4_J,axiom,
    ! [M: num,N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = neg_numeral_sub_rat(N,M) ) ).

% diff_numeral_simps(4)
tff(fact_4703_diff__numeral__special_I2_J,axiom,
    ! [M: num] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int) = neg_numeral_sub_int(M,one) ) ).

% diff_numeral_special(2)
tff(fact_4704_diff__numeral__special_I2_J,axiom,
    ! [M: num] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),one_one_Code_integer) = neg_nu5755505904847501662nteger(M,one) ) ).

% diff_numeral_special(2)
tff(fact_4705_diff__numeral__special_I2_J,axiom,
    ! [M: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,numeral_numeral_rat(M)),one_one_rat) = neg_numeral_sub_rat(M,one) ) ).

% diff_numeral_special(2)
tff(fact_4706_diff__numeral__special_I1_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = neg_numeral_sub_int(one,N) ) ).

% diff_numeral_special(1)
tff(fact_4707_diff__numeral__special_I1_J,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N)) = neg_nu5755505904847501662nteger(one,N) ) ).

% diff_numeral_special(1)
tff(fact_4708_diff__numeral__special_I1_J,axiom,
    ! [N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),numeral_numeral_rat(N)) = neg_numeral_sub_rat(one,N) ) ).

% diff_numeral_special(1)
tff(fact_4709_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = neg_numeral_sub_int(N,one) ) ).

% add_neg_numeral_special(4)
tff(fact_4710_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = neg_nu5755505904847501662nteger(N,one) ) ).

% add_neg_numeral_special(4)
tff(fact_4711_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),numeral_numeral_rat(N)) = neg_numeral_sub_rat(N,one) ) ).

% add_neg_numeral_special(4)
tff(fact_4712_add__neg__numeral__special_I3_J,axiom,
    ! [M: num] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) = neg_numeral_sub_int(M,one) ) ).

% add_neg_numeral_special(3)
tff(fact_4713_add__neg__numeral__special_I3_J,axiom,
    ! [M: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = neg_nu5755505904847501662nteger(M,one) ) ).

% add_neg_numeral_special(3)
tff(fact_4714_add__neg__numeral__special_I3_J,axiom,
    ! [M: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = neg_numeral_sub_rat(M,one) ) ).

% add_neg_numeral_special(3)
tff(fact_4715_add__neg__numeral__special_I2_J,axiom,
    ! [M: num] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) = neg_numeral_sub_int(one,M) ) ).

% add_neg_numeral_special(2)
tff(fact_4716_add__neg__numeral__special_I2_J,axiom,
    ! [M: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer) = neg_nu5755505904847501662nteger(one,M) ) ).

% add_neg_numeral_special(2)
tff(fact_4717_add__neg__numeral__special_I2_J,axiom,
    ! [M: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),one_one_rat) = neg_numeral_sub_rat(one,M) ) ).

% add_neg_numeral_special(2)
tff(fact_4718_add__neg__numeral__special_I1_J,axiom,
    ! [M: num] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) = neg_numeral_sub_int(one,M) ) ).

% add_neg_numeral_special(1)
tff(fact_4719_add__neg__numeral__special_I1_J,axiom,
    ! [M: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))) = neg_nu5755505904847501662nteger(one,M) ) ).

% add_neg_numeral_special(1)
tff(fact_4720_add__neg__numeral__special_I1_J,axiom,
    ! [M: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))) = neg_numeral_sub_rat(one,M) ) ).

% add_neg_numeral_special(1)
tff(fact_4721_diff__numeral__special_I8_J,axiom,
    ! [M: num] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) = neg_numeral_sub_int(one,M) ) ).

% diff_numeral_special(8)
tff(fact_4722_diff__numeral__special_I8_J,axiom,
    ! [M: num] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = neg_nu5755505904847501662nteger(one,M) ) ).

% diff_numeral_special(8)
tff(fact_4723_diff__numeral__special_I8_J,axiom,
    ! [M: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = neg_numeral_sub_rat(one,M) ) ).

% diff_numeral_special(8)
tff(fact_4724_diff__numeral__special_I7_J,axiom,
    ! [N: num] : ( 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,aa_num_int(numeral_numeral_int,N))) = neg_numeral_sub_int(N,one) ) ).

% diff_numeral_special(7)
tff(fact_4725_diff__numeral__special_I7_J,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = neg_nu5755505904847501662nteger(N,one) ) ).

% diff_numeral_special(7)
tff(fact_4726_diff__numeral__special_I7_J,axiom,
    ! [N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = neg_numeral_sub_rat(N,one) ) ).

% diff_numeral_special(7)
tff(fact_4727_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = M )
      <=> ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_4728_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M) ) ) ).

% div_less_dividend
tff(fact_4729_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),X)
    <=> ( X != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
tff(fact_4730_eq__numeral__iff__iszero_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Y)) )
    <=> aa_int_o(ring_1_iszero_int,neg_numeral_sub_int(Y,X)) ) ).

% eq_numeral_iff_iszero(4)
tff(fact_4731_eq__numeral__iff__iszero_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Y)) )
    <=> aa_Code_integer_o(ring_16219924574208605041nteger,neg_nu5755505904847501662nteger(Y,X)) ) ).

% eq_numeral_iff_iszero(4)
tff(fact_4732_eq__numeral__iff__iszero_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(X)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(Y)) )
    <=> aa_rat_o(ring_1_iszero_rat,neg_numeral_sub_rat(Y,X)) ) ).

% eq_numeral_iff_iszero(4)
tff(fact_4733_eq__numeral__iff__iszero_I6_J,axiom,
    ! [Y: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,Y) )
    <=> aa_int_o(ring_1_iszero_int,neg_numeral_sub_int(one,Y)) ) ).

% eq_numeral_iff_iszero(6)
tff(fact_4734_eq__numeral__iff__iszero_I6_J,axiom,
    ! [Y: num] :
      ( ( one_one_Code_integer = aa_num_Code_integer(numera6620942414471956472nteger,Y) )
    <=> aa_Code_integer_o(ring_16219924574208605041nteger,neg_nu5755505904847501662nteger(one,Y)) ) ).

% eq_numeral_iff_iszero(6)
tff(fact_4735_eq__numeral__iff__iszero_I6_J,axiom,
    ! [Y: num] :
      ( ( one_one_rat = numeral_numeral_rat(Y) )
    <=> aa_rat_o(ring_1_iszero_rat,neg_numeral_sub_rat(one,Y)) ) ).

% eq_numeral_iff_iszero(6)
tff(fact_4736_eq__numeral__iff__iszero_I5_J,axiom,
    ! [X: num] :
      ( ( aa_num_int(numeral_numeral_int,X) = one_one_int )
    <=> aa_int_o(ring_1_iszero_int,neg_numeral_sub_int(X,one)) ) ).

% eq_numeral_iff_iszero(5)
tff(fact_4737_eq__numeral__iff__iszero_I5_J,axiom,
    ! [X: num] :
      ( ( aa_num_Code_integer(numera6620942414471956472nteger,X) = one_one_Code_integer )
    <=> aa_Code_integer_o(ring_16219924574208605041nteger,neg_nu5755505904847501662nteger(X,one)) ) ).

% eq_numeral_iff_iszero(5)
tff(fact_4738_eq__numeral__iff__iszero_I5_J,axiom,
    ! [X: num] :
      ( ( numeral_numeral_rat(X) = one_one_rat )
    <=> aa_rat_o(ring_1_iszero_rat,neg_numeral_sub_rat(X,one)) ) ).

% eq_numeral_iff_iszero(5)
tff(fact_4739_crossproduct__eq,axiom,
    ! [W: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W),Y)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_4740_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X: rat,Z: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_4741_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_4742_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: 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,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_4743_crossproduct__noteq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( A != B )
        & ( C != D2 ) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4744_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( A != B )
        & ( C != D2 ) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),D2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4745_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ( A != B )
        & ( C != D2 ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4746_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( A != B )
        & ( C != D2 ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4747_less__one,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),one_one_nat)
    <=> ( N = zero_zero_nat ) ) ).

% less_one
tff(fact_4748_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_4749_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_4750_subset__emptyI,axiom,
    ! [A2: set_se6260736226359567993nt_int] :
      ( ! [X2: set_Pr958786334691620121nt_int] : ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),A2)
     => aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A2),bot_bo1488462491386950373nt_int) ) ).

% subset_emptyI
tff(fact_4751_subset__emptyI,axiom,
    ! [A2: set_set_nat] :
      ( ! [X2: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X2),A2)
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),bot_bot_set_set_nat) ) ).

% subset_emptyI
tff(fact_4752_subset__emptyI,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A2)
     => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),bot_bo8632255151191677115_nat_o) ) ).

% subset_emptyI
tff(fact_4753_subset__emptyI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [X2: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),A2)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),bot_bo2099793752762293965at_nat) ) ).

% subset_emptyI
tff(fact_4754_subset__emptyI,axiom,
    ! [A2: set_o] :
      ( ! [X2: $o] : ~ aa_set_o_o(member_o((X2)),A2)
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),bot_bot_set_o) ) ).

% subset_emptyI
tff(fact_4755_subset__emptyI,axiom,
    ! [A2: set_int] :
      ( ! [X2: int] : ~ aa_set_int_o(member_int(X2),A2)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),bot_bot_set_int) ) ).

% subset_emptyI
tff(fact_4756_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X2: nat] : ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),bot_bot_set_nat) ) ).

% subset_emptyI
tff(fact_4757_dbl__simps_I4_J,axiom,
    aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4758_dbl__simps_I4_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu8804712462038260780nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4759_dbl__simps_I4_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4760_dbl__dec__simps_I4_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_4761_dbl__dec__simps_I4_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_4762_dbl__dec__simps_I4_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_4763_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : ( unique5052692396658037445od_int(one,aa_num_num(bit0,N)) = 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_4764_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : ( unique5055182867167087721od_nat(one,aa_num_num(bit0,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),numeral_numeral_nat(one)) ) ).

% divmod_algorithm_code(3)
tff(fact_4765_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : ( unique3479559517661332726nteger(one,aa_num_num(bit0,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ) ).

% divmod_algorithm_code(3)
tff(fact_4766_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_4767_divmod__step__eq,axiom,
    ! [L: num,Q3: nat,R2: nat] :
      ( aa_Pro262675715343573244at_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,numeral_numeral_nat(L)),R2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Q3)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),numeral_numeral_nat(L))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Q3)),R2)) ) ).

% divmod_step_eq
tff(fact_4768_divmod__step__eq,axiom,
    ! [L: num,Q3: code_integer,R2: code_integer] :
      ( aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(L),produc1086072967326762835nteger(Q3,R2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,L)),R2),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q3)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(R2),aa_num_Code_integer(numera6620942414471956472nteger,L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q3),R2)) ) ).

% divmod_step_eq
tff(fact_4769_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = numeral_numeral_nat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_4770_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_4771_one__add__one,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_4772_one__add__one,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_4773_one__add__one,axiom,
    plus_p4538020629002901425atural(one_one_Code_natural,one_one_Code_natural) = numera5444537566228673987atural(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_4774_dbl__simps_I3_J,axiom,
    aa_int_int(neg_numeral_dbl_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_4775_dbl__simps_I3_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu8804712462038260780nteger,one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_4776_dbl__simps_I3_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_4777_dbl__inc__simps_I3_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_4778_dbl__inc__simps_I3_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_4779_dbl__inc__simps_I3_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,one_one_rat) = numeral_numeral_rat(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_4780_one__div__two__eq__zero,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_4781_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_4782_one__div__two__eq__zero,axiom,
    aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% one_div_two_eq_zero
tff(fact_4783_one__div__two__eq__zero,axiom,
    divide5121882707175180666atural(one_one_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))) = zero_z2226904508553997617atural ).

% one_div_two_eq_zero
tff(fact_4784_bits__1__div__2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_4785_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_4786_bits__1__div__2,axiom,
    aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% bits_1_div_2
tff(fact_4787_bits__1__div__2,axiom,
    divide5121882707175180666atural(one_one_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))) = zero_z2226904508553997617atural ).

% bits_1_div_2
tff(fact_4788_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_4789_add__neg__numeral__special_I9_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4790_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_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,numeral_numeral_rat(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4791_diff__numeral__special_I10_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4792_diff__numeral__special_I10_J,axiom,
    aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4793_diff__numeral__special_I10_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4794_diff__numeral__special_I11_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4795_diff__numeral__special_I11_J,axiom,
    aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4796_diff__numeral__special_I11_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4797_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : ( unique5052692396658037445od_int(one,aa_num_num(bit1,N)) = 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_4798_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : ( unique5055182867167087721od_nat(one,aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),numeral_numeral_nat(one)) ) ).

% divmod_algorithm_code(4)
tff(fact_4799_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : ( unique3479559517661332726nteger(one,aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ) ).

% divmod_algorithm_code(4)
tff(fact_4800_sub__num__simps_I3_J,axiom,
    ! [L: num] : ( neg_numeral_sub_int(one,aa_num_num(bit1,L)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,L))) ) ).

% sub_num_simps(3)
tff(fact_4801_sub__num__simps_I3_J,axiom,
    ! [L: num] : ( neg_nu5755505904847501662nteger(one,aa_num_num(bit1,L)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,L))) ) ).

% sub_num_simps(3)
tff(fact_4802_sub__num__simps_I3_J,axiom,
    ! [L: num] : ( neg_numeral_sub_rat(one,aa_num_num(bit1,L)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(bit0,L))) ) ).

% sub_num_simps(3)
tff(fact_4803_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = $ite(ord_less_eq_num(M,N),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,N)),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ) ).

% divmod_algorithm_code(7)
tff(fact_4804_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N)) = $ite(ord_less_eq_num(M,N),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),numeral_numeral_nat(aa_num_num(bit0,M))),aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(aa_num_num(bit1,N)),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ) ).

% divmod_algorithm_code(7)
tff(fact_4805_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = $ite(ord_less_eq_num(M,N),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M))),aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(aa_num_num(bit1,N)),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ) ).

% divmod_algorithm_code(7)
tff(fact_4806_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = $ite(ord_less_num(M,N),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,N)),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ) ).

% divmod_algorithm_code(8)
tff(fact_4807_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = $ite(ord_less_num(M,N),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),numeral_numeral_nat(aa_num_num(bit1,M))),aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(aa_num_num(bit1,N)),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ) ).

% divmod_algorithm_code(8)
tff(fact_4808_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = $ite(ord_less_num(M,N),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M))),aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(aa_num_num(bit1,N)),unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ) ).

% divmod_algorithm_code(8)
tff(fact_4809_bot__nat__0_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(aTP_Lamm_gf,aTP_Lamm_gg),zero_zero_nat) ).

% bot_nat_0.ordering_top_axioms
tff(fact_4810_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_4811_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X != product_Pair_num_num(one,one) )
     => ( ! [N3: num] : ( X != product_Pair_num_num(one,aa_num_num(bit0,N3)) )
       => ( ! [N3: num] : ( X != product_Pair_num_num(one,aa_num_num(bit1,N3)) )
         => ( ! [M3: num] : ( X != product_Pair_num_num(aa_num_num(bit0,M3),one) )
           => ( ! [M3: num,N3: num] : ( X != product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N3)) )
             => ( ! [M3: num,N3: num] : ( X != product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N3)) )
               => ( ! [M3: num] : ( X != product_Pair_num_num(aa_num_num(bit1,M3),one) )
                 => ( ! [M3: num,N3: num] : ( X != product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N3)) )
                   => ~ ! [M3: num,N3: num] : ( X != product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N3)) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_4812_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_4813_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_4814_int__bit__induct,axiom,
    ! [P: fun_int_o,K: int] :
      ( aa_int_o(P,zero_zero_int)
     => ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K2: int] :
              ( aa_int_o(P,K2)
             => ( ( K2 != zero_zero_int )
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
         => ( ! [K2: int] :
                ( aa_int_o(P,K2)
               => ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_4815_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( numeral_numeral_nat(aa_num_num(bit0,N)) = $let(
          m: nat,
          m:= numeral_numeral_nat(N),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m) ) ) ).

% numeral_code(2)
tff(fact_4816_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = $let(
          m: int,
          m:= aa_num_int(numeral_numeral_int,N),
          aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m) ) ) ).

% numeral_code(2)
tff(fact_4817_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N)) = $let(
          m: code_integer,
          m:= aa_num_Code_integer(numera6620942414471956472nteger,N),
          aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,m),m) ) ) ).

% numeral_code(2)
tff(fact_4818_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( numeral_numeral_rat(aa_num_num(bit0,N)) = $let(
          m: rat,
          m:= numeral_numeral_rat(N),
          aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m),m) ) ) ).

% numeral_code(2)
tff(fact_4819_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( numera5444537566228673987atural(aa_num_num(bit0,N)) = $let(
          m: code_natural,
          m:= numera5444537566228673987atural(N),
          plus_p4538020629002901425atural(m,m) ) ) ).

% numeral_code(2)
tff(fact_4820_numeral__Bit1,axiom,
    ! [N: num] : ( numeral_numeral_nat(aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(N)),numeral_numeral_nat(N))),one_one_nat) ) ).

% numeral_Bit1
tff(fact_4821_numeral__Bit1,axiom,
    ! [N: num] : ( aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ) ).

% numeral_Bit1
tff(fact_4822_numeral__Bit1,axiom,
    ! [N: num] : ( aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_num_Code_integer(numera6620942414471956472nteger,N))),one_one_Code_integer) ) ).

% numeral_Bit1
tff(fact_4823_numeral__Bit1,axiom,
    ! [N: num] : ( numeral_numeral_rat(aa_num_num(bit1,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(N)),numeral_numeral_rat(N))),one_one_rat) ) ).

% numeral_Bit1
tff(fact_4824_numeral__Bit1,axiom,
    ! [N: num] : ( numera5444537566228673987atural(aa_num_num(bit1,N)) = plus_p4538020629002901425atural(plus_p4538020629002901425atural(numera5444537566228673987atural(N),numera5444537566228673987atural(N)),one_one_Code_natural) ) ).

% numeral_Bit1
tff(fact_4825_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = numeral_numeral_nat(aa_num_num(bit0,one)) ).

% nat_1_add_1
tff(fact_4826_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr5098669352780829690e_term,S2: produc8923325533196201883nteger,R: set_Pr3486312109254157132nteger,S4: produc8923325533196201883nteger] :
      ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,R2),S2),R)
     => ( ( S4 = S2 )
       => member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_4827_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr6460944699146269935_nat_o,S2: produc3658429121746597890et_nat,R: set_Pr1312406793078187350et_nat,S4: produc3658429121746597890et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,R2),S2),R)
     => ( ( S4 = S2 )
       => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_4828_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr6460944699146269935_nat_o,S2: produc8799533983703829792et_nat,R: set_Pr4040394263813967220et_nat,S4: produc8799533983703829792et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,R2),S2),R)
     => ( ( S4 = S2 )
       => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_4829_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr7763680713840462443e_term,S2: product_prod_int_int,R: set_Pr6700895561065273913nt_int,S4: product_prod_int_int] :
      ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,R2),S2),R)
     => ( ( S4 = S2 )
       => member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_4830_ssubst__Pair__rhs,axiom,
    ! [R2: fun_in5730379987311621695e_term,S2: product_prod_int_int,R: set_Pr4166680992137330533nt_int,S4: product_prod_int_int] :
      ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,R2),S2),R)
     => ( ( S4 = S2 )
       => member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_4831_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( numeral_numeral_nat(aa_num_num(bit1,N)) = $let(
          m: nat,
          m:= numeral_numeral_nat(N),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m)),one_one_nat) ) ) ).

% numeral_code(3)
tff(fact_4832_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = $let(
          m: int,
          m:= aa_num_int(numeral_numeral_int,N),
          aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m)),one_one_int) ) ) ).

% numeral_code(3)
tff(fact_4833_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N)) = $let(
          m: code_integer,
          m:= aa_num_Code_integer(numera6620942414471956472nteger,N),
          aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,m),m)),one_one_Code_integer) ) ) ).

% numeral_code(3)
tff(fact_4834_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( numeral_numeral_rat(aa_num_num(bit1,N)) = $let(
          m: rat,
          m:= numeral_numeral_rat(N),
          aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m),m)),one_one_rat) ) ) ).

% numeral_code(3)
tff(fact_4835_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( numera5444537566228673987atural(aa_num_num(bit1,N)) = $let(
          m: code_natural,
          m:= numera5444537566228673987atural(N),
          plus_p4538020629002901425atural(plus_p4538020629002901425atural(m,m),one_one_Code_natural) ) ) ).

% numeral_code(3)
tff(fact_4836_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_4837_nat__mult__1,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ) ).

% nat_mult_1
tff(fact_4838_nat__mult__1__right,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ) ).

% nat_mult_1_right
tff(fact_4839_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = $ite(M = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N))) ) ).

% mult_eq_if
tff(fact_4840_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( unique5052692396658037445od_int(M,N) = $ite(ord_less_num(M,N),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,M)),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(N),unique5052692396658037445od_int(M,aa_num_num(bit0,N)))) ) ).

% divmod_divmod_step
tff(fact_4841_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( unique5055182867167087721od_nat(M,N) = $ite(ord_less_num(M,N),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),numeral_numeral_nat(M)),aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(N),unique5055182867167087721od_nat(M,aa_num_num(bit0,N)))) ) ).

% divmod_divmod_step
tff(fact_4842_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( unique3479559517661332726nteger(M,N) = $ite(ord_less_num(M,N),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(N),unique3479559517661332726nteger(M,aa_num_num(bit0,N)))) ) ).

% divmod_divmod_step
tff(fact_4843_mult__2,axiom,
    ! [Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_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_4844_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_4845_mult__2,axiom,
    ! [Z: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Z) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),Z) ) ).

% mult_2
tff(fact_4846_mult__2,axiom,
    ! [Z: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Z) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ) ).

% mult_2
tff(fact_4847_mult__2,axiom,
    ! [Z: code_natural] : ( times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),Z) = plus_p4538020629002901425atural(Z,Z) ) ).

% mult_2
tff(fact_4848_mult__2__right,axiom,
    ! [Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),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_4849_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_4850_mult__2__right,axiom,
    ! [Z: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),Z) ) ).

% mult_2_right
tff(fact_4851_mult__2__right,axiom,
    ! [Z: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),numeral_numeral_rat(aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ) ).

% mult_2_right
tff(fact_4852_mult__2__right,axiom,
    ! [Z: code_natural] : ( times_2397367101498566445atural(Z,numera5444537566228673987atural(aa_num_num(bit0,one))) = plus_p4538020629002901425atural(Z,Z) ) ).

% mult_2_right
tff(fact_4853_left__add__twice,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)),B) ) ).

% left_add_twice
tff(fact_4854_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_4855_left__add__twice,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)),B) ) ).

% left_add_twice
tff(fact_4856_left__add__twice,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),A)),B) ) ).

% left_add_twice
tff(fact_4857_left__add__twice,axiom,
    ! [A: code_natural,B: code_natural] : ( plus_p4538020629002901425atural(A,plus_p4538020629002901425atural(A,B)) = plus_p4538020629002901425atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),A),B) ) ).

% left_add_twice
tff(fact_4858_prop__restrict,axiom,
    ! [X: set_Pr958786334691620121nt_int,Z4: set_se6260736226359567993nt_int,X5: set_se6260736226359567993nt_int,P: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),Z4)
     => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(Z4),collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_gh(X5),P)))
       => aa_set6170482117830564987_int_o(P,X) ) ) ).

% prop_restrict
tff(fact_4859_prop__restrict,axiom,
    ! [X: set_nat,Z4: set_set_nat,X5: set_set_nat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(X),Z4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(Z4),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_gi(X5),P)))
       => aa_set_nat_o(P,X) ) ) ).

% prop_restrict
tff(fact_4860_prop__restrict,axiom,
    ! [X: int,Z4: set_int,X5: set_int,P: fun_int_o] :
      ( aa_set_int_o(member_int(X),Z4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Z4),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gj(X5),P)))
       => aa_int_o(P,X) ) ) ).

% prop_restrict
tff(fact_4861_prop__restrict,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Z4: set_fu3555266933746557007_nat_o,X5: set_fu3555266933746557007_nat_o,P: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Z4)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(Z4),collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_gk(X5),P)))
       => aa_fun1675921005893230629at_o_o(P,X) ) ) ).

% prop_restrict
tff(fact_4862_prop__restrict,axiom,
    ! [X: nat,Z4: set_nat,X5: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),Z4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z4),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gl(X5),P)))
       => aa_nat_o(P,X) ) ) ).

% prop_restrict
tff(fact_4863_Collect__restrict,axiom,
    ! [X5: set_se6260736226359567993nt_int,P: fun_se7249947736802449410_int_o] : aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_gh(X5),P))),X5) ).

% Collect_restrict
tff(fact_4864_Collect__restrict,axiom,
    ! [X5: set_set_nat,P: fun_set_nat_o] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_gi(X5),P))),X5) ).

% Collect_restrict
tff(fact_4865_Collect__restrict,axiom,
    ! [X5: set_int,P: fun_int_o] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gj(X5),P))),X5) ).

% Collect_restrict
tff(fact_4866_Collect__restrict,axiom,
    ! [X5: set_fu3555266933746557007_nat_o,P: fun_fu5135969500543783852at_o_o] : aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_gk(X5),P))),X5) ).

% Collect_restrict
tff(fact_4867_Collect__restrict,axiom,
    ! [X5: set_nat,P: fun_nat_o] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gl(X5),P))),X5) ).

% Collect_restrict
tff(fact_4868_minus__1__div__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_4869_minus__1__div__2__eq,axiom,
    aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_4870_set__bit__0,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(zero_zero_nat),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% set_bit_0
tff(fact_4871_set__bit__0,axiom,
    ! [A: code_natural] : ( bit_se1617098188084679374atural(zero_zero_nat,A) = plus_p4538020629002901425atural(one_one_Code_natural,times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))))) ) ).

% set_bit_0
tff(fact_4872_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_4873_set__bit__0,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ).

% set_bit_0
tff(fact_4874_nat__induct2,axiom,
    ! [P: fun_nat_o,N: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N3: nat] :
              ( aa_nat_o(P,N3)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),numeral_numeral_nat(aa_num_num(bit0,one)))) )
         => aa_nat_o(P,N) ) ) ) ).

% nat_induct2
tff(fact_4875_unset__bit__0,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))) ) ).

% unset_bit_0
tff(fact_4876_unset__bit__0,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(zero_zero_nat),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ).

% unset_bit_0
tff(fact_4877_unset__bit__0,axiom,
    ! [A: code_natural] : ( bit_se7083795435491715335atural(zero_zero_nat,A) = times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))) ) ).

% unset_bit_0
tff(fact_4878_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_4879_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_gm),unique5052692396658037445od_int(M,N)) ) ).

% divmod_algorithm_code(6)
tff(fact_4880_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gn),unique5055182867167087721od_nat(M,N)) ) ).

% divmod_algorithm_code(6)
tff(fact_4881_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_go),unique3479559517661332726nteger(M,N)) ) ).

% divmod_algorithm_code(6)
tff(fact_4882_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : ( aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_gp(L)),Qr) ) ).

% divmod_step_def
tff(fact_4883_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : ( aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(L),Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gq(L)),Qr) ) ).

% divmod_step_def
tff(fact_4884_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : ( aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(L),Qr) = aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_gr(L)),Qr) ) ).

% divmod_step_def
tff(fact_4885_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),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)))
         => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)))),one_one_Code_integer) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4886_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(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,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,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_4887_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_4888_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y)))) ) ).

% xor_numerals(6)
tff(fact_4889_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : ( bit_se2046307713759805098atural(numera5444537566228673987atural(aa_num_num(bit1,X)),numera5444537566228673987atural(aa_num_num(bit0,Y))) = plus_p4538020629002901425atural(one_one_Code_natural,times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),bit_se2046307713759805098atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y)))) ) ).

% xor_numerals(6)
tff(fact_4890_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = 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_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ) ).

% xor_numerals(6)
tff(fact_4891_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(aa_num_num(bit1,X))),numeral_numeral_nat(aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ) ).

% xor_numerals(6)
tff(fact_4892_mod__minus__minus,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ).

% mod_minus_minus
tff(fact_4893_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) ) ).

% mod_minus_minus
tff(fact_4894_case__prod__conv,axiom,
    ! [F: fun_na1668100129766719005at_nat,A: nat,B: nat] : ( aa_Pro8023792847356852296at_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_4895_case__prod__conv,axiom,
    ! [F: fun_na5914473681949881652_nat_o,A: nat,B: nat] : ( aa_Pro989595756709598047_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_4896_case__prod__conv,axiom,
    ! [F: fun_in1696471567231710611nt_int,A: int,B: int] : ( aa_Pro6970504127699319548nt_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_4897_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_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_4898_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_int,A: int,B: int] : ( aa_Pro6799258434465214121nt_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_4899_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_4900_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_4901_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),B) = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
tff(fact_4902_mod__mult__self1__is__0,axiom,
    ! [B: code_natural,A: code_natural] : ( modulo8411746178871703098atural(times_2397367101498566445atural(B,A),B) = zero_z2226904508553997617atural ) ).

% mod_mult_self1_is_0
tff(fact_4903_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_4904_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_4905_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),B) = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
tff(fact_4906_mod__mult__self2__is__0,axiom,
    ! [A: code_natural,B: code_natural] : ( modulo8411746178871703098atural(times_2397367101498566445atural(A,B),B) = zero_z2226904508553997617atural ) ).

% mod_mult_self2_is_0
tff(fact_4907_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_4908_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_4909_bits__mod__by__1,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),one_one_Code_integer) = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
tff(fact_4910_bits__mod__by__1,axiom,
    ! [A: code_natural] : ( modulo8411746178871703098atural(A,one_one_Code_natural) = zero_z2226904508553997617atural ) ).

% bits_mod_by_1
tff(fact_4911_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_4912_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_4913_mod__by__1,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),one_one_Code_integer) = zero_z3403309356797280102nteger ) ).

% mod_by_1
tff(fact_4914_mod__by__1,axiom,
    ! [A: code_natural] : ( modulo8411746178871703098atural(A,one_one_Code_natural) = zero_z2226904508553997617atural ) ).

% mod_by_1
tff(fact_4915_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ).

% mod_mult_self4
tff(fact_4916_mod__mult__self4,axiom,
    ! [B: int,C: 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),C)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ).

% mod_mult_self4
tff(fact_4917_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ) ).

% mod_mult_self4
tff(fact_4918_mod__mult__self4,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] : ( modulo8411746178871703098atural(plus_p4538020629002901425atural(times_2397367101498566445atural(B,C),A),B) = modulo8411746178871703098atural(A,B) ) ).

% mod_mult_self4
tff(fact_4919_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ).

% mod_mult_self3
tff(fact_4920_mod__mult__self3,axiom,
    ! [C: 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,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ).

% mod_mult_self3
tff(fact_4921_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),A)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ) ).

% mod_mult_self3
tff(fact_4922_mod__mult__self3,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] : ( modulo8411746178871703098atural(plus_p4538020629002901425atural(times_2397367101498566445atural(C,B),A),B) = modulo8411746178871703098atural(A,B) ) ).

% mod_mult_self3
tff(fact_4923_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ).

% mod_mult_self2
tff(fact_4924_mod__mult__self2,axiom,
    ! [A: int,B: int,C: 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),C))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ).

% mod_mult_self2
tff(fact_4925_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ) ).

% mod_mult_self2
tff(fact_4926_mod__mult__self2,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : ( modulo8411746178871703098atural(plus_p4538020629002901425atural(A,times_2397367101498566445atural(B,C)),B) = modulo8411746178871703098atural(A,B) ) ).

% mod_mult_self2
tff(fact_4927_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ).

% mod_mult_self1
tff(fact_4928_mod__mult__self1,axiom,
    ! [A: int,C: 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,C),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ).

% mod_mult_self1
tff(fact_4929_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ) ).

% mod_mult_self1
tff(fact_4930_mod__mult__self1,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] : ( modulo8411746178871703098atural(plus_p4538020629002901425atural(A,times_2397367101498566445atural(C,B)),B) = modulo8411746178871703098atural(A,B) ) ).

% mod_mult_self1
tff(fact_4931_minus__mod__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(minus_minus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% minus_mod_self1
tff(fact_4932_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).

% minus_mod_self1
tff(fact_4933_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_4934_mod__minus1__right,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
tff(fact_4935_bits__one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_4936_bits__one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_4937_bits__one__mod__two__eq__one,axiom,
    aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_4938_bits__one__mod__two__eq__one,axiom,
    modulo8411746178871703098atural(one_one_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ).

% bits_one_mod_two_eq_one
tff(fact_4939_one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_4940_one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_4941_one__mod__two__eq__one,axiom,
    aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_4942_one__mod__two__eq__one,axiom,
    modulo8411746178871703098atural(one_one_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ).

% one_mod_two_eq_one
tff(fact_4943_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y))) ) ).

% xor_numerals(3)
tff(fact_4944_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] : ( bit_se2046307713759805098atural(numera5444537566228673987atural(aa_num_num(bit0,X)),numera5444537566228673987atural(aa_num_num(bit0,Y))) = times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),bit_se2046307713759805098atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y))) ) ).

% xor_numerals(3)
tff(fact_4945_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = 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_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ) ).

% xor_numerals(3)
tff(fact_4946_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(aa_num_num(bit0,X))),numeral_numeral_nat(aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ) ).

% xor_numerals(3)
tff(fact_4947_xor__numerals_I8_J,axiom,
    ! [X: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,X))),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,X)) ) ).

% xor_numerals(8)
tff(fact_4948_xor__numerals_I8_J,axiom,
    ! [X: num] : ( bit_se2046307713759805098atural(numera5444537566228673987atural(aa_num_num(bit1,X)),one_one_Code_natural) = numera5444537566228673987atural(aa_num_num(bit0,X)) ) ).

% xor_numerals(8)
tff(fact_4949_xor__numerals_I8_J,axiom,
    ! [X: num] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,X)) ) ).

% xor_numerals(8)
tff(fact_4950_xor__numerals_I8_J,axiom,
    ! [X: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(aa_num_num(bit1,X))),one_one_nat) = numeral_numeral_nat(aa_num_num(bit0,X)) ) ).

% xor_numerals(8)
tff(fact_4951_xor__numerals_I5_J,axiom,
    ! [X: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,X))),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,X)) ) ).

% xor_numerals(5)
tff(fact_4952_xor__numerals_I5_J,axiom,
    ! [X: num] : ( bit_se2046307713759805098atural(numera5444537566228673987atural(aa_num_num(bit0,X)),one_one_Code_natural) = numera5444537566228673987atural(aa_num_num(bit1,X)) ) ).

% xor_numerals(5)
tff(fact_4953_xor__numerals_I5_J,axiom,
    ! [X: num] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,X)) ) ).

% xor_numerals(5)
tff(fact_4954_xor__numerals_I5_J,axiom,
    ! [X: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(aa_num_num(bit0,X))),one_one_nat) = numeral_numeral_nat(aa_num_num(bit1,X)) ) ).

% xor_numerals(5)
tff(fact_4955_xor__numerals_I2_J,axiom,
    ! [Y: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y)) ) ).

% xor_numerals(2)
tff(fact_4956_xor__numerals_I2_J,axiom,
    ! [Y: num] : ( bit_se2046307713759805098atural(one_one_Code_natural,numera5444537566228673987atural(aa_num_num(bit1,Y))) = numera5444537566228673987atural(aa_num_num(bit0,Y)) ) ).

% xor_numerals(2)
tff(fact_4957_xor__numerals_I2_J,axiom,
    ! [Y: num] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y)) ) ).

% xor_numerals(2)
tff(fact_4958_xor__numerals_I2_J,axiom,
    ! [Y: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit1,Y))) = numeral_numeral_nat(aa_num_num(bit0,Y)) ) ).

% xor_numerals(2)
tff(fact_4959_xor__numerals_I1_J,axiom,
    ! [Y: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y)) ) ).

% xor_numerals(1)
tff(fact_4960_xor__numerals_I1_J,axiom,
    ! [Y: num] : ( bit_se2046307713759805098atural(one_one_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,Y))) = numera5444537566228673987atural(aa_num_num(bit1,Y)) ) ).

% xor_numerals(1)
tff(fact_4961_xor__numerals_I1_J,axiom,
    ! [Y: num] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y)) ) ).

% xor_numerals(1)
tff(fact_4962_xor__numerals_I1_J,axiom,
    ! [Y: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,Y))) = numeral_numeral_nat(aa_num_num(bit1,Y)) ) ).

% xor_numerals(1)
tff(fact_4963_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4964_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_4965_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4966_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) != zero_z2226904508553997617atural )
    <=> ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4967_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4968_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_4969_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4970_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) != one_one_Code_natural )
    <=> ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = zero_z2226904508553997617atural ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4971_minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_4972_minus__1__mod__2__eq,axiom,
    aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_4973_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_4974_bits__minus__1__mod__2__eq,axiom,
    aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_4975_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),numeral_numeral_nat(aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_4976_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y))) ) ).

% xor_numerals(7)
tff(fact_4977_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] : ( bit_se2046307713759805098atural(numera5444537566228673987atural(aa_num_num(bit1,X)),numera5444537566228673987atural(aa_num_num(bit1,Y))) = times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),bit_se2046307713759805098atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y))) ) ).

% xor_numerals(7)
tff(fact_4978_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = 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_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ) ).

% xor_numerals(7)
tff(fact_4979_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(aa_num_num(bit1,X))),numeral_numeral_nat(aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ) ).

% xor_numerals(7)
tff(fact_4980_zmod__numeral__Bit1,axiom,
    ! [V: num,W: 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,W))) = 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,W)))),one_one_int) ) ).

% zmod_numeral_Bit1
tff(fact_4981_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] : ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,N)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_gs),unique5052692396658037445od_int(M,N)) ) ).

% divmod_algorithm_code(5)
tff(fact_4982_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] : ( unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,N)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gt),unique5055182867167087721od_nat(M,N)) ) ).

% divmod_algorithm_code(5)
tff(fact_4983_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] : ( unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,N)) = aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_gu),unique3479559517661332726nteger(M,N)) ) ).

% divmod_algorithm_code(5)
tff(fact_4984_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y)))) ) ).

% xor_numerals(4)
tff(fact_4985_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : ( bit_se2046307713759805098atural(numera5444537566228673987atural(aa_num_num(bit0,X)),numera5444537566228673987atural(aa_num_num(bit1,Y))) = plus_p4538020629002901425atural(one_one_Code_natural,times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),bit_se2046307713759805098atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y)))) ) ).

% xor_numerals(4)
tff(fact_4986_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = 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_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ) ).

% xor_numerals(4)
tff(fact_4987_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(aa_num_num(bit0,X))),numeral_numeral_nat(aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ) ).

% xor_numerals(4)
tff(fact_4988_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_4989_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_4990_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_4991_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_4992_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_4993_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_4994_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),K)),X) ) ) ).

% int_div_less_self
tff(fact_4995_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,N))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),N)) ) ) ).

% verit_less_mono_div_int2
tff(fact_4996_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N)
       => 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,A2),N)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),N) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),N)) ) ) ).

% verit_le_mono_div_int
tff(fact_4997_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_4998_divmod__int__def,axiom,
    ! [M: num,N: num] : ( unique5052692396658037445od_int(M,N) = 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,M)),aa_num_int(numeral_numeral_int,N))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ) ).

% divmod_int_def
tff(fact_4999_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_5000_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_5001_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_5002_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_5003_prod_Ocase__distrib,axiom,
    ! [H2: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(H2,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_gv(H2),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5004_prod_Ocase__distrib,axiom,
    ! [H2: fun_o_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : ( aa_o_int(H2,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun7323606162024495668nt_int(aTP_Lamm_gw(H2),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5005_prod_Ocase__distrib,axiom,
    ! [H2: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(H2,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun82482788810740534_int_o(aTP_Lamm_gx(H2),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5006_prod_Ocase__distrib,axiom,
    ! [H2: fun_int_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : ( aa_int_int(H2,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun6119123525526402106nt_int(aTP_Lamm_gy(H2),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5007_prod_Ocase__distrib,axiom,
    ! [H2: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(H2,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_gz(H2),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5008_prod_Ocase__distrib,axiom,
    ! [H2: fun_Pr982841511892225912nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : ( aa_Pro6799258434465214121nt_int(H2,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun7461304428735181397nt_int(aTP_Lamm_ha(H2),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5009_prod_Ocase__distrib,axiom,
    ! [H2: fun_o_8988054265482332260nt_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : ( aa_o_P3134438092330336605nt_int(H2,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_hb(H2),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5010_prod_Ocase__distrib,axiom,
    ! [H2: fun_in4211310926729683942nt_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : ( aa_int7462565939943104719nt_int(H2,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun1034830265760468567nt_int(aTP_Lamm_hc(H2),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5011_prod_Ocase__distrib,axiom,
    ! [H2: fun_Pr2478283820680522243nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : ( aa_Pro6970504127699319548nt_int(H2,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_hd(H2),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5012_prod_Ocase__distrib,axiom,
    ! [H2: fun_fu3275390434952296369_nat_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : ( aa_fun359341089595923442_nat_o(H2,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),Prod)) = aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_he(H2),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5013_case__prod__app,axiom,
    ! [F: fun_na1668100129766719005at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] : ( aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),X),Y) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aa_Pro6895555690143131534at_nat(aTP_Lamm_hf(F),Y)),X) ) ).

% case_prod_app
tff(fact_5014_case__prod__app,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),X),Y)
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_Pro1957341097241355813_nat_o(aTP_Lamm_hg(F),Y)),X) ) ).

% case_prod_app
tff(fact_5015_nested__case__prod__simp,axiom,
    ! [F: fun_na1668100129766719005at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] : ( aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),X),Y) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aa_Pro6895555690143131534at_nat(aTP_Lamm_hf(F),Y)),X) ) ).

% nested_case_prod_simp
tff(fact_5016_nested__case__prod__simp,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),X),Y)
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_Pro1957341097241355813_nat_o(aTP_Lamm_hg(F),Y)),X) ) ).

% nested_case_prod_simp
tff(fact_5017_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ).

% mod_mult_eq
tff(fact_5018_mod__mult__eq,axiom,
    ! [A: int,C: 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,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ).

% mod_mult_eq
tff(fact_5019_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ) ).

% mod_mult_eq
tff(fact_5020_mod__mult__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] : ( modulo8411746178871703098atural(times_2397367101498566445atural(modulo8411746178871703098atural(A,C),modulo8411746178871703098atural(B,C)),C) = modulo8411746178871703098atural(times_2397367101498566445atural(A,B),C) ) ).

% mod_mult_eq
tff(fact_5021_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A3: nat,B: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_5022_mod__mult__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A3),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_5023_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A3),C) )
     => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B3),C) )
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A3),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_5024_mod__mult__cong,axiom,
    ! [A: code_natural,C: code_natural,A3: code_natural,B: code_natural,B3: code_natural] :
      ( ( modulo8411746178871703098atural(A,C) = modulo8411746178871703098atural(A3,C) )
     => ( ( modulo8411746178871703098atural(B,C) = modulo8411746178871703098atural(B3,C) )
       => ( modulo8411746178871703098atural(times_2397367101498566445atural(A,B),C) = modulo8411746178871703098atural(times_2397367101498566445atural(A3,B3),C) ) ) ) ).

% mod_mult_cong
tff(fact_5025_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) ) ).

% mod_mult_mult2
tff(fact_5026_mod__mult__mult2,axiom,
    ! [A: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) ) ).

% mod_mult_mult2
tff(fact_5027_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),C) ) ).

% mod_mult_mult2
tff(fact_5028_mod__mult__mult2,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] : ( modulo8411746178871703098atural(times_2397367101498566445atural(A,C),times_2397367101498566445atural(B,C)) = times_2397367101498566445atural(modulo8411746178871703098atural(A,B),C) ) ).

% mod_mult_mult2
tff(fact_5029_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ).

% mult_mod_right
tff(fact_5030_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ).

% mult_mod_right
tff(fact_5031_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ).

% mult_mod_right
tff(fact_5032_mult__mod__right,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] : ( times_2397367101498566445atural(C,modulo8411746178871703098atural(A,B)) = modulo8411746178871703098atural(times_2397367101498566445atural(C,A),times_2397367101498566445atural(C,B)) ) ).

% mult_mod_right
tff(fact_5033_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ).

% mod_mult_left_eq
tff(fact_5034_mod__mult__left__eq,axiom,
    ! [A: int,C: 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,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ).

% mod_mult_left_eq
tff(fact_5035_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C)),B)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ) ).

% mod_mult_left_eq
tff(fact_5036_mod__mult__left__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] : ( modulo8411746178871703098atural(times_2397367101498566445atural(modulo8411746178871703098atural(A,C),B),C) = modulo8411746178871703098atural(times_2397367101498566445atural(A,B),C) ) ).

% mod_mult_left_eq
tff(fact_5037_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ).

% mod_mult_right_eq
tff(fact_5038_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_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),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ).

% mod_mult_right_eq
tff(fact_5039_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ) ).

% mod_mult_right_eq
tff(fact_5040_mod__mult__right__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : ( modulo8411746178871703098atural(times_2397367101498566445atural(A,modulo8411746178871703098atural(B,C)),C) = modulo8411746178871703098atural(times_2397367101498566445atural(A,B),C) ) ).

% mod_mult_right_eq
tff(fact_5041_mod__minus__eq,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% mod_minus_eq
tff(fact_5042_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B))),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).

% mod_minus_eq
tff(fact_5043_mod__minus__cong,axiom,
    ! [A: int,B: int,A3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),B) )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A3)),B) ) ) ).

% mod_minus_cong
tff(fact_5044_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A3),B) )
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A3)),B) ) ) ).

% mod_minus_cong
tff(fact_5045_mod__minus__right,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B)) ) ).

% mod_minus_right
tff(fact_5046_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ).

% mod_minus_right
tff(fact_5047_split__cong,axiom,
    ! [Q3: product_prod_nat_nat,F: fun_na1668100129766719005at_nat,G: fun_na1668100129766719005at_nat,P3: product_prod_nat_nat] :
      ( ! [X2: nat,Y2: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2) = Q3 )
         => ( aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X2),Y2) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(G,X2),Y2) ) )
     => ( ( P3 = Q3 )
       => ( aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),P3) = aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(G),Q3) ) ) ) ).

% split_cong
tff(fact_5048_split__cong,axiom,
    ! [Q3: product_prod_nat_nat,F: fun_na5914473681949881652_nat_o,G: fun_na5914473681949881652_nat_o,P3: product_prod_nat_nat] :
      ( ! [X2: nat,Y2: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2) = Q3 )
         => ( aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X2),Y2) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(G,X2),Y2) ) )
     => ( ( P3 = Q3 )
       => ( aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),P3) = aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(G),Q3) ) ) ) ).

% split_cong
tff(fact_5049_split__cong,axiom,
    ! [Q3: product_prod_int_int,F: fun_in1696471567231710611nt_int,G: fun_in1696471567231710611nt_int,P3: product_prod_int_int] :
      ( ! [X2: int,Y2: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2) = Q3 )
         => ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X2),Y2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(G,X2),Y2) ) )
     => ( ( P3 = Q3 )
       => ( aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),P3) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(G),Q3) ) ) ) ).

% split_cong
tff(fact_5050_split__cong,axiom,
    ! [Q3: product_prod_int_int,F: fun_int_fun_int_o,G: fun_int_fun_int_o,P3: product_prod_int_int] :
      ( ! [X2: int,Y2: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2) = Q3 )
         => ( aa_int_o(aa_int_fun_int_o(F,X2),Y2)
          <=> aa_int_o(aa_int_fun_int_o(G,X2),Y2) ) )
     => ( ( P3 = Q3 )
       => ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),P3)
        <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(G),Q3) ) ) ) ).

% split_cong
tff(fact_5051_split__cong,axiom,
    ! [Q3: product_prod_int_int,F: fun_int_fun_int_int,G: fun_int_fun_int_int,P3: product_prod_int_int] :
      ( ! [X2: int,Y2: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2) = Q3 )
         => ( aa_int_int(aa_int_fun_int_int(F,X2),Y2) = aa_int_int(aa_int_fun_int_int(G,X2),Y2) ) )
     => ( ( P3 = Q3 )
       => ( aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),P3) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(G),Q3) ) ) ) ).

% split_cong
tff(fact_5052_old_Oprod_Ocase,axiom,
    ! [F: fun_na1668100129766719005at_nat,X1: nat,X22: nat] : ( aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X22) ) ).

% old.prod.case
tff(fact_5053_old_Oprod_Ocase,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X1: nat,X22: nat] : ( aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X1),X22) ) ).

% old.prod.case
tff(fact_5054_old_Oprod_Ocase,axiom,
    ! [F: fun_in1696471567231710611nt_int,X1: int,X22: int] : ( aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X1),X22) ) ).

% old.prod.case
tff(fact_5055_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_o,X1: int,X22: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22))
    <=> aa_int_o(aa_int_fun_int_o(F,X1),X22) ) ).

% old.prod.case
tff(fact_5056_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_int,X1: int,X22: int] : ( aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = aa_int_int(aa_int_fun_int_int(F,X1),X22) ) ).

% old.prod.case
tff(fact_5057_case__prod__Pair__iden,axiom,
    ! [P3: produc2453742332425399446nteger] : ( produc697581972998607883nteger(produc565448451787809096nteger,P3) = P3 ) ).

% case_prod_Pair_iden
tff(fact_5058_case__prod__Pair__iden,axiom,
    ! [P3: produc8799533983703829792et_nat] : ( aa_Pro5931322043027775230et_nat(produc1949718844479134567et_nat(produc6017135503889078682et_nat),P3) = P3 ) ).

% case_prod_Pair_iden
tff(fact_5059_case__prod__Pair__iden,axiom,
    ! [P3: produc3112576180687729086et_nat] : ( produc2255458887868536227et_nat(produc7291855603632349880et_nat,P3) = P3 ) ).

% case_prod_Pair_iden
tff(fact_5060_case__prod__Pair__iden,axiom,
    ! [P3: produc3043004577601930755nt_int] : ( produc6823905543215200229nt_int(produc7320969046608631221nt_int,P3) = P3 ) ).

% case_prod_Pair_iden
tff(fact_5061_case__prod__Pair__iden,axiom,
    ! [P3: produc4788131091073775919nt_int] : ( produc7102243260366339645nt_int(produc4729913460834754785nt_int,P3) = P3 ) ).

% case_prod_Pair_iden
tff(fact_5062_case__prod__Pair__iden,axiom,
    ! [P3: product_prod_int_int] : ( aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(product_Pair_int_int),P3) = P3 ) ).

% case_prod_Pair_iden
tff(fact_5063_case__prodE2,axiom,
    ! [Q: fun_fu8391177546919232046_nat_o,P: fun_na1668100129766719005at_nat,Z: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(Q,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(P),Z))
     => ~ ! [X2: nat,Y2: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2) )
           => ~ aa_fun3538214411680374423_nat_o(Q,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(P,X2),Y2)) ) ) ).

% case_prodE2
tff(fact_5064_case__prodE2,axiom,
    ! [Q: fun_fu6891447142836604759at_o_o,P: fun_na5914473681949881652_nat_o,Z: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(Q,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(P),Z))
     => ~ ! [X2: nat,Y2: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2) )
           => ~ aa_fun8408151826566693824at_o_o(Q,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(P,X2),Y2)) ) ) ).

% case_prodE2
tff(fact_5065_case__prodE2,axiom,
    ! [Q: fun_Pr3238374608701176162_int_o,P: fun_in1696471567231710611nt_int,Z: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Q,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(P),Z))
     => ~ ! [X2: int,Y2: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2) )
           => ~ aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P,X2),Y2)) ) ) ).

% case_prodE2
tff(fact_5066_case__prodE2,axiom,
    ! [Q: fun_o_o,P: fun_int_fun_int_o,Z: product_prod_int_int] :
      ( aa_o_o(Q,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(P),Z))
     => ~ ! [X2: int,Y2: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2) )
           => ~ aa_o_o(Q,aa_int_o(aa_int_fun_int_o(P,X2),Y2)) ) ) ).

% case_prodE2
tff(fact_5067_case__prodE2,axiom,
    ! [Q: fun_int_o,P: fun_int_fun_int_int,Z: product_prod_int_int] :
      ( aa_int_o(Q,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(P),Z))
     => ~ ! [X2: int,Y2: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2) )
           => ~ aa_int_o(Q,aa_int_int(aa_int_fun_int_int(P,X2),Y2)) ) ) ).

% case_prodE2
tff(fact_5068_case__prod__eta,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : ( produc4528391901023682385at_nat(aTP_Lamm_hh(F)) = F ) ).

% case_prod_eta
tff(fact_5069_case__prod__eta,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : ( produc6395135316500328296_nat_o(aTP_Lamm_hi(F)) = F ) ).

% case_prod_eta
tff(fact_5070_case__prod__eta,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : ( produc4245557441103728435nt_int(aTP_Lamm_hj(F)) = F ) ).

% case_prod_eta
tff(fact_5071_case__prod__eta,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : ( produc4947309494688390418_int_o(aTP_Lamm_hk(F)) = F ) ).

% case_prod_eta
tff(fact_5072_case__prod__eta,axiom,
    ! [F: fun_Pr982841511892225912nt_int] : ( produc8211389475949308722nt_int(aTP_Lamm_hl(F)) = F ) ).

% case_prod_eta
tff(fact_5073_cond__case__prod__eta,axiom,
    ! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
      ( ! [X2: nat,Y2: nat] : ( aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X2),Y2) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2)) )
     => ( produc4528391901023682385at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5074_cond__case__prod__eta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,G: fun_Pr2028404973800837934_nat_o] :
      ( ! [X2: nat,Y2: nat] : ( aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X2),Y2) = aa_Pro989595756709598047_nat_o(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2)) )
     => ( produc6395135316500328296_nat_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5075_cond__case__prod__eta,axiom,
    ! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X2: int,Y2: int] : ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X2),Y2) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2)) )
     => ( produc4245557441103728435nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5076_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: int,Y2: int] :
          ( aa_int_o(aa_int_fun_int_o(F,X2),Y2)
        <=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2)) )
     => ( produc4947309494688390418_int_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5077_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_int,G: fun_Pr982841511892225912nt_int] :
      ( ! [X2: int,Y2: int] : ( aa_int_int(aa_int_fun_int_int(F,X2),Y2) = aa_Pro6799258434465214121nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2)) )
     => ( produc8211389475949308722nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5078_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
     => ~ ! [D: 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,C),D)) ) ) ).

% mod_eqE
tff(fact_5079_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C) )
     => ~ ! [D: code_integer] : ( B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D)) ) ) ).

% mod_eqE
tff(fact_5080_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_5081_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_5082_divmod_H__nat__def,axiom,
    ! [M: num,N: num] : ( unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,numeral_numeral_nat(M)),numeral_numeral_nat(N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(M)),numeral_numeral_nat(N))) ) ).

% divmod'_nat_def
tff(fact_5083_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : ( aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_gp(L)),Qr) ) ).

% divmod_step_int_def
tff(fact_5084_uncurry__def,axiom,
    ! [F: fun_na1668100129766719005at_nat] : ( uncurr9212869592684144700at_nat(F) = produc4528391901023682385at_nat(F) ) ).

% uncurry_def
tff(fact_5085_uncurry__def,axiom,
    ! [F: fun_na5914473681949881652_nat_o] : ( uncurr8479020417594943315_nat_o(F) = produc6395135316500328296_nat_o(F) ) ).

% uncurry_def
tff(fact_5086_uncurry__def,axiom,
    ! [F: fun_in1696471567231710611nt_int] : ( uncurr7650761721940715016nt_int(F) = produc4245557441103728435nt_int(F) ) ).

% uncurry_def
tff(fact_5087_uncurry__def,axiom,
    ! [F: fun_int_fun_int_o] : ( uncurry_int_int_o(F) = produc4947309494688390418_int_o(F) ) ).

% uncurry_def
tff(fact_5088_uncurry__def,axiom,
    ! [F: fun_int_fun_int_int] : ( uncurry_int_int_int(F) = produc8211389475949308722nt_int(F) ) ).

% uncurry_def
tff(fact_5089_xor_Ocomm__monoid__axioms,axiom,
    aa_Code_integer_o(comm_m1654649897431166824nteger(bit_se3222712562003087583nteger),zero_z3403309356797280102nteger) ).

% xor.comm_monoid_axioms
tff(fact_5090_xor_Ocomm__monoid__axioms,axiom,
    aa_int_o(comm_monoid_int(bit_se6526347334894502574or_int),zero_zero_int) ).

% xor.comm_monoid_axioms
tff(fact_5091_xor_Ocomm__monoid__axioms,axiom,
    aa_nat_o(comm_monoid_nat(bit_se6528837805403552850or_nat),zero_zero_nat) ).

% xor.comm_monoid_axioms
tff(fact_5092_xor_Omonoid__axioms,axiom,
    aa_Code_integer_o(monoid_Code_integer(bit_se3222712562003087583nteger),zero_z3403309356797280102nteger) ).

% xor.monoid_axioms
tff(fact_5093_xor_Omonoid__axioms,axiom,
    aa_int_o(monoid_int(bit_se6526347334894502574or_int),zero_zero_int) ).

% xor.monoid_axioms
tff(fact_5094_xor_Omonoid__axioms,axiom,
    aa_nat_o(monoid_nat(bit_se6528837805403552850or_nat),zero_zero_nat) ).

% xor.monoid_axioms
tff(fact_5095_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_5096_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_5097_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) = A ) ).

% mult_div_mod_eq
tff(fact_5098_mult__div__mod__eq,axiom,
    ! [B: code_natural,A: code_natural] : ( plus_p4538020629002901425atural(times_2397367101498566445atural(B,divide5121882707175180666atural(A,B)),modulo8411746178871703098atural(A,B)) = A ) ).

% mult_div_mod_eq
tff(fact_5099_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_5100_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_5101_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B))) = A ) ).

% mod_mult_div_eq
tff(fact_5102_mod__mult__div__eq,axiom,
    ! [A: code_natural,B: code_natural] : ( plus_p4538020629002901425atural(modulo8411746178871703098atural(A,B),times_2397367101498566445atural(B,divide5121882707175180666atural(A,B))) = A ) ).

% mod_mult_div_eq
tff(fact_5103_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_5104_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_5105_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B)) = A ) ).

% mod_div_mult_eq
tff(fact_5106_mod__div__mult__eq,axiom,
    ! [A: code_natural,B: code_natural] : ( plus_p4538020629002901425atural(modulo8411746178871703098atural(A,B),times_2397367101498566445atural(divide5121882707175180666atural(A,B),B)) = A ) ).

% mod_div_mult_eq
tff(fact_5107_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_5108_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_5109_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) = A ) ).

% div_mult_mod_eq
tff(fact_5110_div__mult__mod__eq,axiom,
    ! [A: code_natural,B: code_natural] : ( plus_p4538020629002901425atural(times_2397367101498566445atural(divide5121882707175180666atural(A,B),B),modulo8411746178871703098atural(A,B)) = A ) ).

% div_mult_mod_eq
tff(fact_5111_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_5112_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_5113_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) ) ).

% mod_div_decomp
tff(fact_5114_mod__div__decomp,axiom,
    ! [A: code_natural,B: code_natural] : ( A = plus_p4538020629002901425atural(times_2397367101498566445atural(divide5121882707175180666atural(A,B),B),modulo8411746178871703098atural(A,B)) ) ).

% mod_div_decomp
tff(fact_5115_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ) ).

% cancel_div_mod_rules(1)
tff(fact_5116_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,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))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ) ).

% cancel_div_mod_rules(1)
tff(fact_5117_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) ) ).

% cancel_div_mod_rules(1)
tff(fact_5118_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : ( plus_p4538020629002901425atural(plus_p4538020629002901425atural(times_2397367101498566445atural(divide5121882707175180666atural(A,B),B),modulo8411746178871703098atural(A,B)),C) = plus_p4538020629002901425atural(A,C) ) ).

% cancel_div_mod_rules(1)
tff(fact_5119_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ) ).

% cancel_div_mod_rules(2)
tff(fact_5120_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,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))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ) ).

% cancel_div_mod_rules(2)
tff(fact_5121_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) ) ).

% cancel_div_mod_rules(2)
tff(fact_5122_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] : ( plus_p4538020629002901425atural(plus_p4538020629002901425atural(times_2397367101498566445atural(B,divide5121882707175180666atural(A,B)),modulo8411746178871703098atural(A,B)),C) = plus_p4538020629002901425atural(A,C) ) ).

% cancel_div_mod_rules(2)
tff(fact_5123_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ) ).

% div_mult1_eq
tff(fact_5124_div__mult1__eq,axiom,
    ! [A: int,B: int,C: 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)),C) = 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),C))),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),C))),C)) ) ).

% div_mult1_eq
tff(fact_5125_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C))),C)) ) ).

% div_mult1_eq
tff(fact_5126_div__mult1__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : ( divide5121882707175180666atural(times_2397367101498566445atural(A,B),C) = plus_p4538020629002901425atural(times_2397367101498566445atural(A,divide5121882707175180666atural(B,C)),divide5121882707175180666atural(times_2397367101498566445atural(A,modulo8411746178871703098atural(B,C)),C)) ) ).

% div_mult1_eq
tff(fact_5127_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ).

% minus_mult_div_eq_mod
tff(fact_5128_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),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) ) ).

% minus_mult_div_eq_mod
tff(fact_5129_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ) ).

% minus_mult_div_eq_mod
tff(fact_5130_minus__mult__div__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] : ( minus_7197305767214868737atural(A,times_2397367101498566445atural(B,divide5121882707175180666atural(A,B))) = modulo8411746178871703098atural(A,B) ) ).

% minus_mult_div_eq_mod
tff(fact_5131_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ).

% minus_mod_eq_mult_div
tff(fact_5132_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),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)) ) ).

% minus_mod_eq_mult_div
tff(fact_5133_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ).

% minus_mod_eq_mult_div
tff(fact_5134_minus__mod__eq__mult__div,axiom,
    ! [A: code_natural,B: code_natural] : ( minus_7197305767214868737atural(A,modulo8411746178871703098atural(A,B)) = times_2397367101498566445atural(B,divide5121882707175180666atural(A,B)) ) ).

% minus_mod_eq_mult_div
tff(fact_5135_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B) ) ).

% minus_mod_eq_div_mult
tff(fact_5136_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),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) ) ).

% minus_mod_eq_div_mult
tff(fact_5137_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B) ) ).

% minus_mod_eq_div_mult
tff(fact_5138_minus__mod__eq__div__mult,axiom,
    ! [A: code_natural,B: code_natural] : ( minus_7197305767214868737atural(A,modulo8411746178871703098atural(A,B)) = times_2397367101498566445atural(divide5121882707175180666atural(A,B),B) ) ).

% minus_mod_eq_div_mult
tff(fact_5139_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ).

% minus_div_mult_eq_mod
tff(fact_5140_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),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) ) ).

% minus_div_mult_eq_mod
tff(fact_5141_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ) ).

% minus_div_mult_eq_mod
tff(fact_5142_minus__div__mult__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] : ( minus_7197305767214868737atural(A,times_2397367101498566445atural(divide5121882707175180666atural(A,B),B)) = modulo8411746178871703098atural(A,B) ) ).

% minus_div_mult_eq_mod
tff(fact_5143_divmod__def,axiom,
    ! [M: num,N: num] : ( unique5052692396658037445od_int(M,N) = 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,M)),aa_num_int(numeral_numeral_int,N))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ) ).

% divmod_def
tff(fact_5144_divmod__def,axiom,
    ! [M: num,N: num] : ( unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,numeral_numeral_nat(M)),numeral_numeral_nat(N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(M)),numeral_numeral_nat(N))) ) ).

% divmod_def
tff(fact_5145_divmod__def,axiom,
    ! [M: num,N: num] : ( unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N))) ) ).

% divmod_def
tff(fact_5146_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5147_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5148_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5149_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N)) ) ) ).

% verit_le_mono_div
tff(fact_5150_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : ( aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(L),Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gq(L)),Qr) ) ).

% divmod_step_nat_def
tff(fact_5151_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(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,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_5152_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_5153_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,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B)
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5154_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_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,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_5155_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_5156_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,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5157_mod__double__modulus,axiom,
    ! [M: code_integer,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
       => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(X),M) )
          | ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5158_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5159_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),M) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5160_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),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)))
         => ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5161_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(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,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_5162_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_5163_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),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,W),Z) ) ).

% zle_add1_eq_le
tff(fact_5164_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),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,W),Z) ) ).

% zle_diff1_eq
tff(fact_5165_one__div__minus__numeral,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(one,N))) ) ).

% one_div_minus_numeral
tff(fact_5166_minus__one__div__numeral,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(one,N))) ) ).

% minus_one_div_numeral
tff(fact_5167_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_5168_signed__take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      ( aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N),A) = $ite(N = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))))) ) ).

% signed_take_bit_rec
tff(fact_5169_signed__take__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),A) = $ite(N = 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,N),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_5170_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(M,N))) ) ).

% numeral_div_minus_numeral
tff(fact_5171_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(M,N))) ) ).

% minus_numeral_div_numeral
tff(fact_5172_case__prodI2,axiom,
    ! [P3: produc2453742332425399446nteger,C: fun_fu7695273699059480439eger_o] :
      ( ! [A4: fun_Pr5098669352780829690e_term,B4: produc8923325533196201883nteger] :
          ( ( P3 = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A4),B4) )
         => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(C,A4),B4) )
     => aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(C),P3) ) ).

% case_prodI2
tff(fact_5173_case__prodI2,axiom,
    ! [P3: produc8799533983703829792et_nat,C: fun_fu8419763957888848259_nat_o] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: produc3658429121746597890et_nat] :
          ( ( P3 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A4),B4) )
         => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(C,A4),B4) )
     => aa_Pro5643009357358181178_nat_o(produc2915205735004773393_nat_o(C),P3) ) ).

% case_prodI2
tff(fact_5174_case__prodI2,axiom,
    ! [P3: produc3112576180687729086et_nat,C: fun_fu2611741329380808165_nat_o] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: produc8799533983703829792et_nat] :
          ( ( P3 = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A4),B4) )
         => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(C,A4),B4) )
     => aa_Pro2671238023309943196_nat_o(produc4465912260561112435_nat_o(C),P3) ) ).

% case_prodI2
tff(fact_5175_case__prodI2,axiom,
    ! [P3: produc3043004577601930755nt_int,C: fun_fu6865642474282033866_int_o] :
      ( ! [A4: fun_Pr7763680713840462443e_term,B4: product_prod_int_int] :
          ( ( P3 = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),B4) )
         => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(C,A4),B4) )
     => aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(C),P3) ) ).

% case_prodI2
tff(fact_5176_case__prodI2,axiom,
    ! [P3: produc4788131091073775919nt_int,C: fun_fu2978539155105838238_int_o] :
      ( ! [A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
          ( ( P3 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4) )
         => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(C,A4),B4) )
     => aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(C),P3) ) ).

% case_prodI2
tff(fact_5177_case__prodI2,axiom,
    ! [P3: product_prod_int_int,C: fun_int_fun_int_o] :
      ( ! [A4: int,B4: int] :
          ( ( P3 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B4) )
         => aa_int_o(aa_int_fun_int_o(C,A4),B4) )
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P3) ) ).

% case_prodI2
tff(fact_5178_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),aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B)) ) ).

% case_prodI
tff(fact_5179_case__prodI,axiom,
    ! [F: fun_fu8419763957888848259_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(F,A),B)
     => aa_Pro5643009357358181178_nat_o(produc2915205735004773393_nat_o(F),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B)) ) ).

% case_prodI
tff(fact_5180_case__prodI,axiom,
    ! [F: fun_fu2611741329380808165_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(F,A),B)
     => aa_Pro2671238023309943196_nat_o(produc4465912260561112435_nat_o(F),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B)) ) ).

% case_prodI
tff(fact_5181_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),aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B)) ) ).

% case_prodI
tff(fact_5182_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),aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B)) ) ).

% case_prodI
tff(fact_5183_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(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% case_prodI
tff(fact_5184_mem__case__prodI2,axiom,
    ! [P3: produc4788131091073775919nt_int,Z: nat,C: fun_fu9079770784915425324et_nat] :
      ( ! [A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
          ( ( P3 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4) )
         => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro3599192995643711491et_nat(aa_fun8409204002232819093et_nat(C,A4),B4)) )
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc5011043943518292116et_nat(C,P3)) ) ).

% mem_case_prodI2
tff(fact_5185_mem__case__prodI2,axiom,
    ! [P3: produc4788131091073775919nt_int,Z: int,C: fun_fu4266222366127506568et_int] :
      ( ! [A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
          ( ( P3 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4) )
         => aa_set_int_o(member_int(Z),aa_Pro8644714012989290591et_int(aa_fun5919120857301092337et_int(C,A4),B4)) )
     => aa_set_int_o(member_int(Z),produc833192924009095408et_int(C,P3)) ) ).

% mem_case_prodI2
tff(fact_5186_mem__case__prodI2,axiom,
    ! [P3: produc4788131091073775919nt_int,Z: set_nat,C: fun_fu5458086908000294626et_nat] :
      ( ! [A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
          ( ( P3 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4) )
         => aa_set_set_nat_o(member_set_nat(Z),aa_Pro1963545105414287545et_nat(aa_fun8341152516354955083et_nat(C,A4),B4)) )
     => aa_set_set_nat_o(member_set_nat(Z),produc4945675248400274762et_nat(C,P3)) ) ).

% mem_case_prodI2
tff(fact_5187_mem__case__prodI2,axiom,
    ! [P3: produc8799533983703829792et_nat,Z: nat,C: fun_fu86038545275896455et_nat] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: produc3658429121746597890et_nat] :
          ( ( P3 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A4),B4) )
         => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro3416150927818442182et_nat(aa_fun6477889762090788408et_nat(C,A4),B4)) )
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc6247649352436904781et_nat(C,P3)) ) ).

% mem_case_prodI2
tff(fact_5188_mem__case__prodI2,axiom,
    ! [P3: produc8799533983703829792et_nat,Z: int,C: fun_fu4495862163342753507et_int] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: produc3658429121746597890et_nat] :
          ( ( P3 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A4),B4) )
         => aa_set_int_o(member_int(Z),aa_Pro8461671945164021282et_int(aa_fun3987806617159061652et_int(C,A4),B4)) )
     => aa_set_int_o(member_int(Z),produc2069798332927708073et_int(C,P3)) ) ).

% mem_case_prodI2
tff(fact_5189_mem__case__prodI2,axiom,
    ! [P3: produc8799533983703829792et_nat,Z: set_nat,C: fun_fu8819589147417191741et_nat] :
      ( ! [A4: fun_Pr6460944699146269935_nat_o,B4: produc3658429121746597890et_nat] :
          ( ( P3 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A4),B4) )
         => aa_set_set_nat_o(member_set_nat(Z),aa_Pro7265560661054114428et_nat(aa_fun6473727797332513774et_nat(C,A4),B4)) )
     => aa_set_set_nat_o(member_set_nat(Z),produc4421399663965592579et_nat(C,P3)) ) ).

% mem_case_prodI2
tff(fact_5190_mem__case__prodI2,axiom,
    ! [P3: produc4788131091073775919nt_int,Z: set_Pr958786334691620121nt_int,C: fun_fu719511775830429311nt_int] :
      ( ! [A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
          ( ( P3 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4) )
         => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z),aa_Pro7039973274325895612nt_int(aa_fun4635030967160187712nt_int(C,A4),B4)) )
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z),produc6653901401015897067nt_int(C,P3)) ) ).

% mem_case_prodI2
tff(fact_5191_mem__case__prodI2,axiom,
    ! [P3: produc2453742332425399446nteger,Z: nat,C: fun_fu2531416508075921811et_nat] :
      ( ! [A4: fun_Pr5098669352780829690e_term,B4: produc8923325533196201883nteger] :
          ( ( P3 = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A4),B4) )
         => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro381016180347496421et_nat(aa_fun4088370000251959292et_nat(C,A4),B4)) )
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc6164657169593426171et_nat(C,P3)) ) ).

% mem_case_prodI2
tff(fact_5192_mem__case__prodI2,axiom,
    ! [P3: produc2453742332425399446nteger,Z: int,C: fun_fu6941240126142778863et_int] :
      ( ! [A4: fun_Pr5098669352780829690e_term,B4: produc8923325533196201883nteger] :
          ( ( P3 = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A4),B4) )
         => aa_set_int_o(member_int(Z),aa_Pro5426537197693075521et_int(aa_fun1598286855320232536et_int(C,A4),B4)) )
     => aa_set_int_o(member_int(Z),produc1986806150084229463et_int(C,P3)) ) ).

% mem_case_prodI2
tff(fact_5193_mem__case__prodI2,axiom,
    ! [P3: produc3043004577601930755nt_int,Z: nat,C: fun_fu1995110232641039616et_nat] :
      ( ! [A4: fun_Pr7763680713840462443e_term,B4: product_prod_int_int] :
          ( ( P3 = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),B4) )
         => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro3599192995643711491et_nat(aa_fun8024691972395008617et_nat(C,A4),B4)) )
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc3638104198992362856et_nat(C,P3)) ) ).

% mem_case_prodI2
tff(fact_5194_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_fu9079770784915425324et_nat,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro3599192995643711491et_nat(aa_fun8409204002232819093et_nat(C,A),B))
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc5011043943518292116et_nat(C,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5195_mem__case__prodI,axiom,
    ! [Z: int,C: fun_fu4266222366127506568et_int,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_set_int_o(member_int(Z),aa_Pro8644714012989290591et_int(aa_fun5919120857301092337et_int(C,A),B))
     => aa_set_int_o(member_int(Z),produc833192924009095408et_int(C,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5196_mem__case__prodI,axiom,
    ! [Z: set_nat,C: fun_fu5458086908000294626et_nat,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_set_set_nat_o(member_set_nat(Z),aa_Pro1963545105414287545et_nat(aa_fun8341152516354955083et_nat(C,A),B))
     => aa_set_set_nat_o(member_set_nat(Z),produc4945675248400274762et_nat(C,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5197_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_fu86038545275896455et_nat,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro3416150927818442182et_nat(aa_fun6477889762090788408et_nat(C,A),B))
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc6247649352436904781et_nat(C,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_5198_mem__case__prodI,axiom,
    ! [Z: int,C: fun_fu4495862163342753507et_int,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_set_int_o(member_int(Z),aa_Pro8461671945164021282et_int(aa_fun3987806617159061652et_int(C,A),B))
     => aa_set_int_o(member_int(Z),produc2069798332927708073et_int(C,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_5199_mem__case__prodI,axiom,
    ! [Z: set_nat,C: fun_fu8819589147417191741et_nat,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_set_set_nat_o(member_set_nat(Z),aa_Pro7265560661054114428et_nat(aa_fun6473727797332513774et_nat(C,A),B))
     => aa_set_set_nat_o(member_set_nat(Z),produc4421399663965592579et_nat(C,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_5200_mem__case__prodI,axiom,
    ! [Z: set_Pr958786334691620121nt_int,C: fun_fu719511775830429311nt_int,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z),aa_Pro7039973274325895612nt_int(aa_fun4635030967160187712nt_int(C,A),B))
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z),produc6653901401015897067nt_int(C,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5201_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_fu2531416508075921811et_nat,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro381016180347496421et_nat(aa_fun4088370000251959292et_nat(C,A),B))
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc6164657169593426171et_nat(C,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B))) ) ).

% mem_case_prodI
tff(fact_5202_mem__case__prodI,axiom,
    ! [Z: int,C: fun_fu6941240126142778863et_int,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_set_int_o(member_int(Z),aa_Pro5426537197693075521et_int(aa_fun1598286855320232536et_int(C,A),B))
     => aa_set_int_o(member_int(Z),produc1986806150084229463et_int(C,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B))) ) ).

% mem_case_prodI
tff(fact_5203_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_fu1995110232641039616et_nat,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro3599192995643711491et_nat(aa_fun8024691972395008617et_nat(C,A),B))
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc3638104198992362856et_nat(C,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5204_case__prodI2_H,axiom,
    ! [P3: product_prod_nat_nat,C: fun_na5914473681949881652_nat_o,X: product_prod_nat_nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B4) = P3 )
         => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,A4),B4),X) )
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P3),X) ) ).

% case_prodI2'
tff(fact_5205_Collect__const__case__prod,axiom,
    ! [P: $o] :
      ( collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_hm((P)))) = $ite((P),top_to4669805908274784177at_nat,bot_bo2099793752762293965at_nat) ) ).

% Collect_const_case_prod
tff(fact_5206_Collect__const__case__prod,axiom,
    ! [P: $o] :
      ( collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_hn((P)))) = $ite((P),top_to4366644338036079209nt_int,bot_bo1796632182523588997nt_int) ) ).

% Collect_const_case_prod
tff(fact_5207_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ).

% signed_take_bit_of_minus_1
tff(fact_5208_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] : ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ) ).

% signed_take_bit_of_minus_1
tff(fact_5209_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : ( aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(numeral_numeral_nat(K)),one_one_Code_integer) = one_one_Code_integer ) ).

% signed_take_bit_numeral_of_1
tff(fact_5210_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,numeral_numeral_nat(K)),one_one_int) = one_one_int ) ).

% signed_take_bit_numeral_of_1
tff(fact_5211_signed__take__bit__0,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(zero_zero_nat),A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ).

% signed_take_bit_0
tff(fact_5212_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_5213_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_fu9079770784915425324et_nat,P3: produc4788131091073775919nt_int] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc5011043943518292116et_nat(C,P3))
     => ~ ! [X2: fun_in5730379987311621695e_term,Y2: product_prod_int_int] :
            ( ( P3 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X2),Y2) )
           => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro3599192995643711491et_nat(aa_fun8409204002232819093et_nat(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_5214_mem__case__prodE,axiom,
    ! [Z: int,C: fun_fu4266222366127506568et_int,P3: produc4788131091073775919nt_int] :
      ( aa_set_int_o(member_int(Z),produc833192924009095408et_int(C,P3))
     => ~ ! [X2: fun_in5730379987311621695e_term,Y2: product_prod_int_int] :
            ( ( P3 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X2),Y2) )
           => ~ aa_set_int_o(member_int(Z),aa_Pro8644714012989290591et_int(aa_fun5919120857301092337et_int(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_5215_mem__case__prodE,axiom,
    ! [Z: set_nat,C: fun_fu5458086908000294626et_nat,P3: produc4788131091073775919nt_int] :
      ( aa_set_set_nat_o(member_set_nat(Z),produc4945675248400274762et_nat(C,P3))
     => ~ ! [X2: fun_in5730379987311621695e_term,Y2: product_prod_int_int] :
            ( ( P3 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X2),Y2) )
           => ~ aa_set_set_nat_o(member_set_nat(Z),aa_Pro1963545105414287545et_nat(aa_fun8341152516354955083et_nat(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_5216_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_fu86038545275896455et_nat,P3: produc8799533983703829792et_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc6247649352436904781et_nat(C,P3))
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] :
            ( ( P3 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X2),Y2) )
           => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro3416150927818442182et_nat(aa_fun6477889762090788408et_nat(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_5217_mem__case__prodE,axiom,
    ! [Z: int,C: fun_fu4495862163342753507et_int,P3: produc8799533983703829792et_nat] :
      ( aa_set_int_o(member_int(Z),produc2069798332927708073et_int(C,P3))
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] :
            ( ( P3 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X2),Y2) )
           => ~ aa_set_int_o(member_int(Z),aa_Pro8461671945164021282et_int(aa_fun3987806617159061652et_int(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_5218_mem__case__prodE,axiom,
    ! [Z: set_nat,C: fun_fu8819589147417191741et_nat,P3: produc8799533983703829792et_nat] :
      ( aa_set_set_nat_o(member_set_nat(Z),produc4421399663965592579et_nat(C,P3))
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] :
            ( ( P3 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X2),Y2) )
           => ~ aa_set_set_nat_o(member_set_nat(Z),aa_Pro7265560661054114428et_nat(aa_fun6473727797332513774et_nat(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_5219_mem__case__prodE,axiom,
    ! [Z: set_Pr958786334691620121nt_int,C: fun_fu719511775830429311nt_int,P3: produc4788131091073775919nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z),produc6653901401015897067nt_int(C,P3))
     => ~ ! [X2: fun_in5730379987311621695e_term,Y2: product_prod_int_int] :
            ( ( P3 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X2),Y2) )
           => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z),aa_Pro7039973274325895612nt_int(aa_fun4635030967160187712nt_int(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_5220_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_fu2531416508075921811et_nat,P3: produc2453742332425399446nteger] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc6164657169593426171et_nat(C,P3))
     => ~ ! [X2: fun_Pr5098669352780829690e_term,Y2: produc8923325533196201883nteger] :
            ( ( P3 = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X2),Y2) )
           => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro381016180347496421et_nat(aa_fun4088370000251959292et_nat(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_5221_mem__case__prodE,axiom,
    ! [Z: int,C: fun_fu6941240126142778863et_int,P3: produc2453742332425399446nteger] :
      ( aa_set_int_o(member_int(Z),produc1986806150084229463et_int(C,P3))
     => ~ ! [X2: fun_Pr5098669352780829690e_term,Y2: produc8923325533196201883nteger] :
            ( ( P3 = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X2),Y2) )
           => ~ aa_set_int_o(member_int(Z),aa_Pro5426537197693075521et_int(aa_fun1598286855320232536et_int(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_5222_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_fu1995110232641039616et_nat,P3: produc3043004577601930755nt_int] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc3638104198992362856et_nat(C,P3))
     => ~ ! [X2: fun_Pr7763680713840462443e_term,Y2: product_prod_int_int] :
            ( ( P3 = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X2),Y2) )
           => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro3599192995643711491et_nat(aa_fun8024691972395008617et_nat(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_5223_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] : ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(uminus_uminus_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(uminus_uminus_int,K)) ) ).

% signed_take_bit_minus
tff(fact_5224_case__prodE,axiom,
    ! [C: fun_fu7695273699059480439eger_o,P3: produc2453742332425399446nteger] :
      ( aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(C),P3)
     => ~ ! [X2: fun_Pr5098669352780829690e_term,Y2: produc8923325533196201883nteger] :
            ( ( P3 = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X2),Y2) )
           => ~ aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(C,X2),Y2) ) ) ).

% case_prodE
tff(fact_5225_case__prodE,axiom,
    ! [C: fun_fu8419763957888848259_nat_o,P3: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(produc2915205735004773393_nat_o(C),P3)
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] :
            ( ( P3 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X2),Y2) )
           => ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(C,X2),Y2) ) ) ).

% case_prodE
tff(fact_5226_case__prodE,axiom,
    ! [C: fun_fu2611741329380808165_nat_o,P3: produc3112576180687729086et_nat] :
      ( aa_Pro2671238023309943196_nat_o(produc4465912260561112435_nat_o(C),P3)
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat] :
            ( ( P3 = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X2),Y2) )
           => ~ aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(C,X2),Y2) ) ) ).

% case_prodE
tff(fact_5227_case__prodE,axiom,
    ! [C: fun_fu6865642474282033866_int_o,P3: produc3043004577601930755nt_int] :
      ( aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(C),P3)
     => ~ ! [X2: fun_Pr7763680713840462443e_term,Y2: product_prod_int_int] :
            ( ( P3 = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X2),Y2) )
           => ~ aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(C,X2),Y2) ) ) ).

% case_prodE
tff(fact_5228_case__prodE,axiom,
    ! [C: fun_fu2978539155105838238_int_o,P3: produc4788131091073775919nt_int] :
      ( aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(C),P3)
     => ~ ! [X2: fun_in5730379987311621695e_term,Y2: product_prod_int_int] :
            ( ( P3 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X2),Y2) )
           => ~ aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(C,X2),Y2) ) ) ).

% case_prodE
tff(fact_5229_case__prodE,axiom,
    ! [C: fun_int_fun_int_o,P3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P3)
     => ~ ! [X2: int,Y2: int] :
            ( ( P3 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2) )
           => ~ aa_int_o(aa_int_fun_int_o(C,X2),Y2) ) ) ).

% case_prodE
tff(fact_5230_case__prodD,axiom,
    ! [F: fun_fu7695273699059480439eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(F),aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B))
     => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(F,A),B) ) ).

% case_prodD
tff(fact_5231_case__prodD,axiom,
    ! [F: fun_fu8419763957888848259_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_Pro5643009357358181178_nat_o(produc2915205735004773393_nat_o(F),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B))
     => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_5232_case__prodD,axiom,
    ! [F: fun_fu2611741329380808165_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat] :
      ( aa_Pro2671238023309943196_nat_o(produc4465912260561112435_nat_o(F),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B))
     => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_5233_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),aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B))
     => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(F,A),B) ) ).

% case_prodD
tff(fact_5234_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),aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))
     => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(F,A),B) ) ).

% case_prodD
tff(fact_5235_case__prodD,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_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_5236_case__prodE_H,axiom,
    ! [C: fun_na5914473681949881652_nat_o,P3: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P3),Z)
     => ~ ! [X2: nat,Y2: nat] :
            ( ( P3 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2) )
           => ~ aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,X2),Y2),Z) ) ) ).

% case_prodE'
tff(fact_5237_case__prodD_H,axiom,
    ! [R: fun_na5914473681949881652_nat_o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(R),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),C)
     => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(R,A),B),C) ) ).

% case_prodD'
tff(fact_5238_Id__on__def_H,axiom,
    ! [A2: fun_se7249947736802449410_int_o] : ( id_on_6410342593070439734nt_int(collec5210948495886036740nt_int(A2)) = collec2889756465124990930nt_int(produc1547782455048856492_int_o(aTP_Lamm_ho(A2))) ) ).

% Id_on_def'
tff(fact_5239_Id__on__def_H,axiom,
    ! [A2: fun_set_nat_o] : ( id_on_set_nat(collect_set_nat(A2)) = collec6662362479098859352et_nat(produc6247414631856714078_nat_o(aTP_Lamm_hp(A2))) ) ).

% Id_on_def'
tff(fact_5240_Id__on__def_H,axiom,
    ! [A2: fun_nat_o] : ( id_on_nat(collect_nat(A2)) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_hq(A2))) ) ).

% Id_on_def'
tff(fact_5241_Id__on__def_H,axiom,
    ! [A2: fun_fu5135969500543783852at_o_o] : ( id_on_7912660205148244492_nat_o(collec6407654713642994138_nat_o(A2)) = collec5129485182825495890_nat_o(produc2619172464267079468at_o_o(aTP_Lamm_hr(A2))) ) ).

% Id_on_def'
tff(fact_5242_Id__on__def_H,axiom,
    ! [A2: fun_int_o] : ( id_on_int(collect_int(A2)) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_hs(A2))) ) ).

% Id_on_def'
tff(fact_5243_rel__restrict__def,axiom,
    ! [R: set_Pr8057934254988874055nt_int,A2: set_se6260736226359567993nt_int] : ( rel_re4349003428253852667nt_int(R,A2) = collec2889756465124990930nt_int(produc1547782455048856492_int_o(aa_set6233473662189019185_int_o(aTP_Lamm_ht(R),A2))) ) ).

% rel_restrict_def
tff(fact_5244_rel__restrict__def,axiom,
    ! [R: set_Pr5488025237498180813et_nat,A2: set_set_nat] : ( rel_restrict_set_nat(R,A2) = collec6662362479098859352et_nat(produc6247414631856714078_nat_o(aa_set550086741393328324_nat_o(aTP_Lamm_hu(R),A2))) ) ).

% rel_restrict_def
tff(fact_5245_rel__restrict__def,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_nat] : ( rel_restrict_nat(R,A2) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aa_set8149993881734264422_nat_o(aTP_Lamm_hv(R),A2))) ) ).

% rel_restrict_def
tff(fact_5246_rel__restrict__def,axiom,
    ! [R: set_Pr2890355626087285447_nat_o,A2: set_fu3555266933746557007_nat_o] : ( rel_re4406848170174997969_nat_o(R,A2) = collec5129485182825495890_nat_o(produc2619172464267079468at_o_o(aa_set2556502855246081627at_o_o(aTP_Lamm_hw(R),A2))) ) ).

% rel_restrict_def
tff(fact_5247_rel__restrict__def,axiom,
    ! [R: set_Pr958786334691620121nt_int,A2: set_int] : ( rel_restrict_int(R,A2) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aa_set5204721951351315026_int_o(aTP_Lamm_hx(R),A2))) ) ).

% rel_restrict_def
tff(fact_5248_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_less_induct
tff(fact_5249_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_5250_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_5251_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_le_induct
tff(fact_5252_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),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,W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_5253_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_gr_induct
tff(fact_5254_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_5255_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_5256_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_5257_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_5258_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z) ) ).

% zless_imp_add1_zle
tff(fact_5259_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_ge_induct
tff(fact_5260_add1__zle__eq,axiom,
    ! [W: 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,W),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% add1_zle_eq
tff(fact_5261_int__induct,axiom,
    ! [P: fun_int_o,K: int,I: int] :
      ( aa_int_o(P,K)
     => ( ! [I2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
           => ( aa_int_o(P,I2)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_induct
tff(fact_5262_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( N = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_5263_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
     => ( ( M = one_one_int )
        | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_5264_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_5265_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(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,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,K))),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_5266_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_5267_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_5268_divmod__BitM__2__eq,axiom,
    ! [M: num] : ( unique5052692396658037445od_int(bitM(M),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,M)),one_one_int)),one_one_int) ) ).

% divmod_BitM_2_eq
tff(fact_5269_split__part,axiom,
    ! [P: $o,Q: fun_int_fun_int_o,X3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_hy((P)),Q)),X3)
    <=> ( (P)
        & aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Q),X3) ) ) ).

% split_part
tff(fact_5270_diff__Suc__1,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),one_one_nat) = N ) ).

% diff_Suc_1
tff(fact_5271_signed__take__bit__Suc__1,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(suc,N)),one_one_Code_integer) = one_one_Code_integer ) ).

% signed_take_bit_Suc_1
tff(fact_5272_signed__take__bit__Suc__1,axiom,
    ! [N: nat] : ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),one_one_int) = one_one_int ) ).

% signed_take_bit_Suc_1
tff(fact_5273_Suc__diff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_5274_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = numeral_numeral_nat(aa_num_num(bit0,one)) ).

% Suc_1
tff(fact_5275_Suc__diff__1,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).

% Suc_diff_1
tff(fact_5276_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( numeral_numeral_nat(K) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(K)),N))),numeral_numeral_nat(K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_5277_sub__num__simps_I2_J,axiom,
    ! [L: num] : ( neg_numeral_sub_int(one,aa_num_num(bit0,L)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bitM(L))) ) ).

% sub_num_simps(2)
tff(fact_5278_sub__num__simps_I2_J,axiom,
    ! [L: num] : ( neg_nu5755505904847501662nteger(one,aa_num_num(bit0,L)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,bitM(L))) ) ).

% sub_num_simps(2)
tff(fact_5279_sub__num__simps_I2_J,axiom,
    ! [L: num] : ( neg_numeral_sub_rat(one,aa_num_num(bit0,L)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(bitM(L))) ) ).

% sub_num_simps(2)
tff(fact_5280_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_5281_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,pred_numeral(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_5282_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ).

% signed_take_bit_Suc_bit1
tff(fact_5283_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,numeral_numeral_nat(L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,pred_numeral(L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ).

% signed_take_bit_numeral_bit1
tff(fact_5284_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_hz),Prod) ).

% prod.disc_eq_case
tff(fact_5285_unique__remainder,axiom,
    ! [A: int,B: int,Q3: int,R2: int,Q4: int,R3: 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,Q4),R3))
       => ( R2 = R3 ) ) ) ).

% unique_remainder
tff(fact_5286_unique__quotient,axiom,
    ! [A: int,B: int,Q3: int,R2: int,Q4: int,R3: 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,Q4),R3))
       => ( Q3 = Q4 ) ) ) ).

% unique_quotient
tff(fact_5287_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_5288_Suc__eq__plus1,axiom,
    ! [N: nat] : ( aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ) ).

% Suc_eq_plus1
tff(fact_5289_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_5290_Suc__eq__plus1__left,axiom,
    ! [N: nat] : ( aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ) ).

% Suc_eq_plus1_left
tff(fact_5291_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N) ) ).

% diff_Suc_eq_diff_pred
tff(fact_5292_eucl__rel__int__by0,axiom,
    ! [K: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K)) ).

% eucl_rel_int_by0
tff(fact_5293_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3)
             => ( aa_nat_o(P,N3)
               => aa_nat_o(P,aa_nat_nat(suc,N3)) ) )
         => aa_nat_o(P,N) ) ) ) ).

% nat_induct_non_zero
tff(fact_5294_div__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = Q3 ) ) ).

% div_int_unique
tff(fact_5295_mod__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = R2 ) ) ).

% mod_int_unique
tff(fact_5296_pred__numeral__def,axiom,
    ! [K: num] : ( pred_numeral(K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,numeral_numeral_nat(K)),one_one_nat) ) ).

% pred_numeral_def
tff(fact_5297_small__lazy_H_Ocases,axiom,
    ! [X: product_prod_int_int] :
      ~ ! [D: int,I2: int] : ( X != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D),I2) ) ).

% small_lazy'.cases
tff(fact_5298_exhaustive__int_H_Ocases,axiom,
    ! [X: produc4788131091073775919nt_int] :
      ~ ! [F2: fun_in5730379987311621695e_term,D: int,I2: int] : ( X != aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,F2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D),I2)) ) ).

% exhaustive_int'.cases
tff(fact_5299_full__exhaustive__int_H_Ocases,axiom,
    ! [X: produc3043004577601930755nt_int] :
      ~ ! [F2: fun_Pr7763680713840462443e_term,D: int,I2: int] : ( X != aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,F2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D),I2)) ) ).

% full_exhaustive_int'.cases
tff(fact_5300_Suc__pred_H,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_5301_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_5302_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = $ite(M = zero_zero_nat,N,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N))) ) ).

% add_eq_if
tff(fact_5303_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_5304_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L) )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_5305_numeral__BitM,axiom,
    ! [N: num] : ( aa_num_int(numeral_numeral_int,bitM(N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),one_one_int) ) ).

% numeral_BitM
tff(fact_5306_numeral__BitM,axiom,
    ! [N: num] : ( aa_num_Code_integer(numera6620942414471956472nteger,bitM(N)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N))),one_one_Code_integer) ) ).

% numeral_BitM
tff(fact_5307_numeral__BitM,axiom,
    ! [N: num] : ( numeral_numeral_rat(bitM(N)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,numeral_numeral_rat(aa_num_num(bit0,N))),one_one_rat) ) ).

% numeral_BitM
tff(fact_5308_eucl__rel__int,axiom,
    ! [K: int,L: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ).

% eucl_rel_int
tff(fact_5309_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_5310_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_5311_signed__take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ) ).

% signed_take_bit_Suc
tff(fact_5312_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] : ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),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,N),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_5313_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ).

% unset_bit_Suc
tff(fact_5314_unset__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ) ).

% unset_bit_Suc
tff(fact_5315_unset__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : ( bit_se7083795435491715335atural(aa_nat_nat(suc,N),A) = plus_p4538020629002901425atural(modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))),times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),bit_se7083795435491715335atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))))) ) ).

% unset_bit_Suc
tff(fact_5316_unset__bit__Suc,axiom,
    ! [N: nat,A: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,N)),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,N),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_5317_set__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ) ).

% set_bit_Suc
tff(fact_5318_set__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : ( bit_se1617098188084679374atural(aa_nat_nat(suc,N),A) = plus_p4538020629002901425atural(modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))),times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),bit_se1617098188084679374atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))))) ) ).

% set_bit_Suc
tff(fact_5319_set__bit__Suc,axiom,
    ! [N: nat,A: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,N)),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,N),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_5320_set__bit__Suc,axiom,
    ! [N: nat,A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ).

% set_bit_Suc
tff(fact_5321_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q3)),R2) )
        & $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_5322_divmod__nat__if,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat7603221163087113075at_nat(divmod_nat(M),N) = $ite(
          ( ( N = zero_zero_nat )
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ),
          aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M),
          aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ia),aa_nat7603221163087113075at_nat(divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ).

% divmod_nat_if
tff(fact_5323_flip__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ) ).

% flip_bit_Suc
tff(fact_5324_flip__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : ( bit_se168947363167071951atural(aa_nat_nat(suc,N),A) = plus_p4538020629002901425atural(modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))),times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),bit_se168947363167071951atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))))) ) ).

% flip_bit_Suc
tff(fact_5325_flip__bit__Suc,axiom,
    ! [N: nat,A: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,aa_nat_nat(suc,N)),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,N),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_5326_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ).

% flip_bit_Suc
tff(fact_5327_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(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,pred_numeral(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ).

% take_bit_numeral_minus_bit1
tff(fact_5328_int__ge__less__than2__def,axiom,
    ! [D2: int] : ( int_ge_less_than2(D2) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_ib(D2))) ) ).

% int_ge_less_than2_def
tff(fact_5329_int__ge__less__than__def,axiom,
    ! [D2: int] : ( int_ge_less_than(D2) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_ic(D2))) ) ).

% int_ge_less_than_def
tff(fact_5330_take__bit__Suc__1,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),one_one_Code_integer) = one_one_Code_integer ) ).

% take_bit_Suc_1
tff(fact_5331_take__bit__Suc__1,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),one_one_nat) = one_one_nat ) ).

% take_bit_Suc_1
tff(fact_5332_take__bit__Suc__1,axiom,
    ! [N: nat] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),one_one_int) = one_one_int ) ).

% take_bit_Suc_1
tff(fact_5333_take__bit__numeral__1,axiom,
    ! [L: num] : ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(numeral_numeral_nat(L)),one_one_Code_integer) = one_one_Code_integer ) ).

% take_bit_numeral_1
tff(fact_5334_take__bit__numeral__1,axiom,
    ! [L: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,numeral_numeral_nat(L)),one_one_nat) = one_one_nat ) ).

% take_bit_numeral_1
tff(fact_5335_take__bit__numeral__1,axiom,
    ! [L: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,numeral_numeral_nat(L)),one_one_int) = one_one_int ) ).

% take_bit_numeral_1
tff(fact_5336_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),one_one_Code_integer) = zero_z3403309356797280102nteger )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_5337_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = zero_zero_nat )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_5338_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),one_one_int) = zero_zero_int )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_5339_take__bit__minus,axiom,
    ! [N: nat,K: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K))) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,K)) ) ).

% take_bit_minus
tff(fact_5340_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) != zero_zero_int )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_5341_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),numeral_numeral_nat(aa_num_num(bit0,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),numeral_numeral_nat(K))),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% take_bit_Suc_bit0
tff(fact_5342_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),aa_num_Code_integer(numera6620942414471956472nteger,K))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ).

% take_bit_Suc_bit0
tff(fact_5343_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : ( bit_se569199155075624693atural(aa_nat_nat(suc,N),numera5444537566228673987atural(aa_num_num(bit0,K))) = times_2397367101498566445atural(bit_se569199155075624693atural(N,numera5444537566228673987atural(K)),numera5444537566228673987atural(aa_num_num(bit0,one))) ) ).

% take_bit_Suc_bit0
tff(fact_5344_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ).

% take_bit_Suc_bit0
tff(fact_5345_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,numeral_numeral_nat(L)),numeral_numeral_nat(aa_num_num(bit0,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,pred_numeral(L)),numeral_numeral_nat(K))),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% take_bit_numeral_bit0
tff(fact_5346_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(numeral_numeral_nat(L)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(pred_numeral(L)),aa_num_Code_integer(numera6620942414471956472nteger,K))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ).

% take_bit_numeral_bit0
tff(fact_5347_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : ( bit_se569199155075624693atural(numeral_numeral_nat(L),numera5444537566228673987atural(aa_num_num(bit0,K))) = times_2397367101498566445atural(bit_se569199155075624693atural(pred_numeral(L),numera5444537566228673987atural(K)),numera5444537566228673987atural(aa_num_num(bit0,one))) ) ).

% take_bit_numeral_bit0
tff(fact_5348_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,numeral_numeral_nat(L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,pred_numeral(L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ).

% take_bit_numeral_bit0
tff(fact_5349_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ).

% take_bit_Suc_minus_bit0
tff(fact_5350_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,pred_numeral(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ).

% take_bit_numeral_minus_bit0
tff(fact_5351_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),numeral_numeral_nat(aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),numeral_numeral_nat(K))),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_nat) ) ).

% take_bit_Suc_bit1
tff(fact_5352_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),aa_num_Code_integer(numera6620942414471956472nteger,K))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) ) ).

% take_bit_Suc_bit1
tff(fact_5353_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : ( bit_se569199155075624693atural(aa_nat_nat(suc,N),numera5444537566228673987atural(aa_num_num(bit1,K))) = plus_p4538020629002901425atural(times_2397367101498566445atural(bit_se569199155075624693atural(N,numera5444537566228673987atural(K)),numera5444537566228673987atural(aa_num_num(bit0,one))),one_one_Code_natural) ) ).

% take_bit_Suc_bit1
tff(fact_5354_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ).

% take_bit_Suc_bit1
tff(fact_5355_take__bit__Suc,axiom,
    ! [N: nat,A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))))),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))) ) ).

% take_bit_Suc
tff(fact_5356_take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ).

% take_bit_Suc
tff(fact_5357_take__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : ( bit_se569199155075624693atural(aa_nat_nat(suc,N),A) = plus_p4538020629002901425atural(times_2397367101498566445atural(bit_se569199155075624693atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))),numera5444537566228673987atural(aa_num_num(bit0,one))),modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))) ) ).

% take_bit_Suc
tff(fact_5358_take__bit__Suc,axiom,
    ! [N: nat,A: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),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_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) ).

% take_bit_Suc
tff(fact_5359_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,numeral_numeral_nat(L)),numeral_numeral_nat(aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,pred_numeral(L)),numeral_numeral_nat(K))),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_nat) ) ).

% take_bit_numeral_bit1
tff(fact_5360_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(numeral_numeral_nat(L)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(pred_numeral(L)),aa_num_Code_integer(numera6620942414471956472nteger,K))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) ) ).

% take_bit_numeral_bit1
tff(fact_5361_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : ( bit_se569199155075624693atural(numeral_numeral_nat(L),numera5444537566228673987atural(aa_num_num(bit1,K))) = plus_p4538020629002901425atural(times_2397367101498566445atural(bit_se569199155075624693atural(pred_numeral(L),numera5444537566228673987atural(K)),numera5444537566228673987atural(aa_num_num(bit0,one))),one_one_Code_natural) ) ).

% take_bit_numeral_bit1
tff(fact_5362_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,numeral_numeral_nat(L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,pred_numeral(L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ).

% take_bit_numeral_bit1
tff(fact_5363_divmod__nat__def,axiom,
    ! [M: nat,N: nat] : ( aa_nat7603221163087113075at_nat(divmod_nat(M),N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ) ).

% divmod_nat_def
tff(fact_5364_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ).

% take_bit_Suc_minus_bit1
tff(fact_5365_take__bit__rec,axiom,
    ! [N: nat,A: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = $ite(N = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))))),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ).

% take_bit_rec
tff(fact_5366_take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),A) = $ite(N = zero_zero_nat,zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% take_bit_rec
tff(fact_5367_take__bit__rec,axiom,
    ! [N: nat,A: code_natural] :
      ( bit_se569199155075624693atural(N,A) = $ite(N = zero_zero_nat,zero_z2226904508553997617atural,plus_p4538020629002901425atural(times_2397367101498566445atural(bit_se569199155075624693atural(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))),numera5444537566228673987atural(aa_num_num(bit0,one))),modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))))) ) ).

% take_bit_rec
tff(fact_5368_take__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = $ite(N = zero_zero_nat,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,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) ).

% take_bit_rec
tff(fact_5369_mask__numeral,axiom,
    ! [N: num] : ( aa_nat_Code_integer(bit_se2119862282449309892nteger,numeral_numeral_nat(N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,pred_numeral(N)))) ) ).

% mask_numeral
tff(fact_5370_mask__numeral,axiom,
    ! [N: num] : ( aa_nat_Code_natural(bit_se943457434206027407atural,numeral_numeral_nat(N)) = plus_p4538020629002901425atural(one_one_Code_natural,times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),aa_nat_Code_natural(bit_se943457434206027407atural,pred_numeral(N)))) ) ).

% mask_numeral
tff(fact_5371_mask__numeral,axiom,
    ! [N: num] : ( aa_nat_int(bit_se2000444600071755411sk_int,numeral_numeral_nat(N)) = 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_nat_int(bit_se2000444600071755411sk_int,pred_numeral(N)))) ) ).

% mask_numeral
tff(fact_5372_mask__numeral,axiom,
    ! [N: num] : ( aa_nat_nat(bit_se2002935070580805687sk_nat,numeral_numeral_nat(N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,pred_numeral(N)))) ) ).

% mask_numeral
tff(fact_5373_and__int__unfold,axiom,
    ! [K: int,L: int] :
      ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
          ( ( K = zero_zero_int )
          | ( L = zero_zero_int ) ),
          zero_zero_int,
          $ite(
            K = aa_int_int(uminus_uminus_int,one_one_int),
            L,
            $ite(L = aa_int_int(uminus_uminus_int,one_one_int),K,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ).

% and_int_unfold
tff(fact_5374_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_5375_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5376_normalize__negative,axiom,
    ! [Q3: int,P3: 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,P3),Q3)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,P3)),aa_int_int(uminus_uminus_int,Q3))) ) ) ).

% normalize_negative
tff(fact_5377_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_5378_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_5379_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_5380_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_5381_odd__two__times__div__two__succ,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)
     => ( plus_p4538020629002901425atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))),one_one_Code_natural) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_5382_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),aa_int_int(uminus_uminus_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y) ) ).

% dvd_minus_iff
tff(fact_5383_dvd__minus__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Y) ) ).

% dvd_minus_iff
tff(fact_5384_dvd__minus__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),aa_rat_rat(uminus_uminus_rat,Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),Y) ) ).

% dvd_minus_iff
tff(fact_5385_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,X)),Y)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y) ) ).

% minus_dvd_iff
tff(fact_5386_minus__dvd__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),Y)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Y) ) ).

% minus_dvd_iff
tff(fact_5387_minus__dvd__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(uminus_uminus_rat,X)),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),Y) ) ).

% minus_dvd_iff
tff(fact_5388_power__one,axiom,
    ! [N: nat] : ( aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),N) = one_one_Code_integer ) ).

% power_one
tff(fact_5389_power__one,axiom,
    ! [N: nat] : ( aa_nat_assn(power_power_assn(one_one_assn),N) = one_one_assn ) ).

% power_one
tff(fact_5390_power__one,axiom,
    ! [N: nat] : ( aa_nat_rat(power_power_rat(one_one_rat),N) = one_one_rat ) ).

% power_one
tff(fact_5391_power__one,axiom,
    ! [N: nat] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,one_one_int),N) = one_one_int ) ).

% power_one
tff(fact_5392_power__one,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),N) = one_one_nat ) ).

% power_one
tff(fact_5393_power__one__right,axiom,
    ! [A: int] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),one_one_nat) = A ) ).

% power_one_right
tff(fact_5394_power__one__right,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),one_one_nat) = A ) ).

% power_one_right
tff(fact_5395_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),one_one_nat)
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_5396_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_5397_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_5398_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_5399_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_5400_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_5401_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_5402_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5403_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5404_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5405_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5406_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5407_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5408_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5409_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5410_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5411_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5412_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5413_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5414_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5415_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5416_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_5417_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ).

% unit_prod
tff(fact_5418_unit__prod,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ).

% unit_prod
tff(fact_5419_power__inject__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),M) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_5420_power__inject__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( ( aa_nat_rat(power_power_rat(A),M) = aa_nat_rat(power_power_rat(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_5421_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_5422_power__inject__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),M) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_5423_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_5424_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_5425_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_5426_dvd__div__mult__self,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),B)
     => ( times_2397367101498566445atural(divide5121882707175180666atural(B,A),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_5427_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_5428_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_5429_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_5430_dvd__mult__div__cancel,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),B)
     => ( times_2397367101498566445atural(A,divide5121882707175180666atural(B,A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_5431_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% unit_div
tff(fact_5432_unit__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% unit_div
tff(fact_5433_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),one_one_Code_integer) ) ) ).

% unit_div
tff(fact_5434_unit__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
       => aa_Code_natural_o(dvd_dvd_Code_natural(divide5121882707175180666atural(A,B)),one_one_Code_natural) ) ) ).

% unit_div
tff(fact_5435_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_5436_unit__div__1__unit,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_5437_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A)),one_one_Code_integer) ) ).

% unit_div_1_unit
tff(fact_5438_unit__div__1__unit,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => aa_Code_natural_o(dvd_dvd_Code_natural(divide5121882707175180666atural(one_one_Code_natural,A)),one_one_Code_natural) ) ).

% unit_div_1_unit
tff(fact_5439_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_5440_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_5441_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_5442_unit__div__1__div__1,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ( divide5121882707175180666atural(one_one_Code_natural,divide5121882707175180666atural(one_one_Code_natural,A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_5443_and_Oleft__neutral,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),A) = A ) ).

% and.left_neutral
tff(fact_5444_and_Oleft__neutral,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,one_one_int)),A) = A ) ).

% and.left_neutral
tff(fact_5445_and_Oright__neutral,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = A ) ).

% and.right_neutral
tff(fact_5446_and_Oright__neutral,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = A ) ).

% and.right_neutral
tff(fact_5447_bit_Oconj__one__right,axiom,
    ! [X: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = X ) ).

% bit.conj_one_right
tff(fact_5448_bit_Oconj__one__right,axiom,
    ! [X: int] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),aa_int_int(uminus_uminus_int,one_one_int)) = X ) ).

% bit.conj_one_right
tff(fact_5449_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).

% unit_mult_div_div
tff(fact_5450_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% unit_mult_div_div
tff(fact_5451_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A) ) ) ).

% unit_mult_div_div
tff(fact_5452_unit__mult__div__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ( times_2397367101498566445atural(B,divide5121882707175180666atural(one_one_Code_natural,A)) = divide5121882707175180666atural(B,A) ) ) ).

% unit_mult_div_div
tff(fact_5453_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_5454_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_5455_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_5456_unit__div__mult__self,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ( times_2397367101498566445atural(divide5121882707175180666atural(B,A),A) = B ) ) ).

% unit_div_mult_self
tff(fact_5457_power__strict__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),X)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_5458_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_5459_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_5460_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_5461_minus__one__mult__self,axiom,
    ! [N: nat] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N)) = one_one_int ) ).

% minus_one_mult_self
tff(fact_5462_minus__one__mult__self,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)) = one_one_Code_integer ) ).

% minus_one_mult_self
tff(fact_5463_minus__one__mult__self,axiom,
    ! [N: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)) = one_one_rat ) ).

% minus_one_mult_self
tff(fact_5464_left__minus__one__mult__self,axiom,
    ! [N: nat,A: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N)),A)) = A ) ).

% left_minus_one_mult_self
tff(fact_5465_left__minus__one__mult__self,axiom,
    ! [N: nat,A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),A)) = A ) ).

% left_minus_one_mult_self
tff(fact_5466_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),A)) = A ) ).

% left_minus_one_mult_self
tff(fact_5467_and__numerals_I8_J,axiom,
    ! [X: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,X))),one_one_Code_integer) = one_one_Code_integer ) ).

% and_numerals(8)
tff(fact_5468_and__numerals_I8_J,axiom,
    ! [X: num] : ( bit_se2773287842338716102atural(numera5444537566228673987atural(aa_num_num(bit1,X)),one_one_Code_natural) = one_one_Code_natural ) ).

% and_numerals(8)
tff(fact_5469_and__numerals_I8_J,axiom,
    ! [X: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),one_one_int) = one_one_int ) ).

% and_numerals(8)
tff(fact_5470_and__numerals_I8_J,axiom,
    ! [X: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit1,X))),one_one_nat) = one_one_nat ) ).

% and_numerals(8)
tff(fact_5471_and__numerals_I2_J,axiom,
    ! [Y: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y))) = one_one_Code_integer ) ).

% and_numerals(2)
tff(fact_5472_and__numerals_I2_J,axiom,
    ! [Y: num] : ( bit_se2773287842338716102atural(one_one_Code_natural,numera5444537566228673987atural(aa_num_num(bit1,Y))) = one_one_Code_natural ) ).

% and_numerals(2)
tff(fact_5473_and__numerals_I2_J,axiom,
    ! [Y: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = one_one_int ) ).

% and_numerals(2)
tff(fact_5474_and__numerals_I2_J,axiom,
    ! [Y: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit1,Y))) = one_one_nat ) ).

% and_numerals(2)
tff(fact_5475_mask__Suc__0,axiom,
    aa_nat_Code_integer(bit_se2119862282449309892nteger,aa_nat_nat(suc,zero_zero_nat)) = one_one_Code_integer ).

% mask_Suc_0
tff(fact_5476_mask__Suc__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% mask_Suc_0
tff(fact_5477_mask__Suc__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% mask_Suc_0
tff(fact_5478_power__add__numeral,axiom,
    ! [A: code_integer,M: num,N: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(M))),aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(N))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(plus_plus_num(M,N))) ) ).

% power_add_numeral
tff(fact_5479_power__add__numeral,axiom,
    ! [A: assn,M: num,N: num] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(M))),aa_nat_assn(power_power_assn(A),numeral_numeral_nat(N))) = aa_nat_assn(power_power_assn(A),numeral_numeral_nat(plus_plus_num(M,N))) ) ).

% power_add_numeral
tff(fact_5480_power__add__numeral,axiom,
    ! [A: rat,M: num,N: num] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),numeral_numeral_nat(M))),aa_nat_rat(power_power_rat(A),numeral_numeral_nat(N))) = aa_nat_rat(power_power_rat(A),numeral_numeral_nat(plus_plus_num(M,N))) ) ).

% power_add_numeral
tff(fact_5481_power__add__numeral,axiom,
    ! [A: nat,M: num,N: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(plus_plus_num(M,N))) ) ).

% power_add_numeral
tff(fact_5482_power__add__numeral,axiom,
    ! [A: int,M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),numeral_numeral_nat(M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),numeral_numeral_nat(N))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),numeral_numeral_nat(plus_plus_num(M,N))) ) ).

% power_add_numeral
tff(fact_5483_power__add__numeral2,axiom,
    ! [A: code_integer,M: num,N: num,B: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(M))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(N))),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(plus_plus_num(M,N)))),B) ) ).

% power_add_numeral2
tff(fact_5484_power__add__numeral2,axiom,
    ! [A: assn,M: num,N: num,B: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(M))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(N))),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(plus_plus_num(M,N)))),B) ) ).

% power_add_numeral2
tff(fact_5485_power__add__numeral2,axiom,
    ! [A: rat,M: num,N: num,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),numeral_numeral_nat(M))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),numeral_numeral_nat(N))),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),numeral_numeral_nat(plus_plus_num(M,N)))),B) ) ).

% power_add_numeral2
tff(fact_5486_power__add__numeral2,axiom,
    ! [A: nat,M: num,N: num,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(M))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(N))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(plus_plus_num(M,N)))),B) ) ).

% power_add_numeral2
tff(fact_5487_power__add__numeral2,axiom,
    ! [A: int,M: num,N: num,B: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),numeral_numeral_nat(M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),numeral_numeral_nat(N))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),numeral_numeral_nat(plus_plus_num(M,N)))),B) ) ).

% power_add_numeral2
tff(fact_5488_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_nat_Code_integer(bit_se2119862282449309892nteger,N) ) ).

% take_bit_minus_one_eq_mask
tff(fact_5489_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_nat_int(bit_se2000444600071755411sk_int,N) ) ).

% take_bit_minus_one_eq_mask
tff(fact_5490_normalize__denom__zero,axiom,
    ! [P3: int] : ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P3),zero_zero_int)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ) ).

% normalize_denom_zero
tff(fact_5491_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_5492_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_5493_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_5494_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N: 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(aa_int_fun_nat_int(power_power_int,B),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_5495_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_5496_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_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_mult_iff
tff(fact_5497_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_5498_even__mult__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),times_2397367101498566445atural(A,B))
    <=> ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)
        | aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_5499_power__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),X)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_5500_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_5501_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_5502_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_5503_and__numerals_I5_J,axiom,
    ! [X: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,X))),one_one_Code_integer) = zero_z3403309356797280102nteger ) ).

% and_numerals(5)
tff(fact_5504_and__numerals_I5_J,axiom,
    ! [X: num] : ( bit_se2773287842338716102atural(numera5444537566228673987atural(aa_num_num(bit0,X)),one_one_Code_natural) = zero_z2226904508553997617atural ) ).

% and_numerals(5)
tff(fact_5505_and__numerals_I5_J,axiom,
    ! [X: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),one_one_int) = zero_zero_int ) ).

% and_numerals(5)
tff(fact_5506_and__numerals_I5_J,axiom,
    ! [X: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit0,X))),one_one_nat) = zero_zero_nat ) ).

% and_numerals(5)
tff(fact_5507_and__numerals_I1_J,axiom,
    ! [Y: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y))) = zero_z3403309356797280102nteger ) ).

% and_numerals(1)
tff(fact_5508_and__numerals_I1_J,axiom,
    ! [Y: num] : ( bit_se2773287842338716102atural(one_one_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,Y))) = zero_z2226904508553997617atural ) ).

% and_numerals(1)
tff(fact_5509_and__numerals_I1_J,axiom,
    ! [Y: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = zero_zero_int ) ).

% and_numerals(1)
tff(fact_5510_and__numerals_I1_J,axiom,
    ! [Y: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,Y))) = zero_zero_nat ) ).

% and_numerals(1)
tff(fact_5511_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y))) ) ).

% and_numerals(3)
tff(fact_5512_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] : ( bit_se2773287842338716102atural(numera5444537566228673987atural(aa_num_num(bit0,X)),numera5444537566228673987atural(aa_num_num(bit0,Y))) = times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),bit_se2773287842338716102atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y))) ) ).

% and_numerals(3)
tff(fact_5513_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = 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,X)),aa_num_int(numeral_numeral_int,Y))) ) ).

% and_numerals(3)
tff(fact_5514_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit0,X))),numeral_numeral_nat(aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ) ).

% and_numerals(3)
tff(fact_5515_power2__minus,axiom,
    ! [A: int] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% power2_minus
tff(fact_5516_power2__minus,axiom,
    ! [A: code_integer] : ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% power2_minus
tff(fact_5517_power2__minus,axiom,
    ! [A: rat] : ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% power2_minus
tff(fact_5518_and__minus__numerals_I6_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = one_one_int ) ).

% and_minus_numerals(6)
tff(fact_5519_and__minus__numerals_I2_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = one_one_int ) ).

% and_minus_numerals(2)
tff(fact_5520_power__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ) ) ).

% power_decreasing_iff
tff(fact_5521_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ) ) ).

% power_decreasing_iff
tff(fact_5522_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ) ) ).

% power_decreasing_iff
tff(fact_5523_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N: 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(aa_int_fun_nat_int(power_power_int,B),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ) ) ).

% power_decreasing_iff
tff(fact_5524_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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,numeral_numeral_nat(aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_5525_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_5526_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_5527_even__plus__one__iff,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),plus_p4538020629002901425atural(A,one_one_Code_natural))
    <=> ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_5528_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y))) ) ).

% and_numerals(4)
tff(fact_5529_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] : ( bit_se2773287842338716102atural(numera5444537566228673987atural(aa_num_num(bit0,X)),numera5444537566228673987atural(aa_num_num(bit1,Y))) = times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),bit_se2773287842338716102atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y))) ) ).

% and_numerals(4)
tff(fact_5530_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = 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,X)),aa_num_int(numeral_numeral_int,Y))) ) ).

% and_numerals(4)
tff(fact_5531_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit0,X))),numeral_numeral_nat(aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ) ).

% and_numerals(4)
tff(fact_5532_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y))) ) ).

% and_numerals(6)
tff(fact_5533_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] : ( bit_se2773287842338716102atural(numera5444537566228673987atural(aa_num_num(bit1,X)),numera5444537566228673987atural(aa_num_num(bit0,Y))) = times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),bit_se2773287842338716102atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y))) ) ).

% and_numerals(6)
tff(fact_5534_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = 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,X)),aa_num_int(numeral_numeral_int,Y))) ) ).

% and_numerals(6)
tff(fact_5535_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit1,X))),numeral_numeral_nat(aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ) ).

% and_numerals(6)
tff(fact_5536_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_5537_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,N: nat] : ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_5538_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,N: nat] : ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_5539_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),N) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5540_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5541_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5542_power__minus__odd,axiom,
    ! [N: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ) ).

% power_minus_odd
tff(fact_5543_power__minus__odd,axiom,
    ! [N: nat,A: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).

% power_minus_odd
tff(fact_5544_power__minus__odd,axiom,
    ! [N: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),N)) ) ) ).

% power_minus_odd
tff(fact_5545_and__minus__numerals_I5_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = zero_zero_int ) ).

% and_minus_numerals(5)
tff(fact_5546_and__minus__numerals_I1_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = zero_zero_int ) ).

% and_minus_numerals(1)
tff(fact_5547_even__succ__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_5548_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_5549_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_5550_even__succ__div__two,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)
     => ( divide5121882707175180666atural(plus_p4538020629002901425atural(A,one_one_Code_natural),numera5444537566228673987atural(aa_num_num(bit0,one))) = divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_5551_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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)),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),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_5552_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_5553_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_5554_odd__succ__div__two,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)
     => ( divide5121882707175180666atural(plus_p4538020629002901425atural(A,one_one_Code_natural),numera5444537566228673987atural(aa_num_num(bit0,one))) = plus_p4538020629002901425atural(divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))),one_one_Code_natural) ) ) ).

% odd_succ_div_two
tff(fact_5555_even__succ__div__2,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_5556_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_5557_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_5558_even__succ__div__2,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)
     => ( divide5121882707175180666atural(plus_p4538020629002901425atural(one_one_Code_natural,A),numera5444537566228673987atural(aa_num_num(bit0,one))) = divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_5559_power__minus1__even,axiom,
    ! [N: nat] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = one_one_int ) ).

% power_minus1_even
tff(fact_5560_power__minus1__even,axiom,
    ! [N: nat] : ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = one_one_Code_integer ) ).

% power_minus1_even
tff(fact_5561_power__minus1__even,axiom,
    ! [N: nat] : ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = one_one_rat ) ).

% power_minus1_even
tff(fact_5562_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_5563_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_5564_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_5565_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_5566_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_5567_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_5568_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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,numeral_numeral_nat(aa_num_num(bit0,one))),N)),one_one_nat))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5569_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: 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(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5570_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)),one_one_Code_integer))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5571_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N),one_one_Code_natural))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5572_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y)))) ) ).

% and_numerals(7)
tff(fact_5573_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : ( bit_se2773287842338716102atural(numera5444537566228673987atural(aa_num_num(bit1,X)),numera5444537566228673987atural(aa_num_num(bit1,Y))) = plus_p4538020629002901425atural(one_one_Code_natural,times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),bit_se2773287842338716102atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y)))) ) ).

% and_numerals(7)
tff(fact_5574_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = 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,X)),aa_num_int(numeral_numeral_int,Y)))) ) ).

% and_numerals(7)
tff(fact_5575_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit1,X))),numeral_numeral_nat(aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ) ).

% and_numerals(7)
tff(fact_5576_even__succ__div__exp,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5577_even__succ__div__exp,axiom,
    ! [A: int,N: 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),N)
       => ( 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(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5578_even__succ__div__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5579_even__succ__div__exp,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( divide5121882707175180666atural(plus_p4538020629002901425atural(one_one_Code_natural,A),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = divide5121882707175180666atural(A,aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5580_even__succ__mod__exp,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5581_even__succ__mod__exp,axiom,
    ! [A: int,N: 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),N)
       => ( 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(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5582_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5583_even__succ__mod__exp,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( modulo8411746178871703098atural(plus_p4538020629002901425atural(one_one_Code_natural,A),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = plus_p4538020629002901425atural(one_one_Code_natural,modulo8411746178871703098atural(A,aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5584_is__unit__power__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5585_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5586_is__unit__power__iff,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5587_power__commuting__commutes,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_5588_power__commuting__commutes,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Y),X) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(X),N)),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Y),aa_nat_assn(power_power_assn(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_5589_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),N)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(power_power_rat(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_5590_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),X) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)) ) ) ).

% power_commuting_commutes
tff(fact_5591_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),X) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N)) ) ) ).

% power_commuting_commutes
tff(fact_5592_power__mult__distrib,axiom,
    ! [A: code_integer,B: code_integer,N: nat] : ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)) ) ).

% power_mult_distrib
tff(fact_5593_power__mult__distrib,axiom,
    ! [A: assn,B: assn,N: nat] : ( aa_nat_assn(power_power_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),N) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N)),aa_nat_assn(power_power_assn(B),N)) ) ).

% power_mult_distrib
tff(fact_5594_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N: nat] : ( aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)) ) ).

% power_mult_distrib
tff(fact_5595_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)) ) ).

% power_mult_distrib
tff(fact_5596_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N)) ) ).

% power_mult_distrib
tff(fact_5597_power__commutes,axiom,
    ! [A: code_integer,N: nat] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ).

% power_commutes
tff(fact_5598_power__commutes,axiom,
    ! [A: assn,N: nat] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N)),A) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(A),N)) ) ).

% power_commutes
tff(fact_5599_power__commutes,axiom,
    ! [A: rat,N: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N)),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)) ) ).

% power_commutes
tff(fact_5600_power__commutes,axiom,
    ! [A: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ).

% power_commutes
tff(fact_5601_power__commutes,axiom,
    ! [A: int,N: nat] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ).

% power_commutes
tff(fact_5602_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ~ ! [K2: code_integer] : ( A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K2) ) ) ).

% dvdE
tff(fact_5603_dvdE,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,B),A)
     => ~ ! [K2: assn] : ( A != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),K2) ) ) ).

% dvdE
tff(fact_5604_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ~ ! [K2: rat] : ( A != aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K2) ) ) ).

% dvdE
tff(fact_5605_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ~ ! [K2: nat] : ( A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ) ).

% dvdE
tff(fact_5606_dvdE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ~ ! [K2: int] : ( A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) ) ) ).

% dvdE
tff(fact_5607_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% dvdI
tff(fact_5608_dvdI,axiom,
    ! [A: assn,B: assn,K: assn] :
      ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),K) )
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,B),A) ) ).

% dvdI
tff(fact_5609_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A) ) ).

% dvdI
tff(fact_5610_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% dvdI
tff(fact_5611_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% dvdI
tff(fact_5612_dvd__def,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
    <=> ? [K4: code_integer] : ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K4) ) ) ).

% dvd_def
tff(fact_5613_dvd__def,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,B),A)
    <=> ? [K4: assn] : ( A = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),K4) ) ) ).

% dvd_def
tff(fact_5614_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
    <=> ? [K4: rat] : ( A = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K4) ) ) ).

% dvd_def
tff(fact_5615_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ? [K4: nat] : ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K4) ) ) ).

% dvd_def
tff(fact_5616_dvd__def,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
    <=> ? [K4: int] : ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K4) ) ) ).

% dvd_def
tff(fact_5617_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult
tff(fact_5618_dvd__mult,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),C)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ) ).

% dvd_mult
tff(fact_5619_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% dvd_mult
tff(fact_5620_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult
tff(fact_5621_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult
tff(fact_5622_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult2
tff(fact_5623_dvd__mult2,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),B)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ) ).

% dvd_mult2
tff(fact_5624_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% dvd_mult2
tff(fact_5625_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult2
tff(fact_5626_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult2
tff(fact_5627_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ).

% dvd_mult_left
tff(fact_5628_dvd__mult__left,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),C) ) ).

% dvd_mult_left
tff(fact_5629_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ).

% dvd_mult_left
tff(fact_5630_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).

% dvd_mult_left
tff(fact_5631_dvd__mult__left,axiom,
    ! [A: int,B: int,C: 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)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).

% dvd_mult_left
tff(fact_5632_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% dvd_triv_left
tff(fact_5633_dvd__triv__left,axiom,
    ! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)) ).

% dvd_triv_left
tff(fact_5634_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% dvd_triv_left
tff(fact_5635_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ).

% dvd_triv_left
tff(fact_5636_dvd__triv__left,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(times_times_int,A),B)) ).

% dvd_triv_left
tff(fact_5637_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_5638_mult__dvd__mono,axiom,
    ! [A: assn,B: assn,C: assn,D2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,C),D2)
       => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_5639_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: 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,C),D2)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_5640_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_5641_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_5642_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ).

% dvd_mult_right
tff(fact_5643_dvd__mult__right,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,B),C) ) ).

% dvd_mult_right
tff(fact_5644_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),C) ) ).

% dvd_mult_right
tff(fact_5645_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).

% dvd_mult_right
tff(fact_5646_dvd__mult__right,axiom,
    ! [A: int,B: int,C: 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)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).

% dvd_mult_right
tff(fact_5647_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) ).

% dvd_triv_right
tff(fact_5648_dvd__triv__right,axiom,
    ! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A)) ).

% dvd_triv_right
tff(fact_5649_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)) ).

% dvd_triv_right
tff(fact_5650_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) ).

% dvd_triv_right
tff(fact_5651_dvd__triv__right,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(times_times_int,B),A)) ).

% dvd_triv_right
tff(fact_5652_one__dvd,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),A) ).

% one_dvd
tff(fact_5653_one__dvd,axiom,
    ! [A: assn] : aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,one_one_assn),A) ).

% one_dvd
tff(fact_5654_one__dvd,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),A) ).

% one_dvd
tff(fact_5655_one__dvd,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).

% one_dvd
tff(fact_5656_one__dvd,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).

% one_dvd
tff(fact_5657_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% unit_imp_dvd
tff(fact_5658_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% unit_imp_dvd
tff(fact_5659_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% unit_imp_dvd
tff(fact_5660_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: 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,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer) ) ) ).

% dvd_unit_imp_unit
tff(fact_5661_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_5662_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_5663_uminus__dvd__conv_I1_J,axiom,
    ! [D2: int,T2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),T2)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,D2)),T2) ) ).

% uminus_dvd_conv(1)
tff(fact_5664_uminus__dvd__conv_I2_J,axiom,
    ! [D2: int,T2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),T2)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(uminus_uminus_int,T2)) ) ).

% uminus_dvd_conv(2)
tff(fact_5665_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,N: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),M)),aa_nat_Code_integer(power_8256067586552552935nteger(X),N))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ) ).

% dvd_power_iff
tff(fact_5666_dvd__power__iff,axiom,
    ! [X: int,M: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ) ).

% dvd_power_iff
tff(fact_5667_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N: nat] :
      ( ( X != 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,X),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ) ).

% dvd_power_iff
tff(fact_5668_dvd__power,axiom,
    ! [N: nat,X: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( X = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) ) ).

% dvd_power
tff(fact_5669_dvd__power,axiom,
    ! [N: nat,X: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( X = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),aa_nat_rat(power_power_rat(X),N)) ) ).

% dvd_power
tff(fact_5670_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( X = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N)) ) ).

% dvd_power
tff(fact_5671_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( X = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)) ) ).

% dvd_power
tff(fact_5672_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(aa_int_fun_int_o(aTP_Lamm_id,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_id,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_5673_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ie,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ie,B)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% subset_divisors_dvd
tff(fact_5674_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_if,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_if,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_5675_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_if,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_if,B)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5676_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,collect_int(aa_int_fun_int_o(aTP_Lamm_id,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_id,B)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5677_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ie,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ie,B)))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5678_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
        & ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_5679_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( ( A = aa_int_int(uminus_uminus_int,one_one_int) )
        & ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_5680_uminus__power__if,axiom,
    ! [A: int,N: nat] :
      ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N),aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N))) ) ).

% uminus_power_if
tff(fact_5681_uminus__power__if,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N),aa_nat_Code_integer(power_8256067586552552935nteger(A),N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).

% uminus_power_if
tff(fact_5682_uminus__power__if,axiom,
    ! [A: rat,N: nat] :
      ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N),aa_nat_rat(power_power_rat(A),N),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),N))) ) ).

% uminus_power_if
tff(fact_5683_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_5684_one__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ).

% one_le_power
tff(fact_5685_one__le__power,axiom,
    ! [A: rat,N: 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),N)) ) ).

% one_le_power
tff(fact_5686_one__le__power,axiom,
    ! [A: nat,N: 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),N)) ) ).

% one_le_power
tff(fact_5687_one__le__power,axiom,
    ! [A: int,N: 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(aa_int_fun_nat_int(power_power_int,A),N)) ) ).

% one_le_power
tff(fact_5688_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_5689_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_5690_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_5691_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y) = one_one_Code_integer )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N)) = one_one_Code_integer ) ) ).

% left_right_inverse_power
tff(fact_5692_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y) = one_one_assn )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(X),N)),aa_nat_assn(power_power_assn(Y),N)) = one_one_assn ) ) ).

% left_right_inverse_power
tff(fact_5693_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y) = one_one_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(Y),N)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_5694_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_5695_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),N)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_5696_power__Suc,axiom,
    ! [A: code_integer,N: nat] : ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ).

% power_Suc
tff(fact_5697_power__Suc,axiom,
    ! [A: assn,N: nat] : ( aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(A),N)) ) ).

% power_Suc
tff(fact_5698_power__Suc,axiom,
    ! [A: rat,N: nat] : ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)) ) ).

% power_Suc
tff(fact_5699_power__Suc,axiom,
    ! [A: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ).

% power_Suc
tff(fact_5700_power__Suc,axiom,
    ! [A: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ).

% power_Suc
tff(fact_5701_power__Suc2,axiom,
    ! [A: code_integer,N: nat] : ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),A) ) ).

% power_Suc2
tff(fact_5702_power__Suc2,axiom,
    ! [A: assn,N: nat] : ( aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N)),A) ) ).

% power_Suc2
tff(fact_5703_power__Suc2,axiom,
    ! [A: rat,N: nat] : ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N)),A) ) ).

% power_Suc2
tff(fact_5704_power__Suc2,axiom,
    ! [A: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) ) ).

% power_Suc2
tff(fact_5705_power__Suc2,axiom,
    ! [A: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),A) ) ).

% power_Suc2
tff(fact_5706_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer) ) ) ).

% is_unit_mult_iff
tff(fact_5707_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_5708_is__unit__mult__iff,axiom,
    ! [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,A),B)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_5709_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_5710_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_5711_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_5712_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_5713_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_5714_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_5715_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_5716_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_5717_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_5718_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_5719_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_5720_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_5721_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_5722_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_5723_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_5724_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_5725_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_5726_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_5727_power__0,axiom,
    ! [A: code_integer] : ( aa_nat_Code_integer(power_8256067586552552935nteger(A),zero_zero_nat) = one_one_Code_integer ) ).

% power_0
tff(fact_5728_power__0,axiom,
    ! [A: assn] : ( aa_nat_assn(power_power_assn(A),zero_zero_nat) = one_one_assn ) ).

% power_0
tff(fact_5729_power__0,axiom,
    ! [A: rat] : ( aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ) ).

% power_0
tff(fact_5730_power__0,axiom,
    ! [A: int] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),zero_zero_nat) = one_one_int ) ).

% power_0
tff(fact_5731_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_5732_power__one__over,axiom,
    ! [A: rat,N: nat] : ( aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A)),N) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N)) ) ).

% power_one_over
tff(fact_5733_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_5734_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_5735_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)),A) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_5736_dvd__div__mult,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(C),B)
     => ( times_2397367101498566445atural(divide5121882707175180666atural(B,C),A) = divide5121882707175180666atural(times_2397367101498566445atural(B,A),C) ) ) ).

% dvd_div_mult
tff(fact_5737_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_5738_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_5739_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_5740_div__mult__swap,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(C),B)
     => ( times_2397367101498566445atural(A,divide5121882707175180666atural(B,C)) = divide5121882707175180666atural(times_2397367101498566445atural(A,B),C) ) ) ).

% div_mult_swap
tff(fact_5741_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_5742_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_5743_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_5744_div__div__eq__right,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(C),B)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),A)
       => ( divide5121882707175180666atural(A,divide5121882707175180666atural(B,C)) = times_2397367101498566445atural(divide5121882707175180666atural(A,B),C) ) ) ) ).

% div_div_eq_right
tff(fact_5745_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_5746_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_5747_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_5748_dvd__div__mult2__eq,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(times_2397367101498566445atural(B,C)),A)
     => ( divide5121882707175180666atural(A,times_2397367101498566445atural(B,C)) = divide5121882707175180666atural(divide5121882707175180666atural(A,B),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_5749_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_5750_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),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),C)) ) ).

% dvd_mult_imp_div
tff(fact_5751_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) ) ).

% dvd_mult_imp_div
tff(fact_5752_dvd__mult__imp__div,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(times_2397367101498566445atural(A,C)),B)
     => aa_Code_natural_o(dvd_dvd_Code_natural(A),divide5121882707175180666atural(B,C)) ) ).

% dvd_mult_imp_div
tff(fact_5753_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),C)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5754_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),C)
       => ( aa_int_int(aa_int_fun_int_int(times_times_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,C),D2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5755_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),C)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),D2)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5756_div__mult__div__if__dvd,axiom,
    ! [B: code_natural,A: code_natural,D2: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),A)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(D2),C)
       => ( times_2397367101498566445atural(divide5121882707175180666atural(A,B),divide5121882707175180666atural(C,D2)) = divide5121882707175180666atural(times_2397367101498566445atural(A,C),times_2397367101498566445atural(B,D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5757_power__add,axiom,
    ! [A: code_integer,M: nat,N: nat] : ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ).

% power_add
tff(fact_5758_power__add,axiom,
    ! [A: assn,M: nat,N: nat] : ( aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),M)),aa_nat_assn(power_power_assn(A),N)) ) ).

% power_add
tff(fact_5759_power__add,axiom,
    ! [A: rat,M: nat,N: nat] : ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)) ) ).

% power_add
tff(fact_5760_power__add,axiom,
    ! [A: nat,M: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ).

% power_add
tff(fact_5761_power__add,axiom,
    ! [A: int,M: nat,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ).

% power_add
tff(fact_5762_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_5763_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_5764_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_5765_unit__div__cancel,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ( ( divide5121882707175180666atural(B,A) = divide5121882707175180666atural(C,A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_5766_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_5767_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_5768_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_5769_div__unit__dvd__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(divide5121882707175180666atural(A,B)),C)
      <=> aa_Code_natural_o(dvd_dvd_Code_natural(A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_5770_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_5771_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_5772_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_5773_dvd__div__unit__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(A),divide5121882707175180666atural(C,B))
      <=> aa_Code_natural_o(dvd_dvd_Code_natural(A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_5774_dvd__div__neg,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),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_div_neg
tff(fact_5775_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).

% dvd_div_neg
tff(fact_5776_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) ) ) ).

% dvd_div_neg
tff(fact_5777_dvd__neg__div,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,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_neg_div
tff(fact_5778_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).

% dvd_neg_div
tff(fact_5779_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) ) ) ).

% dvd_neg_div
tff(fact_5780_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% minus_one_power_iff
tff(fact_5781_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ) ).

% minus_one_power_iff
tff(fact_5782_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% minus_one_power_iff
tff(fact_5783_mask__int__def,axiom,
    ! [N: nat] : ( aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ) ).

% mask_int_def
tff(fact_5784_mask__eq__exp__minus__1,axiom,
    ! [N: nat] : ( aa_nat_Code_integer(bit_se2119862282449309892nteger,N) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)),one_one_Code_integer) ) ).

% mask_eq_exp_minus_1
tff(fact_5785_mask__eq__exp__minus__1,axiom,
    ! [N: nat] : ( aa_nat_Code_natural(bit_se943457434206027407atural,N) = minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N),one_one_Code_natural) ) ).

% mask_eq_exp_minus_1
tff(fact_5786_mask__eq__exp__minus__1,axiom,
    ! [N: nat] : ( aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ) ).

% mask_eq_exp_minus_1
tff(fact_5787_mask__eq__exp__minus__1,axiom,
    ! [N: nat] : ( aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),one_one_nat) ) ).

% mask_eq_exp_minus_1
tff(fact_5788_power__le__one,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),one_one_Code_integer) ) ) ).

% power_le_one
tff(fact_5789_power__le__one,axiom,
    ! [A: rat,N: 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),N)),one_one_rat) ) ) ).

% power_le_one
tff(fact_5790_power__le__one,axiom,
    ! [A: nat,N: 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),N)),one_one_nat) ) ) ).

% power_le_one
tff(fact_5791_power__le__one,axiom,
    ! [A: int,N: 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(aa_int_fun_nat_int(power_power_int,A),N)),one_one_int) ) ) ).

% power_le_one
tff(fact_5792_power__gt1__lemma,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).

% power_gt1_lemma
tff(fact_5793_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N))) ) ).

% power_gt1_lemma
tff(fact_5794_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% power_gt1_lemma
tff(fact_5795_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N))) ) ).

% power_gt1_lemma
tff(fact_5796_power__less__power__Suc,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).

% power_less_power_Suc
tff(fact_5797_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N))) ) ).

% power_less_power_Suc
tff(fact_5798_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% power_less_power_Suc
tff(fact_5799_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N))) ) ).

% power_less_power_Suc
tff(fact_5800_unity__coeff__ex,axiom,
    ! [P: fun_Code_integer_o,L: code_integer] :
      ( ? [X4: code_integer] : aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X4))
    <=> ? [X4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X4),zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_5801_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X4: rat] : aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,L),X4))
    <=> ? [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),zero_zero_rat))
          & aa_rat_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_5802_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X4: nat] : aa_nat_o(P,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(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_5803_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X4: int] : aa_int_o(P,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(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_5804_power__0__left,axiom,
    ! [N: nat] :
      ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = $ite(N = zero_zero_nat,one_one_Code_integer,zero_z3403309356797280102nteger) ) ).

% power_0_left
tff(fact_5805_power__0__left,axiom,
    ! [N: nat] :
      ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = $ite(N = zero_zero_nat,one_one_rat,zero_zero_rat) ) ).

% power_0_left
tff(fact_5806_power__0__left,axiom,
    ! [N: nat] :
      ( aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),N) = $ite(N = zero_zero_nat,one_one_int,zero_zero_int) ) ).

% power_0_left
tff(fact_5807_power__0__left,axiom,
    ! [N: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = $ite(N = zero_zero_nat,one_one_nat,zero_zero_nat) ) ).

% power_0_left
tff(fact_5808_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] : ( B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C3) ) ) ) ).

% unit_dvdE
tff(fact_5809_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] : ( B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C3) ) ) ) ).

% unit_dvdE
tff(fact_5810_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] : ( B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C3) ) ) ) ).

% unit_dvdE
tff(fact_5811_power__gt1,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_5812_power__gt1,axiom,
    ! [A: rat,N: 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,N))) ) ).

% power_gt1
tff(fact_5813_power__gt1,axiom,
    ! [A: nat,N: 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,N))) ) ).

% power_gt1
tff(fact_5814_power__gt1,axiom,
    ! [A: int,N: 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(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_5815_and_Ocomm__monoid__axioms,axiom,
    aa_Code_integer_o(comm_m1654649897431166824nteger(bit_se3949692690581998587nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% and.comm_monoid_axioms
tff(fact_5816_and_Ocomm__monoid__axioms,axiom,
    aa_int_o(comm_monoid_int(bit_se725231765392027082nd_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% and.comm_monoid_axioms
tff(fact_5817_power__less__imp__less__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% power_less_imp_less_exp
tff(fact_5818_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% power_less_imp_less_exp
tff(fact_5819_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% power_less_imp_less_exp
tff(fact_5820_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% power_less_imp_less_exp
tff(fact_5821_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N4)) ) ) ).

% power_strict_increasing
tff(fact_5822_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N4)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N4)) ) ) ).

% power_strict_increasing
tff(fact_5823_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).

% power_strict_increasing
tff(fact_5824_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N4)) ) ) ).

% power_strict_increasing
tff(fact_5825_power__increasing,axiom,
    ! [N: nat,N4: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N4)) ) ) ).

% power_increasing
tff(fact_5826_power__increasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),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),N)),aa_nat_rat(power_power_rat(A),N4)) ) ) ).

% power_increasing
tff(fact_5827_power__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),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),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).

% power_increasing
tff(fact_5828_power__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),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(aa_int_fun_nat_int(power_power_int,A),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N4)) ) ) ).

% power_increasing
tff(fact_5829_power__minus,axiom,
    ! [A: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ).

% power_minus
tff(fact_5830_power__minus,axiom,
    ! [A: code_integer,N: nat] : ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ).

% power_minus
tff(fact_5831_power__minus,axiom,
    ! [A: rat,N: nat] : ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(A),N)) ) ).

% power_minus
tff(fact_5832_inf__period_I4_J,axiom,
    ! [D2: code_integer,D3: code_integer,T2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),D3)
     => ! [X3: code_integer,K3: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),T2))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3))),T2)) ) ) ).

% inf_period(4)
tff(fact_5833_inf__period_I4_J,axiom,
    ! [D2: rat,D3: rat,T2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),D3)
     => ! [X3: rat,K3: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),T2))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3))),T2)) ) ) ).

% inf_period(4)
tff(fact_5834_inf__period_I4_J,axiom,
    ! [D2: int,D3: int,T2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D3)
     => ! [X3: int,K3: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T2))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3))),T2)) ) ) ).

% inf_period(4)
tff(fact_5835_inf__period_I3_J,axiom,
    ! [D2: code_integer,D3: code_integer,T2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),D3)
     => ! [X3: code_integer,K3: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),T2))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3))),T2)) ) ) ).

% inf_period(3)
tff(fact_5836_inf__period_I3_J,axiom,
    ! [D2: rat,D3: rat,T2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),D3)
     => ! [X3: rat,K3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),T2))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3))),T2)) ) ) ).

% inf_period(3)
tff(fact_5837_inf__period_I3_J,axiom,
    ! [D2: int,D3: int,T2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D3)
     => ! [X3: int,K3: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T2))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3))),T2)) ) ) ).

% inf_period(3)
tff(fact_5838_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D2)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D2),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5839_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D2)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D2),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5840_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D2)
           => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(D2),C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5841_dvd__div__div__eq__mult,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural,D2: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( C != zero_z2226904508553997617atural )
       => ( aa_Code_natural_o(dvd_dvd_Code_natural(A),B)
         => ( aa_Code_natural_o(dvd_dvd_Code_natural(C),D2)
           => ( ( divide5121882707175180666atural(B,A) = divide5121882707175180666atural(D2,C) )
            <=> ( times_2397367101498566445atural(B,C) = times_2397367101498566445atural(A,D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5842_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5843_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5844_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5845_dvd__div__iff__mult,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(C),B)
       => ( aa_Code_natural_o(dvd_dvd_Code_natural(A),divide5121882707175180666atural(B,C))
        <=> aa_Code_natural_o(dvd_dvd_Code_natural(times_2397367101498566445atural(A,C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5846_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5847_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5848_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5849_div__dvd__iff__mult,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),A)
       => ( aa_Code_natural_o(dvd_dvd_Code_natural(divide5121882707175180666atural(A,B)),C)
        <=> aa_Code_natural_o(dvd_dvd_Code_natural(A),times_2397367101498566445atural(C,B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5850_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5851_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = C )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5852_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
       => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A) = C )
        <=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5853_dvd__div__eq__mult,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(A),B)
       => ( ( divide5121882707175180666atural(B,A) = C )
        <=> ( B = times_2397367101498566445atural(C,A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5854_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_5855_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_5856_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)
     => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_5857_unit__div__eq__0__iff,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( ( divide5121882707175180666atural(A,B) = zero_z2226904508553997617atural )
      <=> ( A = zero_z2226904508553997617atural ) ) ) ).

% unit_div_eq_0_iff
tff(fact_5858_and_Omonoid__axioms,axiom,
    aa_Code_integer_o(monoid_Code_integer(bit_se3949692690581998587nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% and.monoid_axioms
tff(fact_5859_and_Omonoid__axioms,axiom,
    aa_int_o(monoid_int(bit_se725231765392027082nd_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% and.monoid_axioms
tff(fact_5860_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_5861_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = C )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_5862_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) = C )
      <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_5863_unit__eq__div1,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( ( divide5121882707175180666atural(A,B) = C )
      <=> ( A = times_2397367101498566445atural(C,B) ) ) ) ).

% unit_eq_div1
tff(fact_5864_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_5865_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( A = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_5866_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( A = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),B) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_5867_unit__eq__div2,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( ( A = divide5121882707175180666atural(C,B) )
      <=> ( times_2397367101498566445atural(A,B) = C ) ) ) ).

% unit_eq_div2
tff(fact_5868_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_5869_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_5870_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_5871_div__mult__unit2,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(C),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),A)
       => ( divide5121882707175180666atural(A,times_2397367101498566445atural(B,C)) = divide5121882707175180666atural(divide5121882707175180666atural(A,B),C) ) ) ) ).

% div_mult_unit2
tff(fact_5872_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_5873_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_5874_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_5875_unit__div__commute,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( times_2397367101498566445atural(divide5121882707175180666atural(A,B),C) = divide5121882707175180666atural(times_2397367101498566445atural(A,C),B) ) ) ).

% unit_div_commute
tff(fact_5876_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_5877_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_5878_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_5879_unit__div__mult__swap,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(C),one_one_Code_natural)
     => ( times_2397367101498566445atural(A,divide5121882707175180666atural(B,C)) = divide5121882707175180666atural(times_2397367101498566445atural(A,B),C) ) ) ).

% unit_div_mult_swap
tff(fact_5880_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5881_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5882_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5883_is__unit__div__mult2__eq,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(C),one_one_Code_natural)
       => ( divide5121882707175180666atural(A,times_2397367101498566445atural(B,C)) = divide5121882707175180666atural(divide5121882707175180666atural(A,B),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5884_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_5885_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_5886_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)
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_5887_unit__imp__mod__eq__0,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( modulo8411746178871703098atural(A,B) = zero_z2226904508553997617atural ) ) ).

% unit_imp_mod_eq_0
tff(fact_5888_power__minus__Bit0,axiom,
    ! [X: int,K: num] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,X)),numeral_numeral_nat(aa_num_num(bit0,K))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,X),numeral_numeral_nat(aa_num_num(bit0,K))) ) ).

% power_minus_Bit0
tff(fact_5889_power__minus__Bit0,axiom,
    ! [X: code_integer,K: num] : ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),numeral_numeral_nat(aa_num_num(bit0,K))) = aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,K))) ) ).

% power_minus_Bit0
tff(fact_5890_power__minus__Bit0,axiom,
    ! [X: rat,K: num] : ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,X)),numeral_numeral_nat(aa_num_num(bit0,K))) = aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,K))) ) ).

% power_minus_Bit0
tff(fact_5891_and_Osemilattice__neutr__axioms,axiom,
    aa_Code_integer_o(semila6106414119144251899nteger(bit_se3949692690581998587nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% and.semilattice_neutr_axioms
tff(fact_5892_and_Osemilattice__neutr__axioms,axiom,
    aa_int_o(semila9079005292280841162tr_int(bit_se725231765392027082nd_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% and.semilattice_neutr_axioms
tff(fact_5893_power__minus__Bit1,axiom,
    ! [X: int,K: num] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,X)),numeral_numeral_nat(aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),numeral_numeral_nat(aa_num_num(bit1,K)))) ) ).

% power_minus_Bit1
tff(fact_5894_power__minus__Bit1,axiom,
    ! [X: code_integer,K: num] : ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),numeral_numeral_nat(aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit1,K)))) ) ).

% power_minus_Bit1
tff(fact_5895_power__minus__Bit1,axiom,
    ! [X: rat,K: num] : ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,X)),numeral_numeral_nat(aa_num_num(bit1,K))) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit1,K)))) ) ).

% power_minus_Bit1
tff(fact_5896_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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,numeral_numeral_nat(aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% even_mask_div_iff'
tff(fact_5897_even__mask__div__iff_H,axiom,
    ! [M: nat,N: 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(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% even_mask_div_iff'
tff(fact_5898_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),one_one_Code_integer)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% even_mask_div_iff'
tff(fact_5899_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),divide5121882707175180666atural(minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M),one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% even_mask_div_iff'
tff(fact_5900_power__numeral__even,axiom,
    ! [Z: code_integer,W: num] :
      ( aa_nat_Code_integer(power_8256067586552552935nteger(Z),numeral_numeral_nat(aa_num_num(bit0,W))) = $let(
          w: code_integer,
          w:= aa_nat_Code_integer(power_8256067586552552935nteger(Z),numeral_numeral_nat(W)),
          aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,w),w) ) ) ).

% power_numeral_even
tff(fact_5901_power__numeral__even,axiom,
    ! [Z: assn,W: num] :
      ( aa_nat_assn(power_power_assn(Z),numeral_numeral_nat(aa_num_num(bit0,W))) = $let(
          w: assn,
          w:= aa_nat_assn(power_power_assn(Z),numeral_numeral_nat(W)),
          aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,w),w) ) ) ).

% power_numeral_even
tff(fact_5902_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      ( aa_nat_rat(power_power_rat(Z),numeral_numeral_nat(aa_num_num(bit0,W))) = $let(
          w: rat,
          w:= aa_nat_rat(power_power_rat(Z),numeral_numeral_nat(W)),
          aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,w),w) ) ) ).

% power_numeral_even
tff(fact_5903_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),numeral_numeral_nat(aa_num_num(bit0,W))) = $let(
          w: nat,
          w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),numeral_numeral_nat(W)),
          aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,w),w) ) ) ).

% power_numeral_even
tff(fact_5904_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),numeral_numeral_nat(aa_num_num(bit0,W))) = $let(
          w: int,
          w:= aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),numeral_numeral_nat(W)),
          aa_int_int(aa_int_fun_int_int(times_times_int,w),w) ) ) ).

% power_numeral_even
tff(fact_5905_power__numeral__odd,axiom,
    ! [Z: code_integer,W: num] :
      ( aa_nat_Code_integer(power_8256067586552552935nteger(Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
          w: code_integer,
          w:= aa_nat_Code_integer(power_8256067586552552935nteger(Z),numeral_numeral_nat(W)),
          aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),w)),w) ) ) ).

% power_numeral_odd
tff(fact_5906_power__numeral__odd,axiom,
    ! [Z: assn,W: num] :
      ( aa_nat_assn(power_power_assn(Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
          w: assn,
          w:= aa_nat_assn(power_power_assn(Z),numeral_numeral_nat(W)),
          aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Z),w)),w) ) ) ).

% power_numeral_odd
tff(fact_5907_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      ( aa_nat_rat(power_power_rat(Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
          w: rat,
          w:= aa_nat_rat(power_power_rat(Z),numeral_numeral_nat(W)),
          aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),w)),w) ) ) ).

% power_numeral_odd
tff(fact_5908_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
          w: nat,
          w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),numeral_numeral_nat(W)),
          aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),w)),w) ) ) ).

% power_numeral_odd
tff(fact_5909_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
          w: int,
          w:= aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),numeral_numeral_nat(W)),
          aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),w)),w) ) ) ).

% power_numeral_odd
tff(fact_5910_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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,numeral_numeral_nat(aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% even_mask_div_iff
tff(fact_5911_even__mask__div__iff,axiom,
    ! [M: nat,N: 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(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
    <=> ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% even_mask_div_iff
tff(fact_5912_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),one_one_Code_integer)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% even_mask_div_iff
tff(fact_5913_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),divide5121882707175180666atural(minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M),one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)))
    <=> ( ( aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N) = zero_z2226904508553997617atural )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% even_mask_div_iff
tff(fact_5914_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),one_one_nat)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5915_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A )
     => ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)),one_one_Code_integer)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5916_stable__imp__take__bit__eq,axiom,
    ! [A: code_natural,N: nat] :
      ( ( divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = A )
     => ( bit_se569199155075624693atural(N,A) = $ite(aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A),zero_z2226904508553997617atural,minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N),one_one_Code_natural)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5917_stable__imp__take__bit__eq,axiom,
    ! [A: int,N: nat] :
      ( ( 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_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = $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,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5918_power__Suc__less,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).

% power_Suc_less
tff(fact_5919_power__Suc__less,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N))),aa_nat_rat(power_power_rat(A),N)) ) ) ).

% power_Suc_less
tff(fact_5920_power__Suc__less,axiom,
    ! [A: nat,N: 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),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ).

% power_Suc_less
tff(fact_5921_power__Suc__less,axiom,
    ! [A: int,N: 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(aa_int_fun_nat_int(power_power_int,A),N))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ) ).

% power_Suc_less
tff(fact_5922_power__Suc__le__self,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),A) ) ) ).

% power_Suc_le_self
tff(fact_5923_power__Suc__le__self,axiom,
    ! [A: rat,N: 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,N))),A) ) ) ).

% power_Suc_le_self
tff(fact_5924_power__Suc__le__self,axiom,
    ! [A: nat,N: 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,N))),A) ) ) ).

% power_Suc_le_self
tff(fact_5925_power__Suc__le__self,axiom,
    ! [A: int,N: 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(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,N))),A) ) ) ).

% power_Suc_le_self
tff(fact_5926_power__Suc__less__one,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),one_one_Code_integer) ) ) ).

% power_Suc_less_one
tff(fact_5927_power__Suc__less__one,axiom,
    ! [A: rat,N: 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,N))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_5928_power__Suc__less__one,axiom,
    ! [A: nat,N: 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,N))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_5929_power__Suc__less__one,axiom,
    ! [A: int,N: 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(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,N))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_5930_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N4)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_5931_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),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),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_5932_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),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),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_5933_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),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(aa_int_fun_nat_int(power_power_int,A),N4)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_5934_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N4)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).

% power_decreasing
tff(fact_5935_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),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),N)) ) ) ) ).

% power_decreasing
tff(fact_5936_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),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),N)) ) ) ) ).

% power_decreasing
tff(fact_5937_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),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(aa_int_fun_nat_int(power_power_int,A),N4)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ) ) ).

% power_decreasing
tff(fact_5938_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_le_imp_le_exp
tff(fact_5939_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_le_imp_le_exp
tff(fact_5940_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_le_imp_le_exp
tff(fact_5941_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N: 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(aa_int_fun_nat_int(power_power_int,A),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_le_imp_le_exp
tff(fact_5942_self__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).

% self_le_power
tff(fact_5943_self__le__power,axiom,
    ! [A: rat,N: 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),N)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),N)) ) ) ).

% self_le_power
tff(fact_5944_self__le__power,axiom,
    ! [A: nat,N: 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),N)
       => 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),N)) ) ) ).

% self_le_power
tff(fact_5945_self__le__power,axiom,
    ! [A: int,N: 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),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ) ).

% self_le_power
tff(fact_5946_one__less__power,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).

% one_less_power
tff(fact_5947_one__less__power,axiom,
    ! [A: rat,N: 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),N)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N)) ) ) ).

% one_less_power
tff(fact_5948_one__less__power,axiom,
    ! [A: nat,N: 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),N)
       => 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),N)) ) ) ).

% one_less_power
tff(fact_5949_one__less__power,axiom,
    ! [A: int,N: 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),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ) ).

% one_less_power
tff(fact_5950_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_5951_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_5952_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)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_5953_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
       => ( divide5121882707175180666atural(A,times_2397367101498566445atural(B,A)) = divide5121882707175180666atural(one_one_Code_natural,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_5954_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_5955_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_5956_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)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_5957_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
       => ( divide5121882707175180666atural(A,times_2397367101498566445atural(A,B)) = divide5121882707175180666atural(one_one_Code_natural,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_5958_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [B4: nat] :
              ( ( B4 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B4),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B4 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B4) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B4) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5959_is__unitE,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [B4: int] :
              ( ( B4 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B4),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B4 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B4) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B4) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5960_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B4: code_integer] :
              ( ( B4 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B4),one_one_Code_integer)
               => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A) = B4 )
                 => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),B4) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B4) = one_one_Code_integer )
                     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5961_is__unitE,axiom,
    ! [A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ~ ( ( A != zero_z2226904508553997617atural )
         => ! [B4: code_natural] :
              ( ( B4 != zero_z2226904508553997617atural )
             => ( aa_Code_natural_o(dvd_dvd_Code_natural(B4),one_one_Code_natural)
               => ( ( divide5121882707175180666atural(one_one_Code_natural,A) = B4 )
                 => ( ( divide5121882707175180666atural(one_one_Code_natural,B4) = A )
                   => ( ( times_2397367101498566445atural(A,B4) = one_one_Code_natural )
                     => ( divide5121882707175180666atural(C,A) != times_2397367101498566445atural(C,B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5962_evenE,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
     => ~ ! [B4: nat] : ( A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B4) ) ) ).

% evenE
tff(fact_5963_evenE,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)
     => ~ ! [B4: int] : ( A != aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B4) ) ) ).

% evenE
tff(fact_5964_evenE,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ~ ! [B4: code_integer] : ( A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B4) ) ) ).

% evenE
tff(fact_5965_evenE,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)
     => ~ ! [B4: code_natural] : ( A != times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),B4) ) ) ).

% evenE
tff(fact_5966_odd__one,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),one_one_nat) ).

% odd_one
tff(fact_5967_odd__one,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int) ).

% odd_one
tff(fact_5968_odd__one,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),one_one_Code_integer) ).

% odd_one
tff(fact_5969_odd__one,axiom,
    ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),one_one_Code_natural) ).

% odd_one
tff(fact_5970_even__minus,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(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_5971_even__minus,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_5972_power4__eq__xxxx,axiom,
    ! [X: code_integer] : ( aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),X)),X) ) ).

% power4_eq_xxxx
tff(fact_5973_power4__eq__xxxx,axiom,
    ! [X: assn] : ( aa_nat_assn(power_power_assn(X),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),X)),X)),X) ) ).

% power4_eq_xxxx
tff(fact_5974_power4__eq__xxxx,axiom,
    ! [X: rat] : ( aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),X)),X) ) ).

% power4_eq_xxxx
tff(fact_5975_power4__eq__xxxx,axiom,
    ! [X: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),X)),X)),X) ) ).

% power4_eq_xxxx
tff(fact_5976_power4__eq__xxxx,axiom,
    ! [X: int] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,X),numeral_numeral_nat(aa_num_num(bit0,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_int_int(aa_int_fun_int_int(times_times_int,X),X)),X)),X) ) ).

% power4_eq_xxxx
tff(fact_5977_power2__eq__square,axiom,
    ! [A: code_integer] : ( aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ) ).

% power2_eq_square
tff(fact_5978_power2__eq__square,axiom,
    ! [A: assn] : ( aa_nat_assn(power_power_assn(A),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),A) ) ).

% power2_eq_square
tff(fact_5979_power2__eq__square,axiom,
    ! [A: rat] : ( aa_nat_rat(power_power_rat(A),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ) ).

% power2_eq_square
tff(fact_5980_power2__eq__square,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ) ).

% power2_eq_square
tff(fact_5981_power2__eq__square,axiom,
    ! [A: int] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ) ).

% power2_eq_square
tff(fact_5982_one__power2,axiom,
    aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_power2
tff(fact_5983_one__power2,axiom,
    aa_nat_rat(power_power_rat(one_one_rat),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_rat ).

% one_power2
tff(fact_5984_one__power2,axiom,
    aa_nat_int(aa_int_fun_nat_int(power_power_int,one_one_int),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_int ).

% one_power2
tff(fact_5985_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_nat ).

% one_power2
tff(fact_5986_power2__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,X),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),numeral_numeral_nat(aa_num_num(bit0,one))) )
    <=> ( ( X = Y )
        | ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5987_power2__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(aa_num_num(bit0,one))) )
    <=> ( ( X = Y )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5988_power2__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(aa_num_num(bit0,one))) )
    <=> ( ( X = Y )
        | ( X = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5989_power3__eq__cube,axiom,
    ! [A: code_integer] : ( aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(aa_num_num(bit1,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A)),A) ) ).

% power3_eq_cube
tff(fact_5990_power3__eq__cube,axiom,
    ! [A: assn] : ( aa_nat_assn(power_power_assn(A),numeral_numeral_nat(aa_num_num(bit1,one))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),A)),A) ) ).

% power3_eq_cube
tff(fact_5991_power3__eq__cube,axiom,
    ! [A: rat] : ( aa_nat_rat(power_power_rat(A),numeral_numeral_nat(aa_num_num(bit1,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),A) ) ).

% power3_eq_cube
tff(fact_5992_power3__eq__cube,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(aa_num_num(bit1,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A)),A) ) ).

% power3_eq_cube
tff(fact_5993_power3__eq__cube,axiom,
    ! [A: int] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),numeral_numeral_nat(aa_num_num(bit1,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),A) ) ).

% power3_eq_cube
tff(fact_5994_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),M)
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_5995_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),M)
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_5996_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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,numeral_numeral_nat(aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5997_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N: 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(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & 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(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5998_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5999_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_natural,M: nat,N: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),divide5121882707175180666atural(times_2397367101498566445atural(A,aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M)),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | ( aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N) = zero_z2226904508553997617atural )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),divide5121882707175180666atural(A,aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6000_power__minus_H,axiom,
    ! [X: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,one_one_int),X)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,X)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N)) ) ) ).

% power_minus'
tff(fact_6001_power__minus_H,axiom,
    ! [X: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(nO_MAT8252062027627875367nteger,one_one_Code_integer),X)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) ) ) ).

% power_minus'
tff(fact_6002_power__minus_H,axiom,
    ! [X: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(nO_MATCH_rat_rat,one_one_rat),X)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,X)),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(X),N)) ) ) ).

% power_minus'
tff(fact_6003_and__one__eq,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),one_one_Code_integer) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ).

% and_one_eq
tff(fact_6004_and__one__eq,axiom,
    ! [A: code_natural] : ( bit_se2773287842338716102atural(A,one_one_Code_natural) = modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) ) ).

% and_one_eq
tff(fact_6005_and__one__eq,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),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))) ) ).

% and_one_eq
tff(fact_6006_and__one__eq,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% and_one_eq
tff(fact_6007_one__and__eq,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,one_one_Code_integer),A) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ).

% one_and_eq
tff(fact_6008_one__and__eq,axiom,
    ! [A: code_natural] : ( bit_se2773287842338716102atural(one_one_Code_natural,A) = modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) ) ).

% one_and_eq
tff(fact_6009_one__and__eq,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_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))) ) ).

% one_and_eq
tff(fact_6010_one__and__eq,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% one_and_eq
tff(fact_6011_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
    <=> ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_6012_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_6013_even__two__times__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,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)))) = A ) ) ).

% even_two_times_div_two
tff(fact_6014_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_6015_even__two__times__div__two,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)
     => ( times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_6016_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_int )
    <=> ( ( A = one_one_int )
        | ( A = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_6017_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( ( A = one_one_Code_integer )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_6018_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_6019_power__diff__power__eq,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% power_diff_power_eq
tff(fact_6020_power__diff__power__eq,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% power_diff_power_eq
tff(fact_6021_power__diff__power__eq,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% power_diff_power_eq
tff(fact_6022_power__diff__power__eq,axiom,
    ! [A: code_natural,M: nat,N: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_nat_Code_natural(power_7079662738309270450atural(A),M),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M),aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),divide5121882707175180666atural(one_one_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% power_diff_power_eq
tff(fact_6023_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6024_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,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 ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6025_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6026_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)
    <=> ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6027_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P3: int,Q3: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P3),Q3) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3) ) ).

% normalize_denom_pos
tff(fact_6028_power__eq__if,axiom,
    ! [P3: code_integer,M: nat] :
      ( aa_nat_Code_integer(power_8256067586552552935nteger(P3),M) = $ite(M = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,P3),aa_nat_Code_integer(power_8256067586552552935nteger(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ) ).

% power_eq_if
tff(fact_6029_power__eq__if,axiom,
    ! [P3: assn,M: nat] :
      ( aa_nat_assn(power_power_assn(P3),M) = $ite(M = zero_zero_nat,one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P3),aa_nat_assn(power_power_assn(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ) ).

% power_eq_if
tff(fact_6030_power__eq__if,axiom,
    ! [P3: rat,M: nat] :
      ( aa_nat_rat(power_power_rat(P3),M) = $ite(M = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P3),aa_nat_rat(power_power_rat(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ) ).

% power_eq_if
tff(fact_6031_power__eq__if,axiom,
    ! [P3: nat,M: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = $ite(M = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ) ).

% power_eq_if
tff(fact_6032_power__eq__if,axiom,
    ! [P3: int,M: nat] :
      ( aa_nat_int(aa_int_fun_nat_int(power_power_int,P3),M) = $ite(M = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P3),aa_nat_int(aa_int_fun_nat_int(power_power_int,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ) ).

% power_eq_if
tff(fact_6033_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6034_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6035_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6036_normalize__crossproduct,axiom,
    ! [Q3: int,S2: int,P3: int,R2: int] :
      ( ( Q3 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P3),Q3)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S2)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P3),S2) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q3) ) ) ) ) ).

% normalize_crossproduct
tff(fact_6037_power__minus__mult,axiom,
    ! [N: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) ) ).

% power_minus_mult
tff(fact_6038_power__minus__mult,axiom,
    ! [N: nat,A: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_assn(power_power_assn(A),N) ) ) ).

% power_minus_mult
tff(fact_6039_power__minus__mult,axiom,
    ! [N: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% power_minus_mult
tff(fact_6040_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) ) ) ).

% power_minus_mult
tff(fact_6041_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N) ) ) ).

% power_minus_mult
tff(fact_6042_power2__sum,axiom,
    ! [X: nat,Y: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),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,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),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,numeral_numeral_nat(aa_num_num(bit0,one))),X)),Y)) ) ).

% power2_sum
tff(fact_6043_power2__sum,axiom,
    ! [X: int,Y: int] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),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(aa_int_fun_nat_int(power_power_int,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),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))),X)),Y)) ) ).

% power2_sum
tff(fact_6044_power2__sum,axiom,
    ! [X: code_integer,Y: code_integer] : ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),X)),Y)) ) ).

% power2_sum
tff(fact_6045_power2__sum,axiom,
    ! [X: rat,Y: rat] : ( aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),X)),Y)) ) ).

% power2_sum
tff(fact_6046_power2__sum,axiom,
    ! [X: code_natural,Y: code_natural] : ( aa_nat_Code_natural(power_7079662738309270450atural(plus_p4538020629002901425atural(X,Y)),numeral_numeral_nat(aa_num_num(bit0,one))) = plus_p4538020629002901425atural(plus_p4538020629002901425atural(aa_nat_Code_natural(power_7079662738309270450atural(X),numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_Code_natural(power_7079662738309270450atural(Y),numeral_numeral_nat(aa_num_num(bit0,one)))),times_2397367101498566445atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),X),Y)) ) ).

% power2_sum
tff(fact_6047_oddE,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
     => ~ ! [B4: nat] : ( A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B4)),one_one_nat) ) ) ).

% oddE
tff(fact_6048_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)
     => ~ ! [B4: 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))),B4)),one_one_int) ) ) ).

% oddE
tff(fact_6049_oddE,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ~ ! [B4: code_integer] : ( A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B4)),one_one_Code_integer) ) ) ).

% oddE
tff(fact_6050_oddE,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)
     => ~ ! [B4: code_natural] : ( A != plus_p4538020629002901425atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),B4),one_one_Code_natural) ) ) ).

% oddE
tff(fact_6051_square__le__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_6052_square__le__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_6053_square__le__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_6054_parity__cases,axiom,
    ! [A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_6055_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_6056_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
         => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_6057_parity__cases,axiom,
    ! [A: code_natural] :
      ( ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)
       => ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) != zero_z2226904508553997617atural ) )
     => ~ ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)
         => ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) != one_one_Code_natural ) ) ) ).

% parity_cases
tff(fact_6058_mod2__eq__if,axiom,
    ! [A: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A),zero_zero_nat,one_one_nat) ) ).

% mod2_eq_if
tff(fact_6059_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_6060_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ) ).

% mod2_eq_if
tff(fact_6061_mod2__eq__if,axiom,
    ! [A: code_natural] :
      ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = $ite(aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A),zero_z2226904508553997617atural,one_one_Code_natural) ) ).

% mod2_eq_if
tff(fact_6062_minus__power__mult__self,axiom,
    ! [A: int,N: nat] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ) ).

% minus_power_mult_self
tff(fact_6063_minus__power__mult__self,axiom,
    ! [A: code_integer,N: nat] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ) ).

% minus_power_mult_self
tff(fact_6064_minus__power__mult__self,axiom,
    ! [A: rat,N: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ) ).

% minus_power_mult_self
tff(fact_6065_power__odd__eq,axiom,
    ! [A: code_integer,N: nat] : ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),numeral_numeral_nat(aa_num_num(bit0,one)))) ) ).

% power_odd_eq
tff(fact_6066_power__odd__eq,axiom,
    ! [A: assn,N: nat] : ( aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(aa_nat_assn(power_power_assn(A),N)),numeral_numeral_nat(aa_num_num(bit0,one)))) ) ).

% power_odd_eq
tff(fact_6067_power__odd__eq,axiom,
    ! [A: rat,N: nat] : ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),N)),numeral_numeral_nat(aa_num_num(bit0,one)))) ) ).

% power_odd_eq
tff(fact_6068_power__odd__eq,axiom,
    ! [A: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = 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),N)),numeral_numeral_nat(aa_num_num(bit0,one)))) ) ).

% power_odd_eq
tff(fact_6069_power__odd__eq,axiom,
    ! [A: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),numeral_numeral_nat(aa_num_num(bit0,one)))) ) ).

% power_odd_eq
tff(fact_6070_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(uminus_uminus_int,one_one_int) ) ).

% minus_1_div_exp_eq_int
tff(fact_6071_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_6072_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_6073_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_6074_power2__diff,axiom,
    ! [X: int,Y: int] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),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(aa_int_fun_nat_int(power_power_int,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),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))),X)),Y)) ) ).

% power2_diff
tff(fact_6075_power2__diff,axiom,
    ! [X: code_integer,Y: code_integer] : ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),Y)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),X)),Y)) ) ).

% power2_diff
tff(fact_6076_power2__diff,axiom,
    ! [X: rat,Y: rat] : ( aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),X)),Y)) ) ).

% power2_diff
tff(fact_6077_mask__half__int,axiom,
    ! [N: nat] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ).

% mask_half_int
tff(fact_6078_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6079_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6080_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6081_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( modulo8411746178871703098atural(times_2397367101498566445atural(A,aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M)),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = times_2397367101498566445atural(modulo8411746178871703098atural(A,aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6082_power__minus1__odd,axiom,
    ! [N: nat] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_int_int(uminus_uminus_int,one_one_int) ) ).

% power_minus1_odd
tff(fact_6083_power__minus1__odd,axiom,
    ! [N: 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,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ).

% power_minus1_odd
tff(fact_6084_power__minus1__odd,axiom,
    ! [N: 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,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ).

% power_minus1_odd
tff(fact_6085_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_6086_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_6087_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),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,N),K)) ) ) ).

% take_bit_incr_eq
tff(fact_6088_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(suc,N))),one_one_Code_integer) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_6089_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))),one_one_int) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_6090_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(numeral_numeral_nat(K)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),numeral_numeral_nat(K))),one_one_Code_integer) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_6091_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,numeral_numeral_nat(K)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),numeral_numeral_nat(K))),one_one_int) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_6092_gcd__nat_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_ig),zero_zero_nat) ).

% gcd_nat.ordering_top_axioms
tff(fact_6093_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_6094_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa),Xb),Xc) = Y )
     => ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(X,produc487386426758144856at_nat(Xa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xc))))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa),Xc))) )
           => ~ accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(X,produc487386426758144856at_nat(Xa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_6095_flip__bit__0,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(zero_zero_nat),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% flip_bit_0
tff(fact_6096_flip__bit__0,axiom,
    ! [A: code_natural] : ( bit_se168947363167071951atural(zero_zero_nat,A) = plus_p4538020629002901425atural(aa_o_Code_natural(zero_n8403883297036319079atural,aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)),times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))))) ) ).

% flip_bit_0
tff(fact_6097_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_6098_flip__bit__0,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ).

% flip_bit_0
tff(fact_6099_one__mod__2__pow__eq,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ) ).

% one_mod_2_pow_eq
tff(fact_6100_one__mod__2__pow__eq,axiom,
    ! [N: nat] : ( modulo8411746178871703098atural(one_one_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = aa_o_Code_natural(zero_n8403883297036319079atural,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ) ).

% one_mod_2_pow_eq
tff(fact_6101_one__mod__2__pow__eq,axiom,
    ! [N: nat] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ) ).

% one_mod_2_pow_eq
tff(fact_6102_one__mod__2__pow__eq,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ) ).

% one_mod_2_pow_eq
tff(fact_6103_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6104_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6105_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: 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,numeral_numeral_rat(X))),N)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6106_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6107_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: 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,aa_num_Code_integer(numera6620942414471956472nteger,X))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6108_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: 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,numeral_numeral_rat(X))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6109_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6110_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: 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,numeral_numeral_rat(X))),N)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6111_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6112_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_Code_integer(zero_n356916108424825756nteger,(P)) = one_one_Code_integer )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_6113_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_rat(zero_n2052037380579107095ol_rat,(P)) = one_one_rat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_6114_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = one_one_int )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_6115_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = one_one_nat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_6116_of__bool__eq_I2_J,axiom,
    aa_o_Code_integer(zero_n356916108424825756nteger,$true) = one_one_Code_integer ).

% of_bool_eq(2)
tff(fact_6117_of__bool__eq_I2_J,axiom,
    aa_o_rat(zero_n2052037380579107095ol_rat,$true) = one_one_rat ).

% of_bool_eq(2)
tff(fact_6118_of__bool__eq_I2_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).

% of_bool_eq(2)
tff(fact_6119_of__bool__eq_I2_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$true) = one_one_nat ).

% of_bool_eq(2)
tff(fact_6120_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(P))),one_one_Code_integer)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_6121_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(P))),one_one_rat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_6122_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_6123_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_6124_of__bool__not__iff,axiom,
    ! [P: $o] : ( aa_o_Code_integer(zero_n356916108424825756nteger,~ (P)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_o_Code_integer(zero_n356916108424825756nteger,(P))) ) ).

% of_bool_not_iff
tff(fact_6125_of__bool__not__iff,axiom,
    ! [P: $o] : ( aa_o_rat(zero_n2052037380579107095ol_rat,~ (P)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_o_rat(zero_n2052037380579107095ol_rat,(P))) ) ).

% of_bool_not_iff
tff(fact_6126_of__bool__not__iff,axiom,
    ! [P: $o] : ( aa_o_int(zero_n2684676970156552555ol_int,~ (P)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ) ).

% of_bool_not_iff
tff(fact_6127_of__int__1,axiom,
    aa_int_Code_integer(ring_18347121197199848620nteger,one_one_int) = one_one_Code_integer ).

% of_int_1
tff(fact_6128_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_6129_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_6130_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = one_one_Code_integer )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_6131_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_6132_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_6133_of__int__mult,axiom,
    ! [W: int,Z: int] : ( aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ) ).

% of_int_mult
tff(fact_6134_of__int__mult,axiom,
    ! [W: int,Z: int] : ( aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_mult
tff(fact_6135_of__int__mult,axiom,
    ! [W: int,Z: int] : ( aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_mult
tff(fact_6136_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_6137_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_6138_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_6139_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_6140_and__nat__numerals_I4_J,axiom,
    ! [X: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ) ).

% and_nat_numerals(4)
tff(fact_6141_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(aa_num_num(bit1,Y))) = one_one_nat ) ).

% and_nat_numerals(2)
tff(fact_6142_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),one_one_Code_integer)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_6143_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_6144_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_6145_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_6146_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_6147_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_6148_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),one_one_Code_integer)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_6149_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_6150_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_6151_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_6152_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_6153_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_6154_take__bit__of__1,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),one_one_Code_integer) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ) ).

% take_bit_of_1
tff(fact_6155_take__bit__of__1,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ) ).

% take_bit_of_1
tff(fact_6156_take__bit__of__1,axiom,
    ! [N: nat] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ) ).

% take_bit_of_1
tff(fact_6157_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6158_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6159_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(X))),N) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6160_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) )
    <=> ( Y = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6161_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N) )
    <=> ( Y = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6162_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(X))),N) )
    <=> ( Y = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6163_one__div__2__pow__eq,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_o_Code_integer(zero_n356916108424825756nteger,N = zero_zero_nat) ) ).

% one_div_2_pow_eq
tff(fact_6164_one__div__2__pow__eq,axiom,
    ! [N: nat] : ( divide5121882707175180666atural(one_one_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = aa_o_Code_natural(zero_n8403883297036319079atural,N = zero_zero_nat) ) ).

% one_div_2_pow_eq
tff(fact_6165_one__div__2__pow__eq,axiom,
    ! [N: nat] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_o_int(zero_n2684676970156552555ol_int,N = zero_zero_nat) ) ).

% one_div_2_pow_eq
tff(fact_6166_one__div__2__pow__eq,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_o_nat(zero_n2687167440665602831ol_nat,N = zero_zero_nat) ) ).

% one_div_2_pow_eq
tff(fact_6167_bits__1__div__exp,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_o_Code_integer(zero_n356916108424825756nteger,N = zero_zero_nat) ) ).

% bits_1_div_exp
tff(fact_6168_bits__1__div__exp,axiom,
    ! [N: nat] : ( divide5121882707175180666atural(one_one_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = aa_o_Code_natural(zero_n8403883297036319079atural,N = zero_zero_nat) ) ).

% bits_1_div_exp
tff(fact_6169_bits__1__div__exp,axiom,
    ! [N: nat] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_o_int(zero_n2684676970156552555ol_int,N = zero_zero_nat) ) ).

% bits_1_div_exp
tff(fact_6170_bits__1__div__exp,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_o_nat(zero_n2687167440665602831ol_nat,N = zero_zero_nat) ) ).

% bits_1_div_exp
tff(fact_6171_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] : ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(M),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) ) ).

% take_bit_of_exp
tff(fact_6172_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] : ( bit_se569199155075624693atural(M,aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = times_2397367101498566445atural(aa_o_Code_natural(zero_n8403883297036319079atural,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) ) ).

% take_bit_of_exp
tff(fact_6173_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ) ).

% take_bit_of_exp
tff(fact_6174_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ).

% take_bit_of_exp
tff(fact_6175_take__bit__of__2,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ).

% take_bit_of_2
tff(fact_6176_take__bit__of__2,axiom,
    ! [N: nat] : ( bit_se569199155075624693atural(N,numera5444537566228673987atural(aa_num_num(bit0,one))) = times_2397367101498566445atural(aa_o_Code_natural(zero_n8403883297036319079atural,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),numera5444537566228673987atural(aa_num_num(bit0,one))) ) ).

% take_bit_of_2
tff(fact_6177_take__bit__of__2,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% take_bit_of_2
tff(fact_6178_take__bit__of__2,axiom,
    ! [N: nat] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ).

% take_bit_of_2
tff(fact_6179_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: 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,aa_num_Code_integer(numera6620942414471956472nteger,X))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6180_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: 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,numeral_numeral_rat(X))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6181_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: 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(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6182_mult__of__int__commute,axiom,
    ! [X: int,Y: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(ring_18347121197199848620nteger,X)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_int_Code_integer(ring_18347121197199848620nteger,X)) ) ).

% mult_of_int_commute
tff(fact_6183_mult__of__int__commute,axiom,
    ! [X: int,Y: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,X)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_int_rat(ring_1_of_int_rat,X)) ) ).

% mult_of_int_commute
tff(fact_6184_mult__of__int__commute,axiom,
    ! [X: int,Y: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,X)) ) ).

% mult_of_int_commute
tff(fact_6185_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( aa_o_Code_integer(zero_n356916108424825756nteger,
          ( (P)
          & (Q) )) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(P))),aa_o_Code_integer(zero_n356916108424825756nteger,(Q))) ) ).

% of_bool_conj
tff(fact_6186_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( aa_o_rat(zero_n2052037380579107095ol_rat,
          ( (P)
          & (Q) )) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(P))),aa_o_rat(zero_n2052037380579107095ol_rat,(Q))) ) ).

% of_bool_conj
tff(fact_6187_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( aa_o_int(zero_n2684676970156552555ol_int,
          ( (P)
          & (Q) )) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ) ).

% of_bool_conj
tff(fact_6188_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( aa_o_nat(zero_n2687167440665602831ol_nat,
          ( (P)
          & (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q))) ) ).

% of_bool_conj
tff(fact_6189_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(P))),one_one_Code_integer) ).

% of_bool_less_eq_one
tff(fact_6190_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(P))),one_one_rat) ).

% of_bool_less_eq_one
tff(fact_6191_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int) ).

% of_bool_less_eq_one
tff(fact_6192_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat) ).

% of_bool_less_eq_one
tff(fact_6193_split__of__bool__asm,axiom,
    ! [P: fun_Code_integer_o,P3: $o] :
      ( aa_Code_integer_o(P,aa_o_Code_integer(zero_n356916108424825756nteger,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_Code_integer_o(P,one_one_Code_integer) )
          | ( ~ (P3)
            & ~ aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool_asm
tff(fact_6194_split__of__bool__asm,axiom,
    ! [P: fun_rat_o,P3: $o] :
      ( aa_rat_o(P,aa_o_rat(zero_n2052037380579107095ol_rat,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_rat_o(P,one_one_rat) )
          | ( ~ (P3)
            & ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_6195_split__of__bool__asm,axiom,
    ! [P: fun_int_o,P3: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_int_o(P,one_one_int) )
          | ( ~ (P3)
            & ~ aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_6196_split__of__bool__asm,axiom,
    ! [P: fun_nat_o,P3: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_nat_o(P,one_one_nat) )
          | ( ~ (P3)
            & ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_6197_split__of__bool,axiom,
    ! [P: fun_Code_integer_o,P3: $o] :
      ( aa_Code_integer_o(P,aa_o_Code_integer(zero_n356916108424825756nteger,(P3)))
    <=> ( ( (P3)
         => aa_Code_integer_o(P,one_one_Code_integer) )
        & ( ~ (P3)
         => aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool
tff(fact_6198_split__of__bool,axiom,
    ! [P: fun_rat_o,P3: $o] :
      ( aa_rat_o(P,aa_o_rat(zero_n2052037380579107095ol_rat,(P3)))
    <=> ( ( (P3)
         => aa_rat_o(P,one_one_rat) )
        & ( ~ (P3)
         => aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_6199_split__of__bool,axiom,
    ! [P: fun_int_o,P3: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P3)))
    <=> ( ( (P3)
         => aa_int_o(P,one_one_int) )
        & ( ~ (P3)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_6200_split__of__bool,axiom,
    ! [P: fun_nat_o,P3: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P3)))
    <=> ( ( (P3)
         => aa_nat_o(P,one_one_nat) )
        & ( ~ (P3)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_6201_of__bool__def,axiom,
    ! [P3: $o] :
      ( aa_o_Code_integer(zero_n356916108424825756nteger,(P3)) = $ite((P3),one_one_Code_integer,zero_z3403309356797280102nteger) ) ).

% of_bool_def
tff(fact_6202_of__bool__def,axiom,
    ! [P3: $o] :
      ( aa_o_rat(zero_n2052037380579107095ol_rat,(P3)) = $ite((P3),one_one_rat,zero_zero_rat) ) ).

% of_bool_def
tff(fact_6203_of__bool__def,axiom,
    ! [P3: $o] :
      ( aa_o_int(zero_n2684676970156552555ol_int,(P3)) = $ite((P3),one_one_int,zero_zero_int) ) ).

% of_bool_def
tff(fact_6204_of__bool__def,axiom,
    ! [P3: $o] :
      ( aa_o_nat(zero_n2687167440665602831ol_nat,(P3)) = $ite((P3),one_one_nat,zero_zero_nat) ) ).

% of_bool_def
tff(fact_6205_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : ( aa_Pro6799258434465214121nt_int(adjust_div,Qr) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aTP_Lamm_ih),Qr) ) ).

% Divides.adjust_div_def
tff(fact_6206_of__int__neg__numeral,axiom,
    ! [K: num] : ( aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)) ) ).

% of_int_neg_numeral
tff(fact_6207_of__int__neg__numeral,axiom,
    ! [K: num] : ( aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K)) ) ).

% of_int_neg_numeral
tff(fact_6208_of__int__neg__numeral,axiom,
    ! [K: num] : ( aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(K)) ) ).

% of_int_neg_numeral
tff(fact_6209_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_dvd_imp_le
tff(fact_6210_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_6211_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa),Xb),Xc) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_6212_mask__nat__def,axiom,
    ! [N: nat] : ( aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),one_one_nat) ) ).

% mask_nat_def
tff(fact_6213_bits__induct,axiom,
    ! [P: fun_Code_integer_o,A: code_integer] :
      ( ! [A4: code_integer] :
          ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A4),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A4 )
         => aa_Code_integer_o(P,A4) )
     => ( ! [A4: code_integer,B4: $o] :
            ( aa_Code_integer_o(P,A4)
           => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(B4))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A4))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A4 )
             => aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(B4))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A4))) ) )
       => aa_Code_integer_o(P,A) ) ) ).

% bits_induct
tff(fact_6214_bits__induct,axiom,
    ! [P: fun_Code_natural_o,A: code_natural] :
      ( ! [A4: code_natural] :
          ( ( divide5121882707175180666atural(A4,numera5444537566228673987atural(aa_num_num(bit0,one))) = A4 )
         => aa_Code_natural_o(P,A4) )
     => ( ! [A4: code_natural,B4: $o] :
            ( aa_Code_natural_o(P,A4)
           => ( ( divide5121882707175180666atural(plus_p4538020629002901425atural(aa_o_Code_natural(zero_n8403883297036319079atural,(B4)),times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),A4)),numera5444537566228673987atural(aa_num_num(bit0,one))) = A4 )
             => aa_Code_natural_o(P,plus_p4538020629002901425atural(aa_o_Code_natural(zero_n8403883297036319079atural,(B4)),times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),A4))) ) )
       => aa_Code_natural_o(P,A) ) ) ).

% bits_induct
tff(fact_6215_bits__induct,axiom,
    ! [P: fun_int_o,A: int] :
      ( ! [A4: int] :
          ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A4 )
         => aa_int_o(P,A4) )
     => ( ! [A4: int,B4: $o] :
            ( aa_int_o(P,A4)
           => ( ( 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,(B4))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A4))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A4 )
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B4))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A4))) ) )
       => aa_int_o(P,A) ) ) ).

% bits_induct
tff(fact_6216_bits__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [A4: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),numeral_numeral_nat(aa_num_num(bit0,one))) = A4 )
         => aa_nat_o(P,A4) )
     => ( ! [A4: nat,B4: $o] :
            ( aa_nat_o(P,A4)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B4))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A4))),numeral_numeral_nat(aa_num_num(bit0,one))) = A4 )
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B4))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A4))) ) )
       => aa_nat_o(P,A) ) ) ).

% bits_induct
tff(fact_6217_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) ) ).

% exp_mod_exp
tff(fact_6218_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : ( modulo8411746178871703098atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = times_2397367101498566445atural(aa_o_Code_natural(zero_n8403883297036319079atural,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M)) ) ).

% exp_mod_exp
tff(fact_6219_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ) ).

% exp_mod_exp
tff(fact_6220_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)) ) ).

% exp_mod_exp
tff(fact_6221_dvd__productE,axiom,
    ! [P3: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ~ ! [X2: nat,Y2: nat] :
            ( ( P3 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),Y2) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X2),A)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y2),B) ) ) ) ).

% dvd_productE
tff(fact_6222_dvd__productE,axiom,
    ! [P3: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P3),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ~ ! [X2: int,Y2: int] :
            ( ( P3 = aa_int_int(aa_int_fun_int_int(times_times_int,X2),Y2) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X2),A)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y2),B) ) ) ) ).

% dvd_productE
tff(fact_6223_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ? [B7: nat,C5: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B7),C5) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B7),B)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C5),C) ) ) ).

% division_decomp
tff(fact_6224_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ? [B7: int,C5: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B7),C5) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B7),B)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C5),C) ) ) ).

% division_decomp
tff(fact_6225_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K)
       => ? [N3: 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),N3)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_6226_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),K)
       => ? [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N3)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_6227_xor__one__eq,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,A),one_one_Code_integer) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)))),aa_o_Code_integer(zero_n356916108424825756nteger,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))) ) ).

% xor_one_eq
tff(fact_6228_xor__one__eq,axiom,
    ! [A: code_natural] : ( bit_se2046307713759805098atural(A,one_one_Code_natural) = minus_7197305767214868737atural(plus_p4538020629002901425atural(A,aa_o_Code_natural(zero_n8403883297036319079atural,aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A))),aa_o_Code_natural(zero_n8403883297036319079atural,~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A))) ) ).

% xor_one_eq
tff(fact_6229_xor__one__eq,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(minus_minus_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)))),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))) ) ).

% xor_one_eq
tff(fact_6230_xor__one__eq,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A))) ) ).

% xor_one_eq
tff(fact_6231_one__xor__eq,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,one_one_Code_integer),A) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)))),aa_o_Code_integer(zero_n356916108424825756nteger,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))) ) ).

% one_xor_eq
tff(fact_6232_one__xor__eq,axiom,
    ! [A: code_natural] : ( bit_se2046307713759805098atural(one_one_Code_natural,A) = minus_7197305767214868737atural(plus_p4538020629002901425atural(A,aa_o_Code_natural(zero_n8403883297036319079atural,aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A))),aa_o_Code_natural(zero_n8403883297036319079atural,~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),A))) ) ).

% one_xor_eq
tff(fact_6233_one__xor__eq,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(minus_minus_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)))),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_xor_eq
tff(fact_6234_one__xor__eq,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A))) ) ).

% one_xor_eq
tff(fact_6235_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
            aa_o_Code_integer(zero_n356916108424825756nteger,
              ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M) != zero_z3403309356797280102nteger )
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ).

% exp_div_exp_eq
tff(fact_6236_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( divide5121882707175180666atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = times_2397367101498566445atural(aa_o_Code_natural(zero_n8403883297036319079atural,
            ( ( aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M) != zero_z2226904508553997617atural )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) )),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ).

% exp_div_exp_eq
tff(fact_6237_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,
            aa_o_int(zero_n2684676970156552555ol_int,
              ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M) != zero_zero_int )
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ))),aa_nat_int(aa_int_fun_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,M),N))) ) ).

% exp_div_exp_eq
tff(fact_6238_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
            aa_o_nat(zero_n2687167440665602831ol_nat,
              ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M) != zero_zero_nat )
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ).

% exp_div_exp_eq
tff(fact_6239_floor__exists1,axiom,
    ! [X: rat] :
    ? [X2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X2)),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),one_one_int)))
      & ! [Y4: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y4)),X)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int))) )
         => ( Y4 = X2 ) ) ) ).

% floor_exists1
tff(fact_6240_floor__exists,axiom,
    ! [X: rat] :
    ? [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),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_6241_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
          ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
          & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
          aa_int_int(uminus_uminus_int,
            aa_o_int(zero_n2684676970156552555ol_int,
              ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
              & ~ 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))),K)
                & ~ 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,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ).

% and_int.simps
tff(fact_6242_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa) = Y )
     => ( Y = $ite(
            ( aa_set_int_o(member_int(X),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(Xa),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))),X)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa) ))),
            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))),X)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_6243_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one)))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = Y ) ) ) ).

% round_unique
tff(fact_6244_push__bit__numeral__minus__1,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(numeral_numeral_nat(N)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),numeral_numeral_nat(N))) ) ).

% push_bit_numeral_minus_1
tff(fact_6245_push__bit__numeral__minus__1,axiom,
    ! [N: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,numeral_numeral_nat(N)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),numeral_numeral_nat(N))) ) ).

% push_bit_numeral_minus_1
tff(fact_6246_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : ( aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(L),Qr) = aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_gr(L)),Qr) ) ).

% divmod_step_integer_def
tff(fact_6247_singletonI,axiom,
    ! [A: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)) ).

% singletonI
tff(fact_6248_singletonI,axiom,
    ! [A: set_Pr958786334691620121nt_int] : aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int)) ).

% singletonI
tff(fact_6249_singletonI,axiom,
    ! [A: set_nat] : aa_set_set_nat_o(member_set_nat(A),insert_set_nat(A,bot_bot_set_set_nat)) ).

% singletonI
tff(fact_6250_singletonI,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),insert4923026906511862463_nat_o(A,bot_bo8632255151191677115_nat_o)) ).

% singletonI
tff(fact_6251_singletonI,axiom,
    ! [A: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)) ).

% singletonI
tff(fact_6252_singletonI,axiom,
    ! [A: $o] : aa_set_o_o(member_o((A)),insert_o((A),bot_bot_set_o)) ).

% singletonI
tff(fact_6253_singletonI,axiom,
    ! [A: nat] : aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)) ).

% singletonI
tff(fact_6254_singletonI,axiom,
    ! [A: int] : aa_set_int_o(member_int(A),insert_int(A,bot_bot_set_int)) ).

% singletonI
tff(fact_6255_Int__insert__right__if1,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),insert9069300056098147895at_nat(A,B2)) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_6256_Int__insert__right__if1,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),insert_o((A),B2)) = insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_6257_Int__insert__right__if1,axiom,
    ! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A2)
     => ( inf_in8396524679539076455nt_int(A2,insert8897473484851387113nt_int(A,B2)) = insert8897473484851387113nt_int(A,inf_in8396524679539076455nt_int(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_6258_Int__insert__right__if1,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ( inf_inf_set_set_nat(A2,insert_set_nat(A,B2)) = insert_set_nat(A,inf_inf_set_set_nat(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_6259_Int__insert__right__if1,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),insert_int(A,B2)) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_6260_Int__insert__right__if1,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2)
     => ( inf_in3376754634429691197_nat_o(A2,insert4923026906511862463_nat_o(A,B2)) = insert4923026906511862463_nat_o(A,inf_in3376754634429691197_nat_o(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_6261_Int__insert__right__if1,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(A,B2)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_6262_Int__insert__right__if1,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),B2)) = aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_6263_Int__insert__right__if0,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),insert9069300056098147895at_nat(A,B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2) ) ) ).

% Int_insert_right_if0
tff(fact_6264_Int__insert__right__if0,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ~ aa_set_o_o(member_o((A)),A2)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),insert_o((A),B2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) ) ) ).

% Int_insert_right_if0
tff(fact_6265_Int__insert__right__if0,axiom,
    ! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A2)
     => ( inf_in8396524679539076455nt_int(A2,insert8897473484851387113nt_int(A,B2)) = inf_in8396524679539076455nt_int(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_6266_Int__insert__right__if0,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(A),A2)
     => ( inf_inf_set_set_nat(A2,insert_set_nat(A,B2)) = inf_inf_set_set_nat(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_6267_Int__insert__right__if0,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(A),A2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),insert_int(A,B2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) ) ) ).

% Int_insert_right_if0
tff(fact_6268_Int__insert__right__if0,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2)
     => ( inf_in3376754634429691197_nat_o(A2,insert4923026906511862463_nat_o(A,B2)) = inf_in3376754634429691197_nat_o(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_6269_Int__insert__right__if0,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(A,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ) ) ).

% Int_insert_right_if0
tff(fact_6270_Int__insert__right__if0,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ) ).

% Int_insert_right_if0
tff(fact_6271_insert__inter__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,A2)),insert9069300056098147895at_nat(A,B2)) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2)) ) ).

% insert_inter_insert
tff(fact_6272_insert__inter__insert,axiom,
    ! [A: $o,A2: set_o,B2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,insert_o((A),A2)),insert_o((A),B2)) = insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2)) ) ).

% insert_inter_insert
tff(fact_6273_insert__inter__insert,axiom,
    ! [A: int,A2: set_int,B2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,A2)),insert_int(A,B2)) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) ) ).

% insert_inter_insert
tff(fact_6274_insert__inter__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A2)),insert8211810215607154385at_nat(A,B2)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ) ).

% insert_inter_insert
tff(fact_6275_insert__inter__insert,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(insert_nat(A),A2)),aa_set_nat_set_nat(insert_nat(A),B2)) = aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ).

% insert_inter_insert
tff(fact_6276_Int__insert__left__if1,axiom,
    ! [A: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),C2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,B2)),C2) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_6277_Int__insert__left__if1,axiom,
    ! [A: $o,C2: set_o,B2: set_o] :
      ( aa_set_o_o(member_o((A)),C2)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,insert_o((A),B2)),C2) = insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B2),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_6278_Int__insert__left__if1,axiom,
    ! [A: set_Pr958786334691620121nt_int,C2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),C2)
     => ( inf_in8396524679539076455nt_int(insert8897473484851387113nt_int(A,B2),C2) = insert8897473484851387113nt_int(A,inf_in8396524679539076455nt_int(B2,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_6279_Int__insert__left__if1,axiom,
    ! [A: set_nat,C2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),C2)
     => ( inf_inf_set_set_nat(insert_set_nat(A,B2),C2) = insert_set_nat(A,inf_inf_set_set_nat(B2,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_6280_Int__insert__left__if1,axiom,
    ! [A: int,C2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(A),C2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,B2)),C2) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_6281_Int__insert__left__if1,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,C2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),C2)
     => ( inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,B2),C2) = insert4923026906511862463_nat_o(A,inf_in3376754634429691197_nat_o(B2,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_6282_Int__insert__left__if1,axiom,
    ! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),C2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B2)),C2) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_6283_Int__insert__left__if1,axiom,
    ! [A: nat,C2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),C2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(insert_nat(A),B2)),C2) = aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_6284_Int__insert__left__if0,axiom,
    ! [A: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),C2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,B2)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),C2) ) ) ).

% Int_insert_left_if0
tff(fact_6285_Int__insert__left__if0,axiom,
    ! [A: $o,C2: set_o,B2: set_o] :
      ( ~ aa_set_o_o(member_o((A)),C2)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,insert_o((A),B2)),C2) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B2),C2) ) ) ).

% Int_insert_left_if0
tff(fact_6286_Int__insert__left__if0,axiom,
    ! [A: set_Pr958786334691620121nt_int,C2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),C2)
     => ( inf_in8396524679539076455nt_int(insert8897473484851387113nt_int(A,B2),C2) = inf_in8396524679539076455nt_int(B2,C2) ) ) ).

% Int_insert_left_if0
tff(fact_6287_Int__insert__left__if0,axiom,
    ! [A: set_nat,C2: set_set_nat,B2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(A),C2)
     => ( inf_inf_set_set_nat(insert_set_nat(A,B2),C2) = inf_inf_set_set_nat(B2,C2) ) ) ).

% Int_insert_left_if0
tff(fact_6288_Int__insert__left__if0,axiom,
    ! [A: int,C2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(A),C2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,B2)),C2) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2) ) ) ).

% Int_insert_left_if0
tff(fact_6289_Int__insert__left__if0,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,C2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),C2)
     => ( inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,B2),C2) = inf_in3376754634429691197_nat_o(B2,C2) ) ) ).

% Int_insert_left_if0
tff(fact_6290_Int__insert__left__if0,axiom,
    ! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),C2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2) ) ) ).

% Int_insert_left_if0
tff(fact_6291_Int__insert__left__if0,axiom,
    ! [A: nat,C2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),C2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(insert_nat(A),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2) ) ) ).

% Int_insert_left_if0
tff(fact_6292_Un__insert__right,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),insert8211810215607154385at_nat(A,B2)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) ) ).

% Un_insert_right
tff(fact_6293_Un__insert__right,axiom,
    ! [A2: set_o,A: $o,B2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),insert_o((A),B2)) = insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B2)) ) ).

% Un_insert_right
tff(fact_6294_Un__insert__right,axiom,
    ! [A2: set_int,A: int,B2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),insert_int(A,B2)) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) ) ).

% Un_insert_right
tff(fact_6295_Un__insert__right,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,A: produc4166570645942440679at_nat,B2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),insert6337962749363155127at_nat(A,B2)) = insert6337962749363155127at_nat(A,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) ) ).

% Un_insert_right
tff(fact_6296_Un__insert__right,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),insert9069300056098147895at_nat(A,B2)) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ) ).

% Un_insert_right
tff(fact_6297_Un__insert__right,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),B2)) = aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% Un_insert_right
tff(fact_6298_Un__insert__left,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,insert8211810215607154385at_nat(A,B2)),C2) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2)) ) ).

% Un_insert_left
tff(fact_6299_Un__insert__left,axiom,
    ! [A: $o,B2: set_o,C2: set_o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,insert_o((A),B2)),C2) = insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B2),C2)) ) ).

% Un_insert_left
tff(fact_6300_Un__insert__left,axiom,
    ! [A: int,B2: set_int,C2: set_int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,insert_int(A,B2)),C2) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B2),C2)) ) ).

% Un_insert_left
tff(fact_6301_Un__insert__left,axiom,
    ! [A: produc4166570645942440679at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,insert6337962749363155127at_nat(A,B2)),C2) = insert6337962749363155127at_nat(A,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B2),C2)) ) ).

% Un_insert_left
tff(fact_6302_Un__insert__left,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,insert9069300056098147895at_nat(A,B2)),C2) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) ) ).

% Un_insert_left
tff(fact_6303_Un__insert__left,axiom,
    ! [A: nat,B2: set_nat,C2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(insert_nat(A),B2)),C2) = aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) ) ).

% Un_insert_left
tff(fact_6304_singleton__conv2,axiom,
    ! [A: produc3843707927480180839at_nat] : ( collec6321179662152712658at_nat(aa_Pro1894297876208389727_nat_o(fequal7949547063978738364at_nat,A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ) ).

% singleton_conv2
tff(fact_6305_singleton__conv2,axiom,
    ! [A: set_Pr958786334691620121nt_int] : ( collec5210948495886036740nt_int(aa_set4993530631398190431_int_o(fequal1127456680179375854nt_int,A)) = insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int) ) ).

% singleton_conv2
tff(fact_6306_singleton__conv2,axiom,
    ! [A: set_nat] : ( collect_set_nat(aa_set7861427066675100071_nat_o(fequal_set_nat,A)) = insert_set_nat(A,bot_bot_set_set_nat) ) ).

% singleton_conv2
tff(fact_6307_singleton__conv2,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o] : ( collec6407654713642994138_nat_o(aa_fun3380769697021226975at_o_o(fequal1890233700291625924_nat_o,A)) = insert4923026906511862463_nat_o(A,bot_bo8632255151191677115_nat_o) ) ).

% singleton_conv2
tff(fact_6308_singleton__conv2,axiom,
    ! [A: product_prod_nat_nat] : ( collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ) ).

% singleton_conv2
tff(fact_6309_singleton__conv2,axiom,
    ! [A: $o] : ( collect_o(aa_o_fun_o_o(fequal_o,(A))) = insert_o((A),bot_bot_set_o) ) ).

% singleton_conv2
tff(fact_6310_singleton__conv2,axiom,
    ! [A: nat] : ( collect_nat(aa_nat_fun_nat_o(fequal_nat,A)) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ) ).

% singleton_conv2
tff(fact_6311_singleton__conv2,axiom,
    ! [A: int] : ( collect_int(aa_int_fun_int_o(fequal_int,A)) = insert_int(A,bot_bot_set_int) ) ).

% singleton_conv2
tff(fact_6312_singleton__conv,axiom,
    ! [A: produc3843707927480180839at_nat] : ( collec6321179662152712658at_nat(aa_Pro1894297876208389727_nat_o(aTP_Lamm_ii,A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ) ).

% singleton_conv
tff(fact_6313_singleton__conv,axiom,
    ! [A: set_Pr958786334691620121nt_int] : ( collec5210948495886036740nt_int(aa_set4993530631398190431_int_o(aTP_Lamm_ij,A)) = insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int) ) ).

% singleton_conv
tff(fact_6314_singleton__conv,axiom,
    ! [A: set_nat] : ( collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_ik,A)) = insert_set_nat(A,bot_bot_set_set_nat) ) ).

% singleton_conv
tff(fact_6315_singleton__conv,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o] : ( collec6407654713642994138_nat_o(aa_fun3380769697021226975at_o_o(aTP_Lamm_il,A)) = insert4923026906511862463_nat_o(A,bot_bo8632255151191677115_nat_o) ) ).

% singleton_conv
tff(fact_6316_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] : ( collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(aTP_Lamm_im,A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ) ).

% singleton_conv
tff(fact_6317_singleton__conv,axiom,
    ! [A: $o] : ( collect_o(aa_o_fun_o_o(aTP_Lamm_in,(A))) = insert_o((A),bot_bot_set_o) ) ).

% singleton_conv
tff(fact_6318_singleton__conv,axiom,
    ! [A: nat] : ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_io,A)) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ) ).

% singleton_conv
tff(fact_6319_singleton__conv,axiom,
    ! [A: int] : ( collect_int(aa_int_fun_int_o(aTP_Lamm_ip,A)) = insert_int(A,bot_bot_set_int) ) ).

% singleton_conv
tff(fact_6320_singleton__insert__inj__eq,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) = insert9069300056098147895at_nat(A,A2) )
    <=> ( ( A = B )
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_6321_singleton__insert__inj__eq,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(A,A2) )
    <=> ( ( A = B )
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_6322_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A2: set_o] :
      ( ( insert_o((B),bot_bot_set_o) = insert_o((A),A2) )
    <=> ( ( (A)
        <=> (B) )
        & aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),insert_o((B),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq
tff(fact_6323_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( insert_int(B,bot_bot_set_int) = insert_int(A,A2) )
    <=> ( ( A = B )
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_6324_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat) = aa_set_nat_set_nat(insert_nat(A),A2) )
    <=> ( ( A = B )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_6325_singleton__insert__inj__eq_H,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,A2) = insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) )
    <=> ( ( A = B )
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_6326_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,A2) = insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) )
    <=> ( ( A = B )
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_6327_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A2: set_o,B: $o] :
      ( ( insert_o((A),A2) = insert_o((B),bot_bot_set_o) )
    <=> ( ( (A)
        <=> (B) )
        & aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),insert_o((B),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_6328_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( insert_int(A,A2) = insert_int(B,bot_bot_set_int) )
    <=> ( ( A = B )
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_6329_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( aa_set_nat_set_nat(insert_nat(A),A2) = aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat) )
    <=> ( ( A = B )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_6330_insert__disjoint_I1_J,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,A2)),B2) = bot_bo228742789529271731at_nat )
    <=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B2)
        & ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2) = bot_bo228742789529271731at_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_6331_insert__disjoint_I1_J,axiom,
    ! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( ( inf_in8396524679539076455nt_int(insert8897473484851387113nt_int(A,A2),B2) = bot_bo1488462491386950373nt_int )
    <=> ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),B2)
        & ( inf_in8396524679539076455nt_int(A2,B2) = bot_bo1488462491386950373nt_int ) ) ) ).

% insert_disjoint(1)
tff(fact_6332_insert__disjoint_I1_J,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( inf_inf_set_set_nat(insert_set_nat(A,A2),B2) = bot_bot_set_set_nat )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(A),B2)
        & ( inf_inf_set_set_nat(A2,B2) = bot_bot_set_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_6333_insert__disjoint_I1_J,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,A2),B2) = bot_bo8632255151191677115_nat_o )
    <=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),B2)
        & ( inf_in3376754634429691197_nat_o(A2,B2) = bot_bo8632255151191677115_nat_o ) ) ) ).

% insert_disjoint(1)
tff(fact_6334_insert__disjoint_I1_J,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A2)),B2) = bot_bo2099793752762293965at_nat )
    <=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B2)
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_6335_insert__disjoint_I1_J,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,insert_o((A),A2)),B2) = bot_bot_set_o )
    <=> ( ~ aa_set_o_o(member_o((A)),B2)
        & ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
tff(fact_6336_insert__disjoint_I1_J,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(insert_nat(A),A2)),B2) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),B2)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_6337_insert__disjoint_I1_J,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,A2)),B2) = bot_bot_set_int )
    <=> ( ~ aa_set_int_o(member_int(A),B2)
        & ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
tff(fact_6338_insert__disjoint_I2_J,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,A2)),B2) )
    <=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B2)
        & ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_6339_insert__disjoint_I2_J,axiom,
    ! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( ( bot_bo1488462491386950373nt_int = inf_in8396524679539076455nt_int(insert8897473484851387113nt_int(A,A2),B2) )
    <=> ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),B2)
        & ( bot_bo1488462491386950373nt_int = inf_in8396524679539076455nt_int(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_6340_insert__disjoint_I2_J,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( bot_bot_set_set_nat = inf_inf_set_set_nat(insert_set_nat(A,A2),B2) )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(A),B2)
        & ( bot_bot_set_set_nat = inf_inf_set_set_nat(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_6341_insert__disjoint_I2_J,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,A2),B2) )
    <=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),B2)
        & ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_6342_insert__disjoint_I2_J,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A2)),B2) )
    <=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B2)
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_6343_insert__disjoint_I2_J,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,insert_o((A),A2)),B2) )
    <=> ( ~ aa_set_o_o(member_o((A)),B2)
        & ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_6344_insert__disjoint_I2_J,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(insert_nat(A),A2)),B2) )
    <=> ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),B2)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_6345_insert__disjoint_I2_J,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,A2)),B2) )
    <=> ( ~ aa_set_int_o(member_int(A),B2)
        & ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_6346_disjoint__insert_I1_J,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),insert9069300056098147895at_nat(A,A2)) = bot_bo228742789529271731at_nat )
    <=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B2)
        & ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),A2) = bot_bo228742789529271731at_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_6347_disjoint__insert_I1_J,axiom,
    ! [B2: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
      ( ( inf_in8396524679539076455nt_int(B2,insert8897473484851387113nt_int(A,A2)) = bot_bo1488462491386950373nt_int )
    <=> ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),B2)
        & ( inf_in8396524679539076455nt_int(B2,A2) = bot_bo1488462491386950373nt_int ) ) ) ).

% disjoint_insert(1)
tff(fact_6348_disjoint__insert_I1_J,axiom,
    ! [B2: set_set_nat,A: set_nat,A2: set_set_nat] :
      ( ( inf_inf_set_set_nat(B2,insert_set_nat(A,A2)) = bot_bot_set_set_nat )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(A),B2)
        & ( inf_inf_set_set_nat(B2,A2) = bot_bot_set_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_6349_disjoint__insert_I1_J,axiom,
    ! [B2: set_fu3555266933746557007_nat_o,A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(B2,insert4923026906511862463_nat_o(A,A2)) = bot_bo8632255151191677115_nat_o )
    <=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),B2)
        & ( inf_in3376754634429691197_nat_o(B2,A2) = bot_bo8632255151191677115_nat_o ) ) ) ).

% disjoint_insert(1)
tff(fact_6350_disjoint__insert_I1_J,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),insert8211810215607154385at_nat(A,A2)) = bot_bo2099793752762293965at_nat )
    <=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B2)
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),A2) = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_6351_disjoint__insert_I1_J,axiom,
    ! [B2: set_o,A: $o,A2: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B2),insert_o((A),A2)) = bot_bot_set_o )
    <=> ( ~ aa_set_o_o(member_o((A)),B2)
        & ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B2),A2) = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
tff(fact_6352_disjoint__insert_I1_J,axiom,
    ! [B2: set_nat,A: nat,A2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),aa_set_nat_set_nat(insert_nat(A),A2)) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),B2)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A2) = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_6353_disjoint__insert_I1_J,axiom,
    ! [B2: set_int,A: int,A2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),insert_int(A,A2)) = bot_bot_set_int )
    <=> ( ~ aa_set_int_o(member_int(A),B2)
        & ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),A2) = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
tff(fact_6354_disjoint__insert_I2_J,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),insert9069300056098147895at_nat(B,B2)) )
    <=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),A2)
        & ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_6355_disjoint__insert_I2_J,axiom,
    ! [A2: set_se6260736226359567993nt_int,B: set_Pr958786334691620121nt_int,B2: set_se6260736226359567993nt_int] :
      ( ( bot_bo1488462491386950373nt_int = inf_in8396524679539076455nt_int(A2,insert8897473484851387113nt_int(B,B2)) )
    <=> ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(B),A2)
        & ( bot_bo1488462491386950373nt_int = inf_in8396524679539076455nt_int(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_6356_disjoint__insert_I2_J,axiom,
    ! [A2: set_set_nat,B: set_nat,B2: set_set_nat] :
      ( ( bot_bot_set_set_nat = inf_inf_set_set_nat(A2,insert_set_nat(B,B2)) )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(B),A2)
        & ( bot_bot_set_set_nat = inf_inf_set_set_nat(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_6357_disjoint__insert_I2_J,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(A2,insert4923026906511862463_nat_o(B,B2)) )
    <=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),A2)
        & ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_6358_disjoint__insert_I2_J,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(B,B2)) )
    <=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),A2)
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_6359_disjoint__insert_I2_J,axiom,
    ! [A2: set_o,B: $o,B2: set_o] :
      ( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),insert_o((B),B2)) )
    <=> ( ~ aa_set_o_o(member_o((B)),A2)
        & ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_6360_disjoint__insert_I2_J,axiom,
    ! [A2: set_nat,B: nat,B2: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(insert_nat(B),B2)) )
    <=> ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B),A2)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_6361_disjoint__insert_I2_J,axiom,
    ! [A2: set_int,B: int,B2: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),insert_int(B,B2)) )
    <=> ( ~ aa_set_int_o(member_int(B),A2)
        & ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_6362_insert__Diff__single,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] : ( insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = insert9069300056098147895at_nat(A,A2) ) ).

% insert_Diff_single
tff(fact_6363_insert__Diff__single,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : ( insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = insert8211810215607154385at_nat(A,A2) ) ).

% insert_Diff_single
tff(fact_6364_insert__Diff__single,axiom,
    ! [A: $o,A2: set_o] : ( insert_o((A),minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = insert_o((A),A2) ) ).

% insert_Diff_single
tff(fact_6365_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] : ( insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = insert_int(A,A2) ) ).

% insert_Diff_single
tff(fact_6366_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] : ( aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) = aa_set_nat_set_nat(insert_nat(A),A2) ) ).

% insert_Diff_single
tff(fact_6367_round__1,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,one_one_rat) = one_one_int ).

% round_1
tff(fact_6368_subset__Compl__singleton,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),uminus935396558254630718at_nat(insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)))
    <=> ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),A2) ) ).

% subset_Compl_singleton
tff(fact_6369_subset__Compl__singleton,axiom,
    ! [A2: set_se6260736226359567993nt_int,B: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A2),uminus6423885277529793776nt_int(insert8897473484851387113nt_int(B,bot_bo1488462491386950373nt_int)))
    <=> ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(B),A2) ) ).

% subset_Compl_singleton
tff(fact_6370_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),uminus613421341184616069et_nat(insert_set_nat(B,bot_bot_set_set_nat)))
    <=> ~ aa_set_set_nat_o(member_set_nat(B),A2) ) ).

% subset_Compl_singleton
tff(fact_6371_subset__Compl__singleton,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),uminus2199602881456817606_nat_o(insert4923026906511862463_nat_o(B,bot_bo8632255151191677115_nat_o)))
    <=> ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),A2) ) ).

% subset_Compl_singleton
tff(fact_6372_subset__Compl__singleton,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)))
    <=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),A2) ) ).

% subset_Compl_singleton
tff(fact_6373_subset__Compl__singleton,axiom,
    ! [A2: set_o,B: $o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),aa_set_o_set_o(uminus_uminus_set_o,insert_o((B),bot_bot_set_o)))
    <=> ~ aa_set_o_o(member_o((B)),A2) ) ).

% subset_Compl_singleton
tff(fact_6374_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),aa_set_int_set_int(uminus1532241313380277803et_int,insert_int(B,bot_bot_set_int)))
    <=> ~ aa_set_int_o(member_int(B),A2) ) ).

% subset_Compl_singleton
tff(fact_6375_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)))
    <=> ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B),A2) ) ).

% subset_Compl_singleton
tff(fact_6376_push__bit__Suc__minus__numeral,axiom,
    ! [N: nat,K: num] : ( aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(aa_nat_nat(suc,N)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,K)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_6377_push__bit__Suc__minus__numeral,axiom,
    ! [N: nat,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_6378_round__neg__numeral,axiom,
    ! [N: num] : ( aa_rat_int(archim7778729529865785530nd_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ) ).

% round_neg_numeral
tff(fact_6379_push__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ).

% push_bit_Suc
tff(fact_6380_push__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : ( bit_se6611745700429515170atural(aa_nat_nat(suc,N),A) = bit_se6611745700429515170atural(N,times_2397367101498566445atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))) ) ).

% push_bit_Suc
tff(fact_6381_push__bit__Suc,axiom,
    ! [N: nat,A: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,aa_nat_nat(suc,N)),A) = aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) ).

% push_bit_Suc
tff(fact_6382_push__bit__Suc,axiom,
    ! [N: nat,A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))) ) ).

% push_bit_Suc
tff(fact_6383_push__bit__of__1,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),one_one_Code_integer) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N) ) ).

% push_bit_of_1
tff(fact_6384_push__bit__of__1,axiom,
    ! [N: nat] : ( bit_se6611745700429515170atural(N,one_one_Code_natural) = aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N) ) ).

% push_bit_of_1
tff(fact_6385_push__bit__of__1,axiom,
    ! [N: nat] : ( aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ).

% push_bit_of_1
tff(fact_6386_push__bit__of__1,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N) ) ).

% push_bit_of_1
tff(fact_6387_push__bit__minus__numeral,axiom,
    ! [L: num,K: num] : ( aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(numeral_numeral_nat(L)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(pred_numeral(L)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,K)))) ) ).

% push_bit_minus_numeral
tff(fact_6388_push__bit__minus__numeral,axiom,
    ! [L: num,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,pred_numeral(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) ) ).

% push_bit_minus_numeral
tff(fact_6389_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(zero_z3403309356797280102nteger),L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ) ).

% minus_integer_code(2)
tff(fact_6390_insert__Collect,axiom,
    ! [A: produc3843707927480180839at_nat,P: fun_Pr3059359313727237556_nat_o] : ( insert9069300056098147895at_nat(A,collec6321179662152712658at_nat(P)) = collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_iq(A),P)) ) ).

% insert_Collect
tff(fact_6391_insert__Collect,axiom,
    ! [A: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] : ( insert8211810215607154385at_nat(A,collec3392354462482085612at_nat(P)) = collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_ir(A),P)) ) ).

% insert_Collect
tff(fact_6392_insert__Collect,axiom,
    ! [A: $o,P: fun_o_o] : ( insert_o((A),collect_o(P)) = collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_is((A)),P)) ) ).

% insert_Collect
tff(fact_6393_insert__Collect,axiom,
    ! [A: set_Pr958786334691620121nt_int,P: fun_se7249947736802449410_int_o] : ( insert8897473484851387113nt_int(A,collec5210948495886036740nt_int(P)) = collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_it(A),P)) ) ).

% insert_Collect
tff(fact_6394_insert__Collect,axiom,
    ! [A: set_nat,P: fun_set_nat_o] : ( insert_set_nat(A,collect_set_nat(P)) = collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_iu(A),P)) ) ).

% insert_Collect
tff(fact_6395_insert__Collect,axiom,
    ! [A: nat,P: fun_nat_o] : ( aa_set_nat_set_nat(insert_nat(A),collect_nat(P)) = collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_iv(A),P)) ) ).

% insert_Collect
tff(fact_6396_insert__Collect,axiom,
    ! [A: int,P: fun_int_o] : ( insert_int(A,collect_int(P)) = collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_iw(A),P)) ) ).

% insert_Collect
tff(fact_6397_insert__Collect,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,P: fun_fu5135969500543783852at_o_o] : ( insert4923026906511862463_nat_o(A,collec6407654713642994138_nat_o(P)) = collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_ix(A),P)) ) ).

% insert_Collect
tff(fact_6398_insert__compr,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : ( insert9069300056098147895at_nat(A,B2) = collec6321179662152712658at_nat(aa_set6353399069419443519_nat_o(aTP_Lamm_iy(A),B2)) ) ).

% insert_compr
tff(fact_6399_insert__compr,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( insert8211810215607154385at_nat(A,B2) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_iz(A),B2)) ) ).

% insert_compr
tff(fact_6400_insert__compr,axiom,
    ! [A: $o,B2: set_o] : ( insert_o((A),B2) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_ja((A)),B2)) ) ).

% insert_compr
tff(fact_6401_insert__compr,axiom,
    ! [A: set_Pr958786334691620121nt_int,B2: set_se6260736226359567993nt_int] : ( insert8897473484851387113nt_int(A,B2) = collec5210948495886036740nt_int(aa_set4062574822058255039_int_o(aTP_Lamm_jb(A),B2)) ) ).

% insert_compr
tff(fact_6402_insert__compr,axiom,
    ! [A: set_nat,B2: set_set_nat] : ( insert_set_nat(A,B2) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_jc(A),B2)) ) ).

% insert_compr
tff(fact_6403_insert__compr,axiom,
    ! [A: nat,B2: set_nat] : ( aa_set_nat_set_nat(insert_nat(A),B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_jd(A),B2)) ) ).

% insert_compr
tff(fact_6404_insert__compr,axiom,
    ! [A: int,B2: set_int] : ( insert_int(A,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_je(A),B2)) ) ).

% insert_compr
tff(fact_6405_insert__compr,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o] : ( insert4923026906511862463_nat_o(A,B2) = collec6407654713642994138_nat_o(aa_set7516337885093543743at_o_o(aTP_Lamm_jf(A),B2)) ) ).

% insert_compr
tff(fact_6406_divmod__integer_H__def,axiom,
    ! [M: num,N: num] : ( unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N))) ) ).

% divmod_integer'_def
tff(fact_6407_exhaustive__integer_H_Ocases,axiom,
    ! [X: produc7509114972854644512nteger] :
      ~ ! [F2: fun_Co3460408610320954864e_term,D: code_integer,I2: code_integer] : ( X != produc7264323995540650962nteger(F2,produc1086072967326762835nteger(D,I2)) ) ).

% exhaustive_integer'.cases
tff(fact_6408_full__exhaustive__integer_H_Ocases,axiom,
    ! [X: produc2453742332425399446nteger] :
      ~ ! [F2: fun_Pr5098669352780829690e_term,D: code_integer,I2: code_integer] : ( X != aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,F2),produc1086072967326762835nteger(D,I2)) ) ).

% full_exhaustive_integer'.cases
tff(fact_6409_ID_Opred__cong,axiom,
    ! [X: produc3843707927480180839at_nat,Ya: produc3843707927480180839at_nat,P: fun_Pr3059359313727237556_nat_o,Pa: fun_Pr3059359313727237556_nat_o] :
      ( ( X = Ya )
     => ( ! [Z2: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Z2),insert9069300056098147895at_nat(Ya,bot_bo228742789529271731at_nat))
           => ( aa_Pro5947980084102686253_nat_o(P,Z2)
            <=> aa_Pro5947980084102686253_nat_o(Pa,Z2) ) )
       => ( aa_Pro5947980084102686253_nat_o(bNF_id4812938428820093592_nat_o(P),X)
        <=> aa_Pro5947980084102686253_nat_o(bNF_id4812938428820093592_nat_o(Pa),Ya) ) ) ) ).

% ID.pred_cong
tff(fact_6410_ID_Opred__cong,axiom,
    ! [X: set_Pr958786334691620121nt_int,Ya: set_Pr958786334691620121nt_int,P: fun_se7249947736802449410_int_o,Pa: fun_se7249947736802449410_int_o] :
      ( ( X = Ya )
     => ( ! [Z2: set_Pr958786334691620121nt_int] :
            ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z2),insert8897473484851387113nt_int(Ya,bot_bo1488462491386950373nt_int))
           => ( aa_set6170482117830564987_int_o(P,Z2)
            <=> aa_set6170482117830564987_int_o(Pa,Z2) ) )
       => ( aa_set6170482117830564987_int_o(bNF_id231123401833519846_int_o(P),X)
        <=> aa_set6170482117830564987_int_o(bNF_id231123401833519846_int_o(Pa),Ya) ) ) ) ).

% ID.pred_cong
tff(fact_6411_ID_Opred__cong,axiom,
    ! [X: set_nat,Ya: set_nat,P: fun_set_nat_o,Pa: fun_set_nat_o] :
      ( ( X = Ya )
     => ( ! [Z2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(Z2),insert_set_nat(Ya,bot_bot_set_set_nat))
           => ( aa_set_nat_o(P,Z2)
            <=> aa_set_nat_o(Pa,Z2) ) )
       => ( aa_set_nat_o(bNF_id348094275710791815_nat_o(P),X)
        <=> aa_set_nat_o(bNF_id348094275710791815_nat_o(Pa),Ya) ) ) ) ).

% ID.pred_cong
tff(fact_6412_ID_Opred__cong,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Ya: fun_Pr6460944699146269935_nat_o,P: fun_fu5135969500543783852at_o_o,Pa: fun_fu5135969500543783852at_o_o] :
      ( ( X = Ya )
     => ( ! [Z2: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Z2),insert4923026906511862463_nat_o(Ya,bot_bo8632255151191677115_nat_o))
           => ( aa_fun1675921005893230629at_o_o(P,Z2)
            <=> aa_fun1675921005893230629at_o_o(Pa,Z2) ) )
       => ( aa_fun1675921005893230629at_o_o(bNF_id3856154264776833680at_o_o(P),X)
        <=> aa_fun1675921005893230629at_o_o(bNF_id3856154264776833680at_o_o(Pa),Ya) ) ) ) ).

% ID.pred_cong
tff(fact_6413_ID_Opred__cong,axiom,
    ! [X: product_prod_nat_nat,Ya: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o,Pa: fun_Pr4796943543331071514_nat_o] :
      ( ( X = Ya )
     => ( ! [Z2: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z2),insert8211810215607154385at_nat(Ya,bot_bo2099793752762293965at_nat))
           => ( aa_Pro6150637604445964947_nat_o(P,Z2)
            <=> aa_Pro6150637604445964947_nat_o(Pa,Z2) ) )
       => ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(bNF_id1976805830016039166_nat_o,P),X)
        <=> aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(bNF_id1976805830016039166_nat_o,Pa),Ya) ) ) ) ).

% ID.pred_cong
tff(fact_6414_ID_Opred__cong,axiom,
    ! [X: $o,Ya: $o,P: fun_o_o,Pa: fun_o_o] :
      ( ( (X)
      <=> (Ya) )
     => ( ! [Z2: $o] :
            ( aa_set_o_o(member_o((Z2)),insert_o((Ya),bot_bot_set_o))
           => ( aa_o_o(P,(Z2))
            <=> aa_o_o(Pa,(Z2)) ) )
       => ( aa_o_o(bNF_id_bnf_fun_o_o(P),(X))
        <=> aa_o_o(bNF_id_bnf_fun_o_o(Pa),(Ya)) ) ) ) ).

% ID.pred_cong
tff(fact_6415_ID_Opred__cong,axiom,
    ! [X: nat,Ya: nat,P: fun_nat_o,Pa: fun_nat_o] :
      ( ( X = Ya )
     => ( ! [Z2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z2),aa_set_nat_set_nat(insert_nat(Ya),bot_bot_set_nat))
           => ( aa_nat_o(P,Z2)
            <=> aa_nat_o(Pa,Z2) ) )
       => ( aa_nat_o(bNF_id_bnf_fun_nat_o(P),X)
        <=> aa_nat_o(bNF_id_bnf_fun_nat_o(Pa),Ya) ) ) ) ).

% ID.pred_cong
tff(fact_6416_ID_Opred__cong,axiom,
    ! [X: int,Ya: int,P: fun_int_o,Pa: fun_int_o] :
      ( ( X = Ya )
     => ( ! [Z2: int] :
            ( aa_set_int_o(member_int(Z2),insert_int(Ya,bot_bot_set_int))
           => ( aa_int_o(P,Z2)
            <=> aa_int_o(Pa,Z2) ) )
       => ( aa_int_o(bNF_id_bnf_fun_int_o(P),X)
        <=> aa_int_o(bNF_id_bnf_fun_int_o(Pa),Ya) ) ) ) ).

% ID.pred_cong
tff(fact_6417_ID_Opred__mono__strong,axiom,
    ! [P: fun_Pr3059359313727237556_nat_o,X: produc3843707927480180839at_nat,Pa: fun_Pr3059359313727237556_nat_o] :
      ( aa_Pro5947980084102686253_nat_o(bNF_id4812938428820093592_nat_o(P),X)
     => ( ! [Z2: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Z2),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))
           => ( aa_Pro5947980084102686253_nat_o(P,Z2)
             => aa_Pro5947980084102686253_nat_o(Pa,Z2) ) )
       => aa_Pro5947980084102686253_nat_o(bNF_id4812938428820093592_nat_o(Pa),X) ) ) ).

% ID.pred_mono_strong
tff(fact_6418_ID_Opred__mono__strong,axiom,
    ! [P: fun_se7249947736802449410_int_o,X: set_Pr958786334691620121nt_int,Pa: fun_se7249947736802449410_int_o] :
      ( aa_set6170482117830564987_int_o(bNF_id231123401833519846_int_o(P),X)
     => ( ! [Z2: set_Pr958786334691620121nt_int] :
            ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z2),insert8897473484851387113nt_int(X,bot_bo1488462491386950373nt_int))
           => ( aa_set6170482117830564987_int_o(P,Z2)
             => aa_set6170482117830564987_int_o(Pa,Z2) ) )
       => aa_set6170482117830564987_int_o(bNF_id231123401833519846_int_o(Pa),X) ) ) ).

% ID.pred_mono_strong
tff(fact_6419_ID_Opred__mono__strong,axiom,
    ! [P: fun_set_nat_o,X: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(bNF_id348094275710791815_nat_o(P),X)
     => ( ! [Z2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(Z2),insert_set_nat(X,bot_bot_set_set_nat))
           => ( aa_set_nat_o(P,Z2)
             => aa_set_nat_o(Pa,Z2) ) )
       => aa_set_nat_o(bNF_id348094275710791815_nat_o(Pa),X) ) ) ).

% ID.pred_mono_strong
tff(fact_6420_ID_Opred__mono__strong,axiom,
    ! [P: fun_fu5135969500543783852at_o_o,X: fun_Pr6460944699146269935_nat_o,Pa: fun_fu5135969500543783852at_o_o] :
      ( aa_fun1675921005893230629at_o_o(bNF_id3856154264776833680at_o_o(P),X)
     => ( ! [Z2: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Z2),insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o))
           => ( aa_fun1675921005893230629at_o_o(P,Z2)
             => aa_fun1675921005893230629at_o_o(Pa,Z2) ) )
       => aa_fun1675921005893230629at_o_o(bNF_id3856154264776833680at_o_o(Pa),X) ) ) ).

% ID.pred_mono_strong
tff(fact_6421_ID_Opred__mono__strong,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,X: product_prod_nat_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(bNF_id1976805830016039166_nat_o,P),X)
     => ( ! [Z2: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))
           => ( aa_Pro6150637604445964947_nat_o(P,Z2)
             => aa_Pro6150637604445964947_nat_o(Pa,Z2) ) )
       => aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(bNF_id1976805830016039166_nat_o,Pa),X) ) ) ).

% ID.pred_mono_strong
tff(fact_6422_ID_Opred__mono__strong,axiom,
    ! [P: fun_o_o,X: $o,Pa: fun_o_o] :
      ( aa_o_o(bNF_id_bnf_fun_o_o(P),(X))
     => ( ! [Z2: $o] :
            ( aa_set_o_o(member_o((Z2)),insert_o((X),bot_bot_set_o))
           => ( aa_o_o(P,(Z2))
             => aa_o_o(Pa,(Z2)) ) )
       => aa_o_o(bNF_id_bnf_fun_o_o(Pa),(X)) ) ) ).

% ID.pred_mono_strong
tff(fact_6423_ID_Opred__mono__strong,axiom,
    ! [P: fun_nat_o,X: nat,Pa: fun_nat_o] :
      ( aa_nat_o(bNF_id_bnf_fun_nat_o(P),X)
     => ( ! [Z2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))
           => ( aa_nat_o(P,Z2)
             => aa_nat_o(Pa,Z2) ) )
       => aa_nat_o(bNF_id_bnf_fun_nat_o(Pa),X) ) ) ).

% ID.pred_mono_strong
tff(fact_6424_ID_Opred__mono__strong,axiom,
    ! [P: fun_int_o,X: int,Pa: fun_int_o] :
      ( aa_int_o(bNF_id_bnf_fun_int_o(P),X)
     => ( ! [Z2: int] :
            ( aa_set_int_o(member_int(Z2),insert_int(X,bot_bot_set_int))
           => ( aa_int_o(P,Z2)
             => aa_int_o(Pa,Z2) ) )
       => aa_int_o(bNF_id_bnf_fun_int_o(Pa),X) ) ) ).

% ID.pred_mono_strong
tff(fact_6425_ID_Orel__cong,axiom,
    ! [X: $o,Ya: $o,Y: $o,Xa: $o,R: fun_o_fun_o_o,Ra: fun_o_fun_o_o] :
      ( ( (X)
      <=> (Ya) )
     => ( ( (Y)
        <=> (Xa) )
       => ( ! [Z2: $o,Yb: $o] :
              ( aa_set_o_o(member_o((Z2)),insert_o((Ya),bot_bot_set_o))
             => ( aa_set_o_o(member_o((Yb)),insert_o((Xa),bot_bot_set_o))
               => ( aa_o_o(aa_o_fun_o_o(R,(Z2)),(Yb))
                <=> aa_o_o(aa_o_fun_o_o(Ra,(Z2)),(Yb)) ) ) )
         => ( aa_o_o(aa_o_fun_o_o(bNF_id6346756738013732234un_o_o(R),(X)),(Y))
          <=> aa_o_o(aa_o_fun_o_o(bNF_id6346756738013732234un_o_o(Ra),(Ya)),(Xa)) ) ) ) ) ).

% ID.rel_cong
tff(fact_6426_ID_Orel__cong,axiom,
    ! [X: $o,Ya: $o,Y: nat,Xa: nat,R: fun_o_fun_nat_o,Ra: fun_o_fun_nat_o] :
      ( ( (X)
      <=> (Ya) )
     => ( ( Y = Xa )
       => ( ! [Z2: $o,Yb: nat] :
              ( aa_set_o_o(member_o((Z2)),insert_o((Ya),bot_bot_set_o))
             => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Yb),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat))
               => ( aa_nat_o(aa_o_fun_nat_o(R,(Z2)),Yb)
                <=> aa_nat_o(aa_o_fun_nat_o(Ra,(Z2)),Yb) ) ) )
         => ( aa_nat_o(aa_o_fun_nat_o(bNF_id8527976288260956976_nat_o(R),(X)),Y)
          <=> aa_nat_o(aa_o_fun_nat_o(bNF_id8527976288260956976_nat_o(Ra),(Ya)),Xa) ) ) ) ) ).

% ID.rel_cong
tff(fact_6427_ID_Orel__cong,axiom,
    ! [X: $o,Ya: $o,Y: int,Xa: int,R: fun_o_fun_int_o,Ra: fun_o_fun_int_o] :
      ( ( (X)
      <=> (Ya) )
     => ( ( Y = Xa )
       => ( ! [Z2: $o,Yb: int] :
              ( aa_set_o_o(member_o((Z2)),insert_o((Ya),bot_bot_set_o))
             => ( aa_set_int_o(member_int(Yb),insert_int(Xa,bot_bot_set_int))
               => ( aa_int_o(aa_o_fun_int_o(R,(Z2)),Yb)
                <=> aa_int_o(aa_o_fun_int_o(Ra,(Z2)),Yb) ) ) )
         => ( aa_int_o(aa_o_fun_int_o(bNF_id6282831320590511060_int_o(R),(X)),Y)
          <=> aa_int_o(aa_o_fun_int_o(bNF_id6282831320590511060_int_o(Ra),(Ya)),Xa) ) ) ) ) ).

% ID.rel_cong
tff(fact_6428_ID_Orel__cong,axiom,
    ! [X: nat,Ya: nat,Y: $o,Xa: $o,R: fun_nat_fun_o_o,Ra: fun_nat_fun_o_o] :
      ( ( X = Ya )
     => ( ( (Y)
        <=> (Xa) )
       => ( ! [Z2: nat,Yb: $o] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z2),aa_set_nat_set_nat(insert_nat(Ya),bot_bot_set_nat))
             => ( aa_set_o_o(member_o((Yb)),insert_o((Xa),bot_bot_set_o))
               => ( aa_o_o(aa_nat_fun_o_o(R,Z2),(Yb))
                <=> aa_o_o(aa_nat_fun_o_o(Ra,Z2),(Yb)) ) ) )
         => ( aa_o_o(aa_nat_fun_o_o(bNF_id9145077949732159988un_o_o(R),X),(Y))
          <=> aa_o_o(aa_nat_fun_o_o(bNF_id9145077949732159988un_o_o(Ra),Ya),(Xa)) ) ) ) ) ).

% ID.rel_cong
tff(fact_6429_ID_Orel__cong,axiom,
    ! [X: nat,Ya: nat,Y: nat,Xa: nat,R: fun_nat_fun_nat_o,Ra: fun_nat_fun_nat_o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: nat,Yb: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z2),aa_set_nat_set_nat(insert_nat(Ya),bot_bot_set_nat))
             => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Yb),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat))
               => ( aa_nat_o(aa_nat_fun_nat_o(R,Z2),Yb)
                <=> aa_nat_o(aa_nat_fun_nat_o(Ra,Z2),Yb) ) ) )
         => ( aa_nat_o(aa_nat_fun_nat_o(bNF_id7359269443668684102_nat_o(R),X),Y)
          <=> aa_nat_o(aa_nat_fun_nat_o(bNF_id7359269443668684102_nat_o(Ra),Ya),Xa) ) ) ) ) ).

% ID.rel_cong
tff(fact_6430_ID_Orel__cong,axiom,
    ! [X: nat,Ya: nat,Y: int,Xa: int,R: fun_nat_fun_int_o,Ra: fun_nat_fun_int_o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: nat,Yb: int] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z2),aa_set_nat_set_nat(insert_nat(Ya),bot_bot_set_nat))
             => ( aa_set_int_o(member_int(Yb),insert_int(Xa,bot_bot_set_int))
               => ( aa_int_o(aa_nat_fun_int_o(R,Z2),Yb)
                <=> aa_int_o(aa_nat_fun_int_o(Ra,Z2),Yb) ) ) )
         => ( aa_int_o(aa_nat_fun_int_o(bNF_id5114124475998238186_int_o(R),X),Y)
          <=> aa_int_o(aa_nat_fun_int_o(bNF_id5114124475998238186_int_o(Ra),Ya),Xa) ) ) ) ) ).

% ID.rel_cong
tff(fact_6431_ID_Orel__cong,axiom,
    ! [X: int,Ya: int,Y: $o,Xa: $o,R: fun_int_fun_o_o,Ra: fun_int_fun_o_o] :
      ( ( X = Ya )
     => ( ( (Y)
        <=> (Xa) )
       => ( ! [Z2: int,Yb: $o] :
              ( aa_set_int_o(member_int(Z2),insert_int(Ya,bot_bot_set_int))
             => ( aa_set_o_o(member_o((Yb)),insert_o((Xa),bot_bot_set_o))
               => ( aa_o_o(aa_int_fun_o_o(R,Z2),(Yb))
                <=> aa_o_o(aa_int_fun_o_o(Ra,Z2),(Yb)) ) ) )
         => ( aa_o_o(aa_int_fun_o_o(bNF_id4800617119677697616un_o_o(R),X),(Y))
          <=> aa_o_o(aa_int_fun_o_o(bNF_id4800617119677697616un_o_o(Ra),Ya),(Xa)) ) ) ) ) ).

% ID.rel_cong
tff(fact_6432_ID_Orel__cong,axiom,
    ! [X: int,Ya: int,Y: nat,Xa: nat,R: fun_int_fun_nat_o,Ra: fun_int_fun_nat_o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: int,Yb: nat] :
              ( aa_set_int_o(member_int(Z2),insert_int(Ya,bot_bot_set_int))
             => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Yb),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat))
               => ( aa_nat_o(aa_int_fun_nat_o(R,Z2),Yb)
                <=> aa_nat_o(aa_int_fun_nat_o(Ra,Z2),Yb) ) ) )
         => ( aa_nat_o(aa_int_fun_nat_o(bNF_id8914055639145754602_nat_o(R),X),Y)
          <=> aa_nat_o(aa_int_fun_nat_o(bNF_id8914055639145754602_nat_o(Ra),Ya),Xa) ) ) ) ) ).

% ID.rel_cong
tff(fact_6433_ID_Orel__cong,axiom,
    ! [X: int,Ya: int,Y: int,Xa: int,R: fun_int_fun_int_o,Ra: fun_int_fun_int_o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: int,Yb: int] :
              ( aa_set_int_o(member_int(Z2),insert_int(Ya,bot_bot_set_int))
             => ( aa_set_int_o(member_int(Yb),insert_int(Xa,bot_bot_set_int))
               => ( aa_int_o(aa_int_fun_int_o(R,Z2),Yb)
                <=> aa_int_o(aa_int_fun_int_o(Ra,Z2),Yb) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(bNF_id6668910671475308686_int_o(R),X),Y)
          <=> aa_int_o(aa_int_fun_int_o(bNF_id6668910671475308686_int_o(Ra),Ya),Xa) ) ) ) ) ).

% ID.rel_cong
tff(fact_6434_ID_Orel__cong,axiom,
    ! [X: set_nat,Ya: set_nat,Y: $o,Xa: $o,R: fun_set_nat_fun_o_o,Ra: fun_set_nat_fun_o_o] :
      ( ( X = Ya )
     => ( ( (Y)
        <=> (Xa) )
       => ( ! [Z2: set_nat,Yb: $o] :
              ( aa_set_set_nat_o(member_set_nat(Z2),insert_set_nat(Ya,bot_bot_set_set_nat))
             => ( aa_set_o_o(member_o((Yb)),insert_o((Xa),bot_bot_set_o))
               => ( aa_o_o(aa_set_nat_fun_o_o(R,Z2),(Yb))
                <=> aa_o_o(aa_set_nat_fun_o_o(Ra,Z2),(Yb)) ) ) )
         => ( aa_o_o(aa_set_nat_fun_o_o(bNF_id1035930574019395754un_o_o(R),X),(Y))
          <=> aa_o_o(aa_set_nat_fun_o_o(bNF_id1035930574019395754un_o_o(Ra),Ya),(Xa)) ) ) ) ) ).

% ID.rel_cong
tff(fact_6435_ID_Orel__mono__strong,axiom,
    ! [R: fun_o_fun_o_o,X: $o,Y: $o,Ra: fun_o_fun_o_o] :
      ( aa_o_o(aa_o_fun_o_o(bNF_id6346756738013732234un_o_o(R),(X)),(Y))
     => ( ! [Z2: $o,Yb: $o] :
            ( aa_set_o_o(member_o((Z2)),insert_o((X),bot_bot_set_o))
           => ( aa_set_o_o(member_o((Yb)),insert_o((Y),bot_bot_set_o))
             => ( aa_o_o(aa_o_fun_o_o(R,(Z2)),(Yb))
               => aa_o_o(aa_o_fun_o_o(Ra,(Z2)),(Yb)) ) ) )
       => aa_o_o(aa_o_fun_o_o(bNF_id6346756738013732234un_o_o(Ra),(X)),(Y)) ) ) ).

% ID.rel_mono_strong
tff(fact_6436_ID_Orel__mono__strong,axiom,
    ! [R: fun_o_fun_nat_o,X: $o,Y: nat,Ra: fun_o_fun_nat_o] :
      ( aa_nat_o(aa_o_fun_nat_o(bNF_id8527976288260956976_nat_o(R),(X)),Y)
     => ( ! [Z2: $o,Yb: nat] :
            ( aa_set_o_o(member_o((Z2)),insert_o((X),bot_bot_set_o))
           => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Yb),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat))
             => ( aa_nat_o(aa_o_fun_nat_o(R,(Z2)),Yb)
               => aa_nat_o(aa_o_fun_nat_o(Ra,(Z2)),Yb) ) ) )
       => aa_nat_o(aa_o_fun_nat_o(bNF_id8527976288260956976_nat_o(Ra),(X)),Y) ) ) ).

% ID.rel_mono_strong
tff(fact_6437_ID_Orel__mono__strong,axiom,
    ! [R: fun_o_fun_int_o,X: $o,Y: int,Ra: fun_o_fun_int_o] :
      ( aa_int_o(aa_o_fun_int_o(bNF_id6282831320590511060_int_o(R),(X)),Y)
     => ( ! [Z2: $o,Yb: int] :
            ( aa_set_o_o(member_o((Z2)),insert_o((X),bot_bot_set_o))
           => ( aa_set_int_o(member_int(Yb),insert_int(Y,bot_bot_set_int))
             => ( aa_int_o(aa_o_fun_int_o(R,(Z2)),Yb)
               => aa_int_o(aa_o_fun_int_o(Ra,(Z2)),Yb) ) ) )
       => aa_int_o(aa_o_fun_int_o(bNF_id6282831320590511060_int_o(Ra),(X)),Y) ) ) ).

% ID.rel_mono_strong
tff(fact_6438_ID_Orel__mono__strong,axiom,
    ! [R: fun_nat_fun_o_o,X: nat,Y: $o,Ra: fun_nat_fun_o_o] :
      ( aa_o_o(aa_nat_fun_o_o(bNF_id9145077949732159988un_o_o(R),X),(Y))
     => ( ! [Z2: nat,Yb: $o] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))
           => ( aa_set_o_o(member_o((Yb)),insert_o((Y),bot_bot_set_o))
             => ( aa_o_o(aa_nat_fun_o_o(R,Z2),(Yb))
               => aa_o_o(aa_nat_fun_o_o(Ra,Z2),(Yb)) ) ) )
       => aa_o_o(aa_nat_fun_o_o(bNF_id9145077949732159988un_o_o(Ra),X),(Y)) ) ) ).

% ID.rel_mono_strong
tff(fact_6439_ID_Orel__mono__strong,axiom,
    ! [R: fun_nat_fun_nat_o,X: nat,Y: nat,Ra: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(bNF_id7359269443668684102_nat_o(R),X),Y)
     => ( ! [Z2: nat,Yb: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))
           => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Yb),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat))
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Z2),Yb)
               => aa_nat_o(aa_nat_fun_nat_o(Ra,Z2),Yb) ) ) )
       => aa_nat_o(aa_nat_fun_nat_o(bNF_id7359269443668684102_nat_o(Ra),X),Y) ) ) ).

% ID.rel_mono_strong
tff(fact_6440_ID_Orel__mono__strong,axiom,
    ! [R: fun_nat_fun_int_o,X: nat,Y: int,Ra: fun_nat_fun_int_o] :
      ( aa_int_o(aa_nat_fun_int_o(bNF_id5114124475998238186_int_o(R),X),Y)
     => ( ! [Z2: nat,Yb: int] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))
           => ( aa_set_int_o(member_int(Yb),insert_int(Y,bot_bot_set_int))
             => ( aa_int_o(aa_nat_fun_int_o(R,Z2),Yb)
               => aa_int_o(aa_nat_fun_int_o(Ra,Z2),Yb) ) ) )
       => aa_int_o(aa_nat_fun_int_o(bNF_id5114124475998238186_int_o(Ra),X),Y) ) ) ).

% ID.rel_mono_strong
tff(fact_6441_ID_Orel__mono__strong,axiom,
    ! [R: fun_int_fun_o_o,X: int,Y: $o,Ra: fun_int_fun_o_o] :
      ( aa_o_o(aa_int_fun_o_o(bNF_id4800617119677697616un_o_o(R),X),(Y))
     => ( ! [Z2: int,Yb: $o] :
            ( aa_set_int_o(member_int(Z2),insert_int(X,bot_bot_set_int))
           => ( aa_set_o_o(member_o((Yb)),insert_o((Y),bot_bot_set_o))
             => ( aa_o_o(aa_int_fun_o_o(R,Z2),(Yb))
               => aa_o_o(aa_int_fun_o_o(Ra,Z2),(Yb)) ) ) )
       => aa_o_o(aa_int_fun_o_o(bNF_id4800617119677697616un_o_o(Ra),X),(Y)) ) ) ).

% ID.rel_mono_strong
tff(fact_6442_ID_Orel__mono__strong,axiom,
    ! [R: fun_int_fun_nat_o,X: int,Y: nat,Ra: fun_int_fun_nat_o] :
      ( aa_nat_o(aa_int_fun_nat_o(bNF_id8914055639145754602_nat_o(R),X),Y)
     => ( ! [Z2: int,Yb: nat] :
            ( aa_set_int_o(member_int(Z2),insert_int(X,bot_bot_set_int))
           => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Yb),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat))
             => ( aa_nat_o(aa_int_fun_nat_o(R,Z2),Yb)
               => aa_nat_o(aa_int_fun_nat_o(Ra,Z2),Yb) ) ) )
       => aa_nat_o(aa_int_fun_nat_o(bNF_id8914055639145754602_nat_o(Ra),X),Y) ) ) ).

% ID.rel_mono_strong
tff(fact_6443_ID_Orel__mono__strong,axiom,
    ! [R: fun_int_fun_int_o,X: int,Y: int,Ra: fun_int_fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(bNF_id6668910671475308686_int_o(R),X),Y)
     => ( ! [Z2: int,Yb: int] :
            ( aa_set_int_o(member_int(Z2),insert_int(X,bot_bot_set_int))
           => ( aa_set_int_o(member_int(Yb),insert_int(Y,bot_bot_set_int))
             => ( aa_int_o(aa_int_fun_int_o(R,Z2),Yb)
               => aa_int_o(aa_int_fun_int_o(Ra,Z2),Yb) ) ) )
       => aa_int_o(aa_int_fun_int_o(bNF_id6668910671475308686_int_o(Ra),X),Y) ) ) ).

% ID.rel_mono_strong
tff(fact_6444_ID_Orel__mono__strong,axiom,
    ! [R: fun_set_nat_fun_o_o,X: set_nat,Y: $o,Ra: fun_set_nat_fun_o_o] :
      ( aa_o_o(aa_set_nat_fun_o_o(bNF_id1035930574019395754un_o_o(R),X),(Y))
     => ( ! [Z2: set_nat,Yb: $o] :
            ( aa_set_set_nat_o(member_set_nat(Z2),insert_set_nat(X,bot_bot_set_set_nat))
           => ( aa_set_o_o(member_o((Yb)),insert_o((Y),bot_bot_set_o))
             => ( aa_o_o(aa_set_nat_fun_o_o(R,Z2),(Yb))
               => aa_o_o(aa_set_nat_fun_o_o(Ra,Z2),(Yb)) ) ) )
       => aa_o_o(aa_set_nat_fun_o_o(bNF_id1035930574019395754un_o_o(Ra),X),(Y)) ) ) ).

% ID.rel_mono_strong
tff(fact_6445_ID_Orel__refl__strong,axiom,
    ! [X: produc3843707927480180839at_nat,Ra: fun_Pr5947183769845927726_nat_o] :
      ( ! [Z2: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Z2),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))
         => aa_Pro5947980084102686253_nat_o(aa_Pro1894297876208389727_nat_o(Ra,Z2),Z2) )
     => aa_Pro5947980084102686253_nat_o(aa_Pro1894297876208389727_nat_o(bNF_id8101293743253779274_nat_o(Ra),X),X) ) ).

% ID.rel_refl_strong
tff(fact_6446_ID_Orel__refl__strong,axiom,
    ! [X: set_Pr958786334691620121nt_int,Ra: fun_se2092758488442489134_int_o] :
      ( ! [Z2: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z2),insert8897473484851387113nt_int(X,bot_bo1488462491386950373nt_int))
         => aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(Ra,Z2),Z2) )
     => aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(bNF_id818620265798320458_int_o(Ra),X),X) ) ).

% ID.rel_refl_strong
tff(fact_6447_ID_Orel__refl__strong,axiom,
    ! [X: set_nat,Ra: fun_se2170385732985038782_nat_o] :
      ( ! [Z2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(Z2),insert_set_nat(X,bot_bot_set_set_nat))
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(Ra,Z2),Z2) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(bNF_id2203958827186076122_nat_o(Ra),X),X) ) ).

% ID.rel_refl_strong
tff(fact_6448_ID_Orel__refl__strong,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Ra: fun_fu4164567611050918574at_o_o] :
      ( ! [Z2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Z2),insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o))
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(Ra,Z2),Z2) )
     => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(bNF_id5330520648461660874at_o_o(Ra),X),X) ) ).

% ID.rel_refl_strong
tff(fact_6449_ID_Orel__refl__strong,axiom,
    ! [X: product_prod_nat_nat,Ra: fun_Pr2028404973800837934_nat_o] :
      ( ! [Z2: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))
         => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Ra,Z2),Z2) )
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(bNF_id523184180544252746_nat_o(Ra),X),X) ) ).

% ID.rel_refl_strong
tff(fact_6450_ID_Orel__refl__strong,axiom,
    ! [X: $o,Ra: fun_o_fun_o_o] :
      ( ! [Z2: $o] :
          ( aa_set_o_o(member_o((Z2)),insert_o((X),bot_bot_set_o))
         => aa_o_o(aa_o_fun_o_o(Ra,(Z2)),(Z2)) )
     => aa_o_o(aa_o_fun_o_o(bNF_id6346756738013732234un_o_o(Ra),(X)),(X)) ) ).

% ID.rel_refl_strong
tff(fact_6451_ID_Orel__refl__strong,axiom,
    ! [X: nat,Ra: fun_nat_fun_nat_o] :
      ( ! [Z2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))
         => aa_nat_o(aa_nat_fun_nat_o(Ra,Z2),Z2) )
     => aa_nat_o(aa_nat_fun_nat_o(bNF_id7359269443668684102_nat_o(Ra),X),X) ) ).

% ID.rel_refl_strong
tff(fact_6452_ID_Orel__refl__strong,axiom,
    ! [X: int,Ra: fun_int_fun_int_o] :
      ( ! [Z2: int] :
          ( aa_set_int_o(member_int(Z2),insert_int(X,bot_bot_set_int))
         => aa_int_o(aa_int_fun_int_o(Ra,Z2),Z2) )
     => aa_int_o(aa_int_fun_int_o(bNF_id6668910671475308686_int_o(Ra),X),X) ) ).

% ID.rel_refl_strong
tff(fact_6453_singleton__inject,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) = insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_6454_singleton__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_6455_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( insert_o((A),bot_bot_set_o) = insert_o((B),bot_bot_set_o) )
     => ( (A)
      <=> (B) ) ) ).

% singleton_inject
tff(fact_6456_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) = aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_6457_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( insert_int(A,bot_bot_set_int) = insert_int(B,bot_bot_set_int) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_6458_insert__not__empty,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] : ( insert9069300056098147895at_nat(A,A2) != bot_bo228742789529271731at_nat ) ).

% insert_not_empty
tff(fact_6459_insert__not__empty,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : ( insert8211810215607154385at_nat(A,A2) != bot_bo2099793752762293965at_nat ) ).

% insert_not_empty
tff(fact_6460_insert__not__empty,axiom,
    ! [A: $o,A2: set_o] : ( insert_o((A),A2) != bot_bot_set_o ) ).

% insert_not_empty
tff(fact_6461_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] : ( aa_set_nat_set_nat(insert_nat(A),A2) != bot_bot_set_nat ) ).

% insert_not_empty
tff(fact_6462_insert__not__empty,axiom,
    ! [A: int,A2: set_int] : ( insert_int(A,A2) != bot_bot_set_int ) ).

% insert_not_empty
tff(fact_6463_doubleton__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat,D2: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)) = insert9069300056098147895at_nat(C,insert9069300056098147895at_nat(D2,bot_bo228742789529271731at_nat)) )
    <=> ( ( ( A = C )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_6464_doubleton__eq__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D2: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)) = insert8211810215607154385at_nat(C,insert8211810215607154385at_nat(D2,bot_bo2099793752762293965at_nat)) )
    <=> ( ( ( A = C )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_6465_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C: $o,D2: $o] :
      ( ( insert_o((A),insert_o((B),bot_bot_set_o)) = insert_o((C),insert_o((D2),bot_bot_set_o)) )
    <=> ( ( ( (A)
          <=> (C) )
          & ( (B)
          <=> (D2) ) )
        | ( ( (A)
          <=> (D2) )
          & ( (B)
          <=> (C) ) ) ) ) ).

% doubleton_eq_iff
tff(fact_6466_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)) = aa_set_nat_set_nat(insert_nat(C),aa_set_nat_set_nat(insert_nat(D2),bot_bot_set_nat)) )
    <=> ( ( ( A = C )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_6467_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( insert_int(A,insert_int(B,bot_bot_set_int)) = insert_int(C,insert_int(D2,bot_bot_set_int)) )
    <=> ( ( ( A = C )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_6468_singleton__iff,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_6469_singleton__iff,axiom,
    ! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(B),insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_6470_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(B),insert_set_nat(A,bot_bot_set_set_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_6471_singleton__iff,axiom,
    ! [B: fun_Pr6460944699146269935_nat_o,A: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),insert4923026906511862463_nat_o(A,bot_bo8632255151191677115_nat_o))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_6472_singleton__iff,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_6473_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( aa_set_o_o(member_o((B)),insert_o((A),bot_bot_set_o))
    <=> ( (B)
      <=> (A) ) ) ).

% singleton_iff
tff(fact_6474_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_6475_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( aa_set_int_o(member_int(B),insert_int(A,bot_bot_set_int))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_6476_singletonD,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_6477_singletonD,axiom,
    ! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(B),insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int))
     => ( B = A ) ) ).

% singletonD
tff(fact_6478_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(B),insert_set_nat(A,bot_bot_set_set_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_6479_singletonD,axiom,
    ! [B: fun_Pr6460944699146269935_nat_o,A: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),insert4923026906511862463_nat_o(A,bot_bo8632255151191677115_nat_o))
     => ( B = A ) ) ).

% singletonD
tff(fact_6480_singletonD,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_6481_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( aa_set_o_o(member_o((B)),insert_o((A),bot_bot_set_o))
     => ( (B)
      <=> (A) ) ) ).

% singletonD
tff(fact_6482_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_6483_singletonD,axiom,
    ! [B: int,A: int] :
      ( aa_set_int_o(member_int(B),insert_int(A,bot_bot_set_int))
     => ( B = A ) ) ).

% singletonD
tff(fact_6484_flip__bit__int__def,axiom,
    ! [N: nat,K: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),K) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) ) ).

% flip_bit_int_def
tff(fact_6485_insert__UNIV,axiom,
    ! [X: produc3843707927480180839at_nat] : ( insert9069300056098147895at_nat(X,top_to6833984726390702231at_nat) = top_to6833984726390702231at_nat ) ).

% insert_UNIV
tff(fact_6486_insert__UNIV,axiom,
    ! [X: product_prod_nat_nat] : ( insert8211810215607154385at_nat(X,top_to4669805908274784177at_nat) = top_to4669805908274784177at_nat ) ).

% insert_UNIV
tff(fact_6487_insert__UNIV,axiom,
    ! [X: list_nat] : ( insert_list_nat(X,top_top_set_list_nat) = top_top_set_list_nat ) ).

% insert_UNIV
tff(fact_6488_insert__UNIV,axiom,
    ! [X: $o] : ( insert_o((X),top_top_set_o) = top_top_set_o ) ).

% insert_UNIV
tff(fact_6489_insert__UNIV,axiom,
    ! [X: rat] : ( insert_rat(X,top_top_set_rat) = top_top_set_rat ) ).

% insert_UNIV
tff(fact_6490_insert__UNIV,axiom,
    ! [X: nat] : ( aa_set_nat_set_nat(insert_nat(X),top_top_set_nat) = top_top_set_nat ) ).

% insert_UNIV
tff(fact_6491_insert__UNIV,axiom,
    ! [X: int] : ( insert_int(X,top_top_set_int) = top_top_set_int ) ).

% insert_UNIV
tff(fact_6492_Int__insert__right,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),insert9069300056098147895at_nat(A,B2)) = $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A2),insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2)) ) ).

% Int_insert_right
tff(fact_6493_Int__insert__right,axiom,
    ! [A2: set_o,A: $o,B2: set_o] :
      ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),insert_o((A),B2)) = $ite(aa_set_o_o(member_o((A)),A2),insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2)) ) ).

% Int_insert_right
tff(fact_6494_Int__insert__right,axiom,
    ! [A2: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int,B2: set_se6260736226359567993nt_int] :
      ( inf_in8396524679539076455nt_int(A2,insert8897473484851387113nt_int(A,B2)) = $ite(aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A2),insert8897473484851387113nt_int(A,inf_in8396524679539076455nt_int(A2,B2)),inf_in8396524679539076455nt_int(A2,B2)) ) ).

% Int_insert_right
tff(fact_6495_Int__insert__right,axiom,
    ! [A2: set_set_nat,A: set_nat,B2: set_set_nat] :
      ( inf_inf_set_set_nat(A2,insert_set_nat(A,B2)) = $ite(aa_set_set_nat_o(member_set_nat(A),A2),insert_set_nat(A,inf_inf_set_set_nat(A2,B2)),inf_inf_set_set_nat(A2,B2)) ) ).

% Int_insert_right
tff(fact_6496_Int__insert__right,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),insert_int(A,B2)) = $ite(aa_set_int_o(member_int(A),A2),insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) ) ).

% Int_insert_right
tff(fact_6497_Int__insert__right,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,A: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( inf_in3376754634429691197_nat_o(A2,insert4923026906511862463_nat_o(A,B2)) = $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2),insert4923026906511862463_nat_o(A,inf_in3376754634429691197_nat_o(A2,B2)),inf_in3376754634429691197_nat_o(A2,B2)) ) ).

% Int_insert_right
tff(fact_6498_Int__insert__right,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(A,B2)) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2),insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ) ).

% Int_insert_right
tff(fact_6499_Int__insert__right,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),B2)) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A2),aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ).

% Int_insert_right
tff(fact_6500_Int__insert__left,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,B2)),C2) = $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),C2),insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),C2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),C2)) ) ).

% Int_insert_left
tff(fact_6501_Int__insert__left,axiom,
    ! [A: $o,B2: set_o,C2: set_o] :
      ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,insert_o((A),B2)),C2) = $ite(aa_set_o_o(member_o((A)),C2),insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B2),C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B2),C2)) ) ).

% Int_insert_left
tff(fact_6502_Int__insert__left,axiom,
    ! [A: set_Pr958786334691620121nt_int,B2: set_se6260736226359567993nt_int,C2: set_se6260736226359567993nt_int] :
      ( inf_in8396524679539076455nt_int(insert8897473484851387113nt_int(A,B2),C2) = $ite(aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),C2),insert8897473484851387113nt_int(A,inf_in8396524679539076455nt_int(B2,C2)),inf_in8396524679539076455nt_int(B2,C2)) ) ).

% Int_insert_left
tff(fact_6503_Int__insert__left,axiom,
    ! [A: set_nat,B2: set_set_nat,C2: set_set_nat] :
      ( inf_inf_set_set_nat(insert_set_nat(A,B2),C2) = $ite(aa_set_set_nat_o(member_set_nat(A),C2),insert_set_nat(A,inf_inf_set_set_nat(B2,C2)),inf_inf_set_set_nat(B2,C2)) ) ).

% Int_insert_left
tff(fact_6504_Int__insert__left,axiom,
    ! [A: int,B2: set_int,C2: set_int] :
      ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,B2)),C2) = $ite(aa_set_int_o(member_int(A),C2),insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2)) ) ).

% Int_insert_left
tff(fact_6505_Int__insert__left,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o,C2: set_fu3555266933746557007_nat_o] :
      ( inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,B2),C2) = $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),C2),insert4923026906511862463_nat_o(A,inf_in3376754634429691197_nat_o(B2,C2)),inf_in3376754634429691197_nat_o(B2,C2)) ) ).

% Int_insert_left
tff(fact_6506_Int__insert__left,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B2)),C2) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),C2),insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ) ).

% Int_insert_left
tff(fact_6507_Int__insert__left,axiom,
    ! [A: nat,B2: set_nat,C2: set_nat] :
      ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(insert_nat(A),B2)),C2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),C2),aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ) ).

% Int_insert_left
tff(fact_6508_flip__bit__nat__def,axiom,
    ! [M: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ) ).

% flip_bit_nat_def
tff(fact_6509_push__bit__minus,axiom,
    ! [N: nat,A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),A)) ) ).

% push_bit_minus
tff(fact_6510_push__bit__minus,axiom,
    ! [N: nat,A: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),aa_int_int(uminus_uminus_int,A)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),A)) ) ).

% push_bit_minus
tff(fact_6511_Collect__conv__if2,axiom,
    ! [A: produc3843707927480180839at_nat,P: fun_Pr3059359313727237556_nat_o] :
      ( collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_jg(A),P)) = $ite(aa_Pro5947980084102686253_nat_o(P,A),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat),bot_bo228742789529271731at_nat) ) ).

% Collect_conv_if2
tff(fact_6512_Collect__conv__if2,axiom,
    ! [A: set_Pr958786334691620121nt_int,P: fun_se7249947736802449410_int_o] :
      ( collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_jh(A),P)) = $ite(aa_set6170482117830564987_int_o(P,A),insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int),bot_bo1488462491386950373nt_int) ) ).

% Collect_conv_if2
tff(fact_6513_Collect__conv__if2,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ji(A),P)) = $ite(aa_set_nat_o(P,A),insert_set_nat(A,bot_bot_set_set_nat),bot_bot_set_set_nat) ) ).

% Collect_conv_if2
tff(fact_6514_Collect__conv__if2,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,P: fun_fu5135969500543783852at_o_o] :
      ( collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_jj(A),P)) = $ite(aa_fun1675921005893230629at_o_o(P,A),insert4923026906511862463_nat_o(A,bot_bo8632255151191677115_nat_o),bot_bo8632255151191677115_nat_o) ) ).

% Collect_conv_if2
tff(fact_6515_Collect__conv__if2,axiom,
    ! [A: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_jk(A),P)) = $ite(aa_Pro6150637604445964947_nat_o(P,A),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat),bot_bo2099793752762293965at_nat) ) ).

% Collect_conv_if2
tff(fact_6516_Collect__conv__if2,axiom,
    ! [A: $o,P: fun_o_o] :
      ( collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_jl((A)),P)) = $ite(aa_o_o(P,(A)),insert_o((A),bot_bot_set_o),bot_bot_set_o) ) ).

% Collect_conv_if2
tff(fact_6517_Collect__conv__if2,axiom,
    ! [A: nat,P: fun_nat_o] :
      ( collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_jm(A),P)) = $ite(aa_nat_o(P,A),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat),bot_bot_set_nat) ) ).

% Collect_conv_if2
tff(fact_6518_Collect__conv__if2,axiom,
    ! [A: int,P: fun_int_o] :
      ( collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_jn(A),P)) = $ite(aa_int_o(P,A),insert_int(A,bot_bot_set_int),bot_bot_set_int) ) ).

% Collect_conv_if2
tff(fact_6519_Collect__conv__if,axiom,
    ! [A: produc3843707927480180839at_nat,P: fun_Pr3059359313727237556_nat_o] :
      ( collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_jo(A),P)) = $ite(aa_Pro5947980084102686253_nat_o(P,A),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat),bot_bo228742789529271731at_nat) ) ).

% Collect_conv_if
tff(fact_6520_Collect__conv__if,axiom,
    ! [A: set_Pr958786334691620121nt_int,P: fun_se7249947736802449410_int_o] :
      ( collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_jp(A),P)) = $ite(aa_set6170482117830564987_int_o(P,A),insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int),bot_bo1488462491386950373nt_int) ) ).

% Collect_conv_if
tff(fact_6521_Collect__conv__if,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_jq(A),P)) = $ite(aa_set_nat_o(P,A),insert_set_nat(A,bot_bot_set_set_nat),bot_bot_set_set_nat) ) ).

% Collect_conv_if
tff(fact_6522_Collect__conv__if,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,P: fun_fu5135969500543783852at_o_o] :
      ( collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_jr(A),P)) = $ite(aa_fun1675921005893230629at_o_o(P,A),insert4923026906511862463_nat_o(A,bot_bo8632255151191677115_nat_o),bot_bo8632255151191677115_nat_o) ) ).

% Collect_conv_if
tff(fact_6523_Collect__conv__if,axiom,
    ! [A: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_js(A),P)) = $ite(aa_Pro6150637604445964947_nat_o(P,A),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat),bot_bo2099793752762293965at_nat) ) ).

% Collect_conv_if
tff(fact_6524_Collect__conv__if,axiom,
    ! [A: $o,P: fun_o_o] :
      ( collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_jt((A)),P)) = $ite(aa_o_o(P,(A)),insert_o((A),bot_bot_set_o),bot_bot_set_o) ) ).

% Collect_conv_if
tff(fact_6525_Collect__conv__if,axiom,
    ! [A: nat,P: fun_nat_o] :
      ( collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ju(A),P)) = $ite(aa_nat_o(P,A),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat),bot_bot_set_nat) ) ).

% Collect_conv_if
tff(fact_6526_Collect__conv__if,axiom,
    ! [A: int,P: fun_int_o] :
      ( collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_jv(A),P)) = $ite(aa_int_o(P,A),insert_int(A,bot_bot_set_int),bot_bot_set_int) ) ).

% Collect_conv_if
tff(fact_6527_insert__def,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( insert8211810215607154385at_nat(A,B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(aTP_Lamm_im,A))),B2) ) ).

% insert_def
tff(fact_6528_insert__def,axiom,
    ! [A: $o,B2: set_o] : ( insert_o((A),B2) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,collect_o(aa_o_fun_o_o(aTP_Lamm_in,(A)))),B2) ) ).

% insert_def
tff(fact_6529_insert__def,axiom,
    ! [A: set_Pr958786334691620121nt_int,B2: set_se6260736226359567993nt_int] : ( insert8897473484851387113nt_int(A,B2) = sup_su2047564715030645325nt_int(collec5210948495886036740nt_int(aa_set4993530631398190431_int_o(aTP_Lamm_ij,A)),B2) ) ).

% insert_def
tff(fact_6530_insert__def,axiom,
    ! [A: set_nat,B2: set_set_nat] : ( insert_set_nat(A,B2) = sup_sup_set_set_nat(collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_ik,A)),B2) ) ).

% insert_def
tff(fact_6531_insert__def,axiom,
    ! [A: int,B2: set_int] : ( insert_int(A,B2) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ip,A))),B2) ) ).

% insert_def
tff(fact_6532_insert__def,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o] : ( insert4923026906511862463_nat_o(A,B2) = sup_su2896084399781877795_nat_o(collec6407654713642994138_nat_o(aa_fun3380769697021226975at_o_o(aTP_Lamm_il,A)),B2) ) ).

% insert_def
tff(fact_6533_insert__def,axiom,
    ! [A: produc4166570645942440679at_nat,B2: set_Pr8551490117392284871at_nat] : ( insert6337962749363155127at_nat(A,B2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,collec5204685387357076818at_nat(aTP_Lamm_jw(A))),B2) ) ).

% insert_def
tff(fact_6534_insert__def,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : ( insert9069300056098147895at_nat(A,B2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,collec6321179662152712658at_nat(aa_Pro1894297876208389727_nat_o(aTP_Lamm_ii,A))),B2) ) ).

% insert_def
tff(fact_6535_insert__def,axiom,
    ! [A: nat,B2: set_nat] : ( aa_set_nat_set_nat(insert_nat(A),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_io,A))),B2) ) ).

% insert_def
tff(fact_6536_subset__singletonD,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))
     => ( ( A2 = bot_bo228742789529271731at_nat )
        | ( A2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ).

% subset_singletonD
tff(fact_6537_subset__singletonD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))
     => ( ( A2 = bot_bo2099793752762293965at_nat )
        | ( A2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singletonD
tff(fact_6538_subset__singletonD,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),insert_o((X),bot_bot_set_o))
     => ( ( A2 = bot_bot_set_o )
        | ( A2 = insert_o((X),bot_bot_set_o) ) ) ) ).

% subset_singletonD
tff(fact_6539_subset__singletonD,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),insert_int(X,bot_bot_set_int))
     => ( ( A2 = bot_bot_set_int )
        | ( A2 = insert_int(X,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_6540_subset__singletonD,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))
     => ( ( A2 = bot_bot_set_nat )
        | ( A2 = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_6541_subset__singleton__iff,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X5),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))
    <=> ( ( X5 = bot_bo228742789529271731at_nat )
        | ( X5 = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_6542_subset__singleton__iff,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X5),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))
    <=> ( ( X5 = bot_bo2099793752762293965at_nat )
        | ( X5 = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_6543_subset__singleton__iff,axiom,
    ! [X5: set_o,A: $o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,X5),insert_o((A),bot_bot_set_o))
    <=> ( ( X5 = bot_bot_set_o )
        | ( X5 = insert_o((A),bot_bot_set_o) ) ) ) ).

% subset_singleton_iff
tff(fact_6544_subset__singleton__iff,axiom,
    ! [X5: set_int,A: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X5),insert_int(A,bot_bot_set_int))
    <=> ( ( X5 = bot_bot_set_int )
        | ( X5 = insert_int(A,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_6545_subset__singleton__iff,axiom,
    ! [X5: set_nat,A: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X5),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))
    <=> ( ( X5 = bot_bot_set_nat )
        | ( X5 = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_6546_singleton__Un__iff,axiom,
    ! [X: produc4166570645942440679at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
      ( ( insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2) )
    <=> ( ( ( A2 = bot_bo8422036546324065075at_nat )
          & ( B2 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) ) )
        | ( ( A2 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) )
          & ( B2 = bot_bo8422036546324065075at_nat ) )
        | ( ( A2 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) )
          & ( B2 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_6547_singleton__Un__iff,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) )
    <=> ( ( ( A2 = bot_bo228742789529271731at_nat )
          & ( B2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) )
        | ( ( A2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B2 = bot_bo228742789529271731at_nat ) )
        | ( ( A2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_6548_singleton__Un__iff,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2) )
    <=> ( ( ( A2 = bot_bo2099793752762293965at_nat )
          & ( B2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) )
        | ( ( A2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B2 = bot_bo2099793752762293965at_nat ) )
        | ( ( A2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_6549_singleton__Un__iff,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ( insert_o((X),bot_bot_set_o) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B2) )
    <=> ( ( ( A2 = bot_bot_set_o )
          & ( B2 = insert_o((X),bot_bot_set_o) ) )
        | ( ( A2 = insert_o((X),bot_bot_set_o) )
          & ( B2 = bot_bot_set_o ) )
        | ( ( A2 = insert_o((X),bot_bot_set_o) )
          & ( B2 = insert_o((X),bot_bot_set_o) ) ) ) ) ).

% singleton_Un_iff
tff(fact_6550_singleton__Un__iff,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) )
    <=> ( ( ( A2 = bot_bot_set_nat )
          & ( B2 = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ) )
        | ( ( A2 = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) )
          & ( B2 = bot_bot_set_nat ) )
        | ( ( A2 = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) )
          & ( B2 = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_6551_singleton__Un__iff,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ( insert_int(X,bot_bot_set_int) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2) )
    <=> ( ( ( A2 = bot_bot_set_int )
          & ( B2 = insert_int(X,bot_bot_set_int) ) )
        | ( ( A2 = insert_int(X,bot_bot_set_int) )
          & ( B2 = bot_bot_set_int ) )
        | ( ( A2 = insert_int(X,bot_bot_set_int) )
          & ( B2 = insert_int(X,bot_bot_set_int) ) ) ) ) ).

% singleton_Un_iff
tff(fact_6552_Un__singleton__iff,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,X: produc4166570645942440679at_nat] :
      ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2) = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) )
    <=> ( ( ( A2 = bot_bo8422036546324065075at_nat )
          & ( B2 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) ) )
        | ( ( A2 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) )
          & ( B2 = bot_bo8422036546324065075at_nat ) )
        | ( ( A2 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) )
          & ( B2 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_6553_Un__singleton__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
    <=> ( ( ( A2 = bot_bo228742789529271731at_nat )
          & ( B2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) )
        | ( ( A2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B2 = bot_bo228742789529271731at_nat ) )
        | ( ( A2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_6554_Un__singleton__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2) = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
    <=> ( ( ( A2 = bot_bo2099793752762293965at_nat )
          & ( B2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) )
        | ( ( A2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B2 = bot_bo2099793752762293965at_nat ) )
        | ( ( A2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_6555_Un__singleton__iff,axiom,
    ! [A2: set_o,B2: set_o,X: $o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B2) = insert_o((X),bot_bot_set_o) )
    <=> ( ( ( A2 = bot_bot_set_o )
          & ( B2 = insert_o((X),bot_bot_set_o) ) )
        | ( ( A2 = insert_o((X),bot_bot_set_o) )
          & ( B2 = bot_bot_set_o ) )
        | ( ( A2 = insert_o((X),bot_bot_set_o) )
          & ( B2 = insert_o((X),bot_bot_set_o) ) ) ) ) ).

% Un_singleton_iff
tff(fact_6556_Un__singleton__iff,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) )
    <=> ( ( ( A2 = bot_bot_set_nat )
          & ( B2 = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ) )
        | ( ( A2 = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) )
          & ( B2 = bot_bot_set_nat ) )
        | ( ( A2 = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) )
          & ( B2 = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_6557_Un__singleton__iff,axiom,
    ! [A2: set_int,B2: set_int,X: int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2) = insert_int(X,bot_bot_set_int) )
    <=> ( ( ( A2 = bot_bot_set_int )
          & ( B2 = insert_int(X,bot_bot_set_int) ) )
        | ( ( A2 = insert_int(X,bot_bot_set_int) )
          & ( B2 = bot_bot_set_int ) )
        | ( ( A2 = insert_int(X,bot_bot_set_int) )
          & ( B2 = insert_int(X,bot_bot_set_int) ) ) ) ) ).

% Un_singleton_iff
tff(fact_6558_insert__is__Un,axiom,
    ! [A: produc4166570645942440679at_nat,A2: set_Pr8551490117392284871at_nat] : ( insert6337962749363155127at_nat(A,A2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,insert6337962749363155127at_nat(A,bot_bo8422036546324065075at_nat)),A2) ) ).

% insert_is_Un
tff(fact_6559_insert__is__Un,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] : ( insert9069300056098147895at_nat(A,A2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)),A2) ) ).

% insert_is_Un
tff(fact_6560_insert__is__Un,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : ( insert8211810215607154385at_nat(A,A2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)),A2) ) ).

% insert_is_Un
tff(fact_6561_insert__is__Un,axiom,
    ! [A: $o,A2: set_o] : ( insert_o((A),A2) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,insert_o((A),bot_bot_set_o)),A2) ) ).

% insert_is_Un
tff(fact_6562_insert__is__Un,axiom,
    ! [A: nat,A2: set_nat] : ( aa_set_nat_set_nat(insert_nat(A),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)),A2) ) ).

% insert_is_Un
tff(fact_6563_insert__is__Un,axiom,
    ! [A: int,A2: set_int] : ( insert_int(A,A2) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,insert_int(A,bot_bot_set_int)),A2) ) ).

% insert_is_Un
tff(fact_6564_set__minus__singleton__eq,axiom,
    ! [X: produc3843707927480180839at_nat,X5: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),X5)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,X5),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_6565_set__minus__singleton__eq,axiom,
    ! [X: set_Pr958786334691620121nt_int,X5: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),X5)
     => ( minus_2612819937483484256nt_int(X5,insert8897473484851387113nt_int(X,bot_bo1488462491386950373nt_int)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_6566_set__minus__singleton__eq,axiom,
    ! [X: set_nat,X5: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(X),X5)
     => ( minus_2163939370556025621et_nat(X5,insert_set_nat(X,bot_bot_set_set_nat)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_6567_set__minus__singleton__eq,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,X5: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),X5)
     => ( minus_6456595905716665654_nat_o(X5,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_6568_set__minus__singleton__eq,axiom,
    ! [X: product_prod_nat_nat,X5: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),X5)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,X5),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_6569_set__minus__singleton__eq,axiom,
    ! [X: $o,X5: set_o] :
      ( ~ aa_set_o_o(member_o((X)),X5)
     => ( minus_minus_set_o(X5,insert_o((X),bot_bot_set_o)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_6570_set__minus__singleton__eq,axiom,
    ! [X: int,X5: set_int] :
      ( ~ aa_set_int_o(member_int(X),X5)
     => ( minus_minus_set_int(X5,insert_int(X,bot_bot_set_int)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_6571_set__minus__singleton__eq,axiom,
    ! [X: nat,X5: set_nat] :
      ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),X5)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X5),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_6572_insert__minus__eq,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( X != Y )
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,insert9069300056098147895at_nat(X,A2)),insert9069300056098147895at_nat(Y,bot_bo228742789529271731at_nat)) = insert9069300056098147895at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),insert9069300056098147895at_nat(Y,bot_bo228742789529271731at_nat))) ) ) ).

% insert_minus_eq
tff(fact_6573_insert__minus__eq,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( X != Y )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A2)),insert8211810215607154385at_nat(Y,bot_bo2099793752762293965at_nat)) = insert8211810215607154385at_nat(X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),insert8211810215607154385at_nat(Y,bot_bo2099793752762293965at_nat))) ) ) ).

% insert_minus_eq
tff(fact_6574_insert__minus__eq,axiom,
    ! [X: $o,Y: $o,A2: set_o] :
      ( ~ ( (X)
        <=> (Y) )
     => ( minus_minus_set_o(insert_o((X),A2),insert_o((Y),bot_bot_set_o)) = insert_o((X),minus_minus_set_o(A2,insert_o((Y),bot_bot_set_o))) ) ) ).

% insert_minus_eq
tff(fact_6575_insert__minus__eq,axiom,
    ! [X: int,Y: int,A2: set_int] :
      ( ( X != Y )
     => ( minus_minus_set_int(insert_int(X,A2),insert_int(Y,bot_bot_set_int)) = insert_int(X,minus_minus_set_int(A2,insert_int(Y,bot_bot_set_int))) ) ) ).

% insert_minus_eq
tff(fact_6576_insert__minus__eq,axiom,
    ! [X: nat,Y: nat,A2: set_nat] :
      ( ( X != Y )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(insert_nat(X),A2)),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat)) = aa_set_nat_set_nat(insert_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat))) ) ) ).

% insert_minus_eq
tff(fact_6577_Diff__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),insert9069300056098147895at_nat(A,B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),B2)),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)) ) ).

% Diff_insert
tff(fact_6578_Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),insert8211810215607154385at_nat(A,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)) ) ).

% Diff_insert
tff(fact_6579_Diff__insert,axiom,
    ! [A2: set_o,A: $o,B2: set_o] : ( minus_minus_set_o(A2,insert_o((A),B2)) = minus_minus_set_o(minus_minus_set_o(A2,B2),insert_o((A),bot_bot_set_o)) ) ).

% Diff_insert
tff(fact_6580_Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] : ( minus_minus_set_int(A2,insert_int(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,B2),insert_int(A,bot_bot_set_int)) ) ).

% Diff_insert
tff(fact_6581_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)) ) ).

% Diff_insert
tff(fact_6582_insert__Diff,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A2)
     => ( insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_6583_insert__Diff,axiom,
    ! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A2)
     => ( insert8897473484851387113nt_int(A,minus_2612819937483484256nt_int(A2,insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int))) = A2 ) ) ).

% insert_Diff
tff(fact_6584_insert__Diff,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ( insert_set_nat(A,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_6585_insert__Diff,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2)
     => ( insert4923026906511862463_nat_o(A,minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(A,bot_bo8632255151191677115_nat_o))) = A2 ) ) ).

% insert_Diff
tff(fact_6586_insert__Diff,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2)
     => ( insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_6587_insert__Diff,axiom,
    ! [A: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ( insert_o((A),minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = A2 ) ) ).

% insert_Diff
tff(fact_6588_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = A2 ) ) ).

% insert_Diff
tff(fact_6589_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A2)
     => ( aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_6590_Diff__insert2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),insert9069300056098147895at_nat(A,B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))),B2) ) ).

% Diff_insert2
tff(fact_6591_Diff__insert2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),insert8211810215607154385at_nat(A,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))),B2) ) ).

% Diff_insert2
tff(fact_6592_Diff__insert2,axiom,
    ! [A2: set_o,A: $o,B2: set_o] : ( minus_minus_set_o(A2,insert_o((A),B2)) = minus_minus_set_o(minus_minus_set_o(A2,insert_o((A),bot_bot_set_o)),B2) ) ).

% Diff_insert2
tff(fact_6593_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B2: set_int] : ( minus_minus_set_int(A2,insert_int(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,insert_int(A,bot_bot_set_int)),B2) ) ).

% Diff_insert2
tff(fact_6594_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))),B2) ) ).

% Diff_insert2
tff(fact_6595_Diff__insert__absorb,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,insert9069300056098147895at_nat(X,A2)),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_6596_Diff__insert__absorb,axiom,
    ! [X: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A2)
     => ( minus_2612819937483484256nt_int(insert8897473484851387113nt_int(X,A2),insert8897473484851387113nt_int(X,bot_bo1488462491386950373nt_int)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_6597_Diff__insert__absorb,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(X),A2)
     => ( minus_2163939370556025621et_nat(insert_set_nat(X,A2),insert_set_nat(X,bot_bot_set_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_6598_Diff__insert__absorb,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
     => ( minus_6456595905716665654_nat_o(insert4923026906511862463_nat_o(X,A2),insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_6599_Diff__insert__absorb,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A2)),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_6600_Diff__insert__absorb,axiom,
    ! [X: $o,A2: set_o] :
      ( ~ aa_set_o_o(member_o((X)),A2)
     => ( minus_minus_set_o(insert_o((X),A2),insert_o((X),bot_bot_set_o)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_6601_Diff__insert__absorb,axiom,
    ! [X: int,A2: set_int] :
      ( ~ aa_set_int_o(member_int(X),A2)
     => ( minus_minus_set_int(insert_int(X,A2),insert_int(X,bot_bot_set_int)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_6602_Diff__insert__absorb,axiom,
    ! [X: nat,A2: set_nat] :
      ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(insert_nat(X),A2)),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_6603_vimage__singleton__eq,axiom,
    ! [A: nat,F: fun_nat_o,B: $o] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),vimage_nat_o(F,insert_o((B),bot_bot_set_o)))
    <=> ( aa_nat_o(F,A)
      <=> (B) ) ) ).

% vimage_singleton_eq
tff(fact_6604_vimage__singleton__eq,axiom,
    ! [A: int,F: fun_int_o,B: $o] :
      ( aa_set_int_o(member_int(A),vimage_int_o(F,insert_o((B),bot_bot_set_o)))
    <=> ( aa_int_o(F,A)
      <=> (B) ) ) ).

% vimage_singleton_eq
tff(fact_6605_vimage__singleton__eq,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)))
    <=> ( aa_nat_nat(F,A) = B ) ) ).

% vimage_singleton_eq
tff(fact_6606_vimage__singleton__eq,axiom,
    ! [A: int,F: fun_int_nat,B: nat] :
      ( aa_set_int_o(member_int(A),vimage_int_nat(F,aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)))
    <=> ( aa_int_nat(F,A) = B ) ) ).

% vimage_singleton_eq
tff(fact_6607_vimage__singleton__eq,axiom,
    ! [A: nat,F: fun_nat_int,B: int] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),vimage_nat_int(F,insert_int(B,bot_bot_set_int)))
    <=> ( aa_nat_int(F,A) = B ) ) ).

% vimage_singleton_eq
tff(fact_6608_vimage__singleton__eq,axiom,
    ! [A: int,F: fun_int_int,B: int] :
      ( aa_set_int_o(member_int(A),vimage_int_int(F,insert_int(B,bot_bot_set_int)))
    <=> ( aa_int_int(F,A) = B ) ) ).

% vimage_singleton_eq
tff(fact_6609_vimage__singleton__eq,axiom,
    ! [A: set_nat,F: fun_set_nat_o,B: $o] :
      ( aa_set_set_nat_o(member_set_nat(A),vimage_set_nat_o(F,insert_o((B),bot_bot_set_o)))
    <=> ( aa_set_nat_o(F,A)
      <=> (B) ) ) ).

% vimage_singleton_eq
tff(fact_6610_vimage__singleton__eq,axiom,
    ! [A: set_nat,F: fun_set_nat_nat,B: nat] :
      ( aa_set_set_nat_o(member_set_nat(A),vimage_set_nat_nat(F,aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)))
    <=> ( aa_set_nat_nat(F,A) = B ) ) ).

% vimage_singleton_eq
tff(fact_6611_vimage__singleton__eq,axiom,
    ! [A: set_nat,F: fun_set_nat_int,B: int] :
      ( aa_set_set_nat_o(member_set_nat(A),vimage_set_nat_int(F,insert_int(B,bot_bot_set_int)))
    <=> ( aa_set_nat_int(F,A) = B ) ) ).

% vimage_singleton_eq
tff(fact_6612_vimage__singleton__eq,axiom,
    ! [A: nat,F: fun_na1476630412504255114at_nat,B: product_prod_nat_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),vimage8013328719654469172at_nat(F,insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)))
    <=> ( aa_nat7603221163087113075at_nat(F,A) = B ) ) ).

% vimage_singleton_eq
tff(fact_6613_subset__insert__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(X,B2))
    <=> $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A2),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B2),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),B2)) ) ).

% subset_insert_iff
tff(fact_6614_subset__insert__iff,axiom,
    ! [A2: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A2),insert8897473484851387113nt_int(X,B2))
    <=> $ite(aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A2),aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(minus_2612819937483484256nt_int(A2,insert8897473484851387113nt_int(X,bot_bo1488462491386950373nt_int))),B2),aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A2),B2)) ) ).

% subset_insert_iff
tff(fact_6615_subset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),insert_set_nat(X,B2))
    <=> $ite(aa_set_set_nat_o(member_set_nat(X),A2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))),B2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)) ) ).

% subset_insert_iff
tff(fact_6616_subset__insert__iff,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),insert4923026906511862463_nat_o(X,B2))
    <=> $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2),aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o))),B2),aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),B2)) ) ).

% subset_insert_iff
tff(fact_6617_subset__insert__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),insert8211810215607154385at_nat(X,B2))
    <=> $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2),aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2),aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),B2)) ) ).

% subset_insert_iff
tff(fact_6618_subset__insert__iff,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),insert_o((X),B2))
    <=> $ite(aa_set_o_o(member_o((X)),A2),aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))),B2),aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),B2)) ) ).

% subset_insert_iff
tff(fact_6619_subset__insert__iff,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),insert_int(X,B2))
    <=> $ite(aa_set_int_o(member_int(X),A2),aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B2),aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),B2)) ) ).

% subset_insert_iff
tff(fact_6620_subset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),B2))
    <=> $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))),B2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)) ) ).

% subset_insert_iff
tff(fact_6621_Diff__single__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B2)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(X,B2)) ) ).

% Diff_single_insert
tff(fact_6622_Diff__single__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),insert8211810215607154385at_nat(X,B2)) ) ).

% Diff_single_insert
tff(fact_6623_Diff__single__insert,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))),B2)
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),insert_o((X),B2)) ) ).

% Diff_single_insert
tff(fact_6624_Diff__single__insert,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B2)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),insert_int(X,B2)) ) ).

% Diff_single_insert
tff(fact_6625_Diff__single__insert,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))),B2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),B2)) ) ).

% Diff_single_insert
tff(fact_6626_remove__subset,axiom,
    ! [X: produc3843707927480180839at_nat,S: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),S)
     => aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,S),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),S) ) ).

% remove_subset
tff(fact_6627_remove__subset,axiom,
    ! [X: set_Pr958786334691620121nt_int,S: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),S)
     => aa_set6820988702716878619_int_o(ord_le1924305788584680229nt_int(minus_2612819937483484256nt_int(S,insert8897473484851387113nt_int(X,bot_bo1488462491386950373nt_int))),S) ) ).

% remove_subset
tff(fact_6628_remove__subset,axiom,
    ! [X: set_nat,S: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(X),S)
     => aa_set_set_nat_o(ord_less_set_set_nat(minus_2163939370556025621et_nat(S,insert_set_nat(X,bot_bot_set_set_nat))),S) ) ).

% remove_subset
tff(fact_6629_remove__subset,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,S: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),S)
     => aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(minus_6456595905716665654_nat_o(S,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o))),S) ) ).

% remove_subset
tff(fact_6630_remove__subset,axiom,
    ! [X: product_prod_nat_nat,S: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),S)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,S),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),S) ) ).

% remove_subset
tff(fact_6631_remove__subset,axiom,
    ! [X: $o,S: set_o] :
      ( aa_set_o_o(member_o((X)),S)
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,minus_minus_set_o(S,insert_o((X),bot_bot_set_o))),S) ) ).

% remove_subset
tff(fact_6632_remove__subset,axiom,
    ! [X: int,S: set_int] :
      ( aa_set_int_o(member_int(X),S)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,minus_minus_set_int(S,insert_int(X,bot_bot_set_int))),S) ) ).

% remove_subset
tff(fact_6633_remove__subset,axiom,
    ! [X: nat,S: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),S)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))),S) ) ).

% remove_subset
tff(fact_6634_vimage__insert,axiom,
    ! [F: fun_nat_o,A: $o,B2: set_o] : ( vimage_nat_o(F,insert_o((A),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vimage_nat_o(F,insert_o((A),bot_bot_set_o))),vimage_nat_o(F,B2)) ) ).

% vimage_insert
tff(fact_6635_vimage__insert,axiom,
    ! [F: fun_nat_nat,A: nat,B2: set_nat] : ( aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(insert_nat(A),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))),aa_set_nat_set_nat(vimage_nat_nat(F),B2)) ) ).

% vimage_insert
tff(fact_6636_vimage__insert,axiom,
    ! [F: fun_nat_int,A: int,B2: set_int] : ( vimage_nat_int(F,insert_int(A,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vimage_nat_int(F,insert_int(A,bot_bot_set_int))),vimage_nat_int(F,B2)) ) ).

% vimage_insert
tff(fact_6637_vimage__insert,axiom,
    ! [F: fun_na1476630412504255114at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( vimage8013328719654469172at_nat(F,insert8211810215607154385at_nat(A,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vimage8013328719654469172at_nat(F,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))),vimage8013328719654469172at_nat(F,B2)) ) ).

% vimage_insert
tff(fact_6638_vimage__insert,axiom,
    ! [F: fun_na367238361360896496at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : ( vimage6435164912253009178at_nat(F,insert9069300056098147895at_nat(A,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vimage6435164912253009178at_nat(F,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))),vimage6435164912253009178at_nat(F,B2)) ) ).

% vimage_insert
tff(fact_6639_vimage__insert,axiom,
    ! [F: fun_Pr1099658227000185396_nat_o,A: $o,B2: set_o] : ( vimage2280073292226551852_nat_o(F,insert_o((A),B2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,vimage2280073292226551852_nat_o(F,insert_o((A),bot_bot_set_o))),vimage2280073292226551852_nat_o(F,B2)) ) ).

% vimage_insert
tff(fact_6640_vimage__insert,axiom,
    ! [F: fun_Pr3059359313727237556_nat_o,A: $o,B2: set_o] : ( vimage3307185822755684012_nat_o(F,insert_o((A),B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,vimage3307185822755684012_nat_o(F,insert_o((A),bot_bot_set_o))),vimage3307185822755684012_nat_o(F,B2)) ) ).

% vimage_insert
tff(fact_6641_vimage__insert,axiom,
    ! [F: fun_Pr7574179848310716106at_nat,A: nat,B2: set_nat] : ( vimage7952471809483217980at_nat(F,aa_set_nat_set_nat(insert_nat(A),B2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,vimage7952471809483217980at_nat(F,aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))),vimage7952471809483217980at_nat(F,B2)) ) ).

% vimage_insert
tff(fact_6642_vimage__insert,axiom,
    ! [F: fun_Pr1253225528153422666at_nat,A: nat,B2: set_nat] : ( vimage7134676753939176892at_nat(F,aa_set_nat_set_nat(insert_nat(A),B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,vimage7134676753939176892at_nat(F,aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))),vimage7134676753939176892at_nat(F,B2)) ) ).

% vimage_insert
tff(fact_6643_vimage__insert,axiom,
    ! [F: fun_Pr3396328828801519398at_int,A: int,B2: set_int] : ( vimage7949981338974167704at_int(F,insert_int(A,B2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,vimage7949981338974167704at_int(F,insert_int(A,bot_bot_set_int))),vimage7949981338974167704at_int(F,B2)) ) ).

% vimage_insert
tff(fact_6644_Compl__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] : ( uminus935396558254630718at_nat(insert9069300056098147895at_nat(X,A2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,uminus935396558254630718at_nat(A2)),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) ) ).

% Compl_insert
tff(fact_6645_Compl__insert,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : ( aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,insert8211810215607154385at_nat(X,A2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,A2)),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ) ).

% Compl_insert
tff(fact_6646_Compl__insert,axiom,
    ! [X: $o,A2: set_o] : ( aa_set_o_set_o(uminus_uminus_set_o,insert_o((X),A2)) = minus_minus_set_o(aa_set_o_set_o(uminus_uminus_set_o,A2),insert_o((X),bot_bot_set_o)) ) ).

% Compl_insert
tff(fact_6647_Compl__insert,axiom,
    ! [X: int,A2: set_int] : ( aa_set_int_set_int(uminus1532241313380277803et_int,insert_int(X,A2)) = minus_minus_set_int(aa_set_int_set_int(uminus1532241313380277803et_int,A2),insert_int(X,bot_bot_set_int)) ) ).

% Compl_insert
tff(fact_6648_Compl__insert,axiom,
    ! [X: nat,A2: set_nat] : ( aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(insert_nat(X),A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) ) ).

% Compl_insert
tff(fact_6649_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_6650_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_6651_flip__bit__eq__xor,axiom,
    ! [N: nat,A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,A),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),one_one_Code_integer)) ) ).

% flip_bit_eq_xor
tff(fact_6652_flip__bit__eq__xor,axiom,
    ! [N: nat,A: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),A) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) ) ).

% flip_bit_eq_xor
tff(fact_6653_flip__bit__eq__xor,axiom,
    ! [N: nat,A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat)) ) ).

% flip_bit_eq_xor
tff(fact_6654_psubset__insert__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(A2),insert9069300056098147895at_nat(X,B2))
    <=> $ite(
          aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B2),
          aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(A2),B2),
          $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A2),aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B2),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_6655_psubset__insert__iff,axiom,
    ! [A2: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le1924305788584680229nt_int(A2),insert8897473484851387113nt_int(X,B2))
    <=> $ite(
          aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),B2),
          aa_set6820988702716878619_int_o(ord_le1924305788584680229nt_int(A2),B2),
          $ite(aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A2),aa_set6820988702716878619_int_o(ord_le1924305788584680229nt_int(minus_2612819937483484256nt_int(A2,insert8897473484851387113nt_int(X,bot_bo1488462491386950373nt_int))),B2),aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_6656_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A2),insert_set_nat(X,B2))
    <=> $ite(
          aa_set_set_nat_o(member_set_nat(X),B2),
          aa_set_set_nat_o(ord_less_set_set_nat(A2),B2),
          $ite(aa_set_set_nat_o(member_set_nat(X),A2),aa_set_set_nat_o(ord_less_set_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))),B2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_6657_psubset__insert__iff,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(A2),insert4923026906511862463_nat_o(X,B2))
    <=> $ite(
          aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B2),
          aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(A2),B2),
          $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2),aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o))),B2),aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_6658_psubset__insert__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A2),insert8211810215607154385at_nat(X,B2))
    <=> $ite(
          aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B2),
          aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A2),B2),
          $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2),aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2),aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_6659_psubset__insert__iff,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,A2),insert_o((X),B2))
    <=> $ite(
          aa_set_o_o(member_o((X)),B2),
          aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,A2),B2),
          $ite(aa_set_o_o(member_o((X)),A2),aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))),B2),aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_6660_psubset__insert__iff,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A2),insert_int(X,B2))
    <=> $ite(
          aa_set_int_o(member_int(X),B2),
          aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A2),B2),
          $ite(aa_set_int_o(member_int(X),A2),aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B2),aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_6661_psubset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),B2))
    <=> $ite(
          aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),B2),
          aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2),
          $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))),B2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_6662_push__bit__double,axiom,
    ! [N: nat,A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),A)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ).

% push_bit_double
tff(fact_6663_push__bit__double,axiom,
    ! [N: nat,A: code_natural] : ( bit_se6611745700429515170atural(N,times_2397367101498566445atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))) = times_2397367101498566445atural(bit_se6611745700429515170atural(N,A),numera5444537566228673987atural(aa_num_num(bit0,one))) ) ).

% push_bit_double
tff(fact_6664_push__bit__double,axiom,
    ! [N: nat,A: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),aa_int_int(aa_int_fun_int_int(times_times_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_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ).

% push_bit_double
tff(fact_6665_push__bit__double,axiom,
    ! [N: nat,A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),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(bit_se547839408752420682it_nat,N),A)),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% push_bit_double
tff(fact_6666_push__bit__eq__mult,axiom,
    ! [N: nat,A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) ) ).

% push_bit_eq_mult
tff(fact_6667_push__bit__eq__mult,axiom,
    ! [N: nat,A: code_natural] : ( bit_se6611745700429515170atural(N,A) = times_2397367101498566445atural(A,aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) ) ).

% push_bit_eq_mult
tff(fact_6668_push__bit__eq__mult,axiom,
    ! [N: nat,A: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ).

% push_bit_eq_mult
tff(fact_6669_push__bit__eq__mult,axiom,
    ! [N: nat,A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ) ).

% push_bit_eq_mult
tff(fact_6670_push__bit__minus__one,axiom,
    ! [N: nat] : ( aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ).

% push_bit_minus_one
tff(fact_6671_of__int__round__le,axiom,
    ! [X: 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,X))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_6672_of__int__round__ge,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))) ).

% of_int_round_ge
tff(fact_6673_of__int__round__gt,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))) ).

% of_int_round_gt
tff(fact_6674_integer__of__int__code,axiom,
    ! [K: int] :
      ( aa_int_Code_integer(code_integer_of_int,K) = $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))),
          $ite(
            K = zero_zero_int,
            zero_z3403309356797280102nteger,
            $let(
              l: code_integer,
              l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
              $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,l),one_one_Code_integer)) ) ) ) ) ).

% integer_of_int_code
tff(fact_6675_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L))
     => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
            ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                & ~ 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))),K)
                  & ~ 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,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.psimps
tff(fact_6676_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa))
       => ~ ( ( Y = $ite(
                  ( aa_set_int_o(member_int(X),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(Xa),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))),X)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa) ))),
                  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))),X)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa)) ) ) ) ).

% and_int.pelims
tff(fact_6677_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),one_one_rat)),X) ) ).

% neg_numeral_le_ceiling
tff(fact_6678_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_6679_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),one_one_rat)) ) ).

% floor_le_neg_numeral
tff(fact_6680_neg__numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),one_one_rat)),X) ) ).

% neg_numeral_less_floor
tff(fact_6681_floor__one,axiom,
    aa_rat_int(archim3151403230148437115or_rat,one_one_rat) = one_one_int ).

% floor_one
tff(fact_6682_ceiling__one,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_6683_floor__uminus__of__int,axiom,
    ! [Z: int] : ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(uminus_uminus_int,Z) ) ).

% floor_uminus_of_int
tff(fact_6684_zero__less__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ).

% zero_less_floor
tff(fact_6685_floor__le__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ).

% floor_le_zero
tff(fact_6686_one__le__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ).

% one_le_floor
tff(fact_6687_ceiling__less__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_6688_one__le__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X) ) ).

% one_le_ceiling
tff(fact_6689_floor__less__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ).

% floor_less_one
tff(fact_6690_ceiling__le__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),one_one_rat) ) ).

% ceiling_le_one
tff(fact_6691_one__less__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),X) ) ).

% one_less_ceiling
tff(fact_6692_floor__neg__numeral,axiom,
    ! [V: num] : ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ) ).

% floor_neg_numeral
tff(fact_6693_ceiling__neg__numeral,axiom,
    ! [V: num] : ( aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ) ).

% ceiling_neg_numeral
tff(fact_6694_ceiling__add__one,axiom,
    ! [X: rat] : ( aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int) ) ).

% ceiling_add_one
tff(fact_6695_floor__diff__one,axiom,
    ! [X: rat] : ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) ) ).

% floor_diff_one
tff(fact_6696_ceiling__diff__one,axiom,
    ! [X: rat] : ( aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int) ) ).

% ceiling_diff_one
tff(fact_6697_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_6698_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X) ) ).

% zero_le_ceiling
tff(fact_6699_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),one_one_rat)),X) ) ).

% numeral_less_floor
tff(fact_6700_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),one_one_rat)) ) ).

% floor_le_numeral
tff(fact_6701_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,numeral_numeral_rat(V)),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_6702_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,numeral_numeral_rat(V)),one_one_rat)),X) ) ).

% numeral_le_ceiling
tff(fact_6703_one__less__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(aa_num_num(bit0,one))),X) ) ).

% one_less_floor
tff(fact_6704_floor__le__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),numeral_numeral_rat(aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_6705_neg__numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),X) ) ).

% neg_numeral_le_floor
tff(fact_6706_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))) ) ).

% floor_less_neg_numeral
tff(fact_6707_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))) ) ).

% ceiling_le_neg_numeral
tff(fact_6708_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),X) ) ).

% neg_numeral_less_ceiling
tff(fact_6709_integer__of__int__cases,axiom,
    ! [X: code_integer] :
      ~ ! [Y2: int] :
          ( ( X = aa_int_Code_integer(code_integer_of_int,Y2) )
         => ~ aa_set_int_o(member_int(Y2),top_top_set_int) ) ).

% integer_of_int_cases
tff(fact_6710_integer__of__int__induct,axiom,
    ! [P: fun_Code_integer_o,X: code_integer] :
      ( ! [Y2: int] :
          ( aa_set_int_o(member_int(Y2),top_top_set_int)
         => aa_Code_integer_o(P,aa_int_Code_integer(code_integer_of_int,Y2)) )
     => aa_Code_integer_o(P,X) ) ).

% integer_of_int_induct
tff(fact_6711_integer__of__int__inject,axiom,
    ! [X: int,Y: int] :
      ( aa_set_int_o(member_int(X),top_top_set_int)
     => ( aa_set_int_o(member_int(Y),top_top_set_int)
       => ( ( aa_int_Code_integer(code_integer_of_int,X) = aa_int_Code_integer(code_integer_of_int,Y) )
        <=> ( X = Y ) ) ) ) ).

% integer_of_int_inject
tff(fact_6712_uminus__integer__code_I1_J,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% uminus_integer_code(1)
tff(fact_6713_ceiling__def,axiom,
    ! [X: rat] : ( aa_rat_int(archim2889992004027027881ng_rat,X) = aa_int_int(uminus_uminus_int,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,X))) ) ).

% ceiling_def
tff(fact_6714_floor__minus,axiom,
    ! [X: rat] : ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,X)) = aa_int_int(uminus_uminus_int,aa_rat_int(archim2889992004027027881ng_rat,X)) ) ).

% floor_minus
tff(fact_6715_ceiling__minus,axiom,
    ! [X: rat] : ( aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(uminus_uminus_rat,X)) = aa_int_int(uminus_uminus_int,aa_rat_int(archim3151403230148437115or_rat,X)) ) ).

% ceiling_minus
tff(fact_6716_uminus__integer_Oabs__eq,axiom,
    ! [X: int] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,X)) ) ).

% uminus_integer.abs_eq
tff(fact_6717_ceiling__diff__floor__le__1,axiom,
    ! [X: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim3151403230148437115or_rat,X))),one_one_int) ).

% ceiling_diff_floor_le_1
tff(fact_6718_ceiling__altdef,axiom,
    ! [X: rat] :
      ( aa_rat_int(archim2889992004027027881ng_rat,X) = $ite(X = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)) ) ).

% ceiling_altdef
tff(fact_6719_one__integer__def,axiom,
    one_one_Code_integer = aa_int_Code_integer(code_integer_of_int,one_one_int) ).

% one_integer_def
tff(fact_6720_one__add__floor,axiom,
    ! [X: rat] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) ) ).

% one_add_floor
tff(fact_6721_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,X) = Z ) ) ) ).

% floor_unique
tff(fact_6722_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim3151403230148437115or_rat,X) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)) ) ) ).

% floor_eq_iff
tff(fact_6723_floor__split,axiom,
    ! [P: fun_int_o,T2: rat] :
      ( aa_int_o(P,aa_rat_int(archim3151403230148437115or_rat,T2))
    <=> ! [I3: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I3)),T2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,I3)),one_one_rat)) )
         => aa_int_o(P,I3) ) ) ).

% floor_split
tff(fact_6724_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% le_mult_floor
tff(fact_6725_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X) ) ).

% less_floor_iff
tff(fact_6726_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).

% floor_le_iff
tff(fact_6727_floor__correct,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int))) ) ).

% floor_correct
tff(fact_6728_ceiling__correct,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))),one_one_rat)),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))) ) ).

% ceiling_correct
tff(fact_6729_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = Z ) ) ) ).

% ceiling_unique
tff(fact_6730_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,X) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).

% ceiling_eq_iff
tff(fact_6731_ceiling__split,axiom,
    ! [P: fun_int_o,T2: rat] :
      ( aa_int_o(P,aa_rat_int(archim2889992004027027881ng_rat,T2))
    <=> ! [I3: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,I3)),one_one_rat)),T2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T2),aa_int_rat(ring_1_of_int_rat,I3)) )
         => aa_int_o(P,I3) ) ) ).

% ceiling_split
tff(fact_6732_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B))) ) ) ).

% mult_ceiling_le
tff(fact_6733_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_6734_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X) ) ).

% le_ceiling_iff
tff(fact_6735_floor__divide__lower,axiom,
    ! [Q3: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q3)))),Q3)),P3) ) ).

% floor_divide_lower
tff(fact_6736_ceiling__divide__upper,axiom,
    ! [Q3: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q3)))),Q3)) ) ).

% ceiling_divide_upper
tff(fact_6737_floor__divide__upper,axiom,
    ! [Q3: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q3)))),one_one_rat)),Q3)) ) ).

% floor_divide_upper
tff(fact_6738_round__def,axiom,
    ! [X: rat] : ( aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one))))) ) ).

% round_def
tff(fact_6739_ceiling__divide__lower,axiom,
    ! [Q3: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q3)))),one_one_rat)),Q3)),P3) ) ).

% ceiling_divide_lower
tff(fact_6740_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: 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))
     => ( ! [K2: int,L2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K2),L2))
           => ( ( ~ ( 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(L2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K2),L2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_6741_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: 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(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P,I2),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_6742_round__altdef,axiom,
    ! [X: rat] :
      ( aa_rat_int(archim7778729529865785530nd_rat,X) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,X),aa_rat_int(archim3151403230148437115or_rat,X)) ) ).

% round_altdef
tff(fact_6743_the__elem__eq,axiom,
    ! [X: produc3843707927480180839at_nat] : ( the_el221668144340439132at_nat(insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) = X ) ).

% the_elem_eq
tff(fact_6744_the__elem__eq,axiom,
    ! [X: product_prod_nat_nat] : ( the_el2281957884133575798at_nat(insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = X ) ).

% the_elem_eq
tff(fact_6745_the__elem__eq,axiom,
    ! [X: $o] :
      ( aa_set_o_o(the_elem_o,insert_o((X),bot_bot_set_o))
    <=> (X) ) ).

% the_elem_eq
tff(fact_6746_the__elem__eq,axiom,
    ! [X: nat] : ( aa_set_nat_nat(the_elem_nat,aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = X ) ).

% the_elem_eq
tff(fact_6747_the__elem__eq,axiom,
    ! [X: int] : ( the_elem_int(insert_int(X,bot_bot_set_int)) = X ) ).

% the_elem_eq
tff(fact_6748_round__unique_H,axiom,
    ! [X: rat,N: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_int_rat(ring_1_of_int_rat,N)))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,X) = N ) ) ).

% round_unique'
tff(fact_6749_of__int__round__abs__le,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))),X))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_6750_num__of__integer__code,axiom,
    ! [K: code_integer] :
      ( code_num_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_jx,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% num_of_integer_code
tff(fact_6751_bit__cut__integer__def,axiom,
    ! [K: code_integer] : ( code_bit_cut_integer(K) = produc6677183202524767010eger_o(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(K),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),K)) ) ).

% bit_cut_integer_def
tff(fact_6752_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_6753_abs__idempotent,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ) ).

% abs_idempotent
tff(fact_6754_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_6755_abs__zero,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_6756_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_6757_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_6758_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_6759_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_6760_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_6761_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_6762_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_6763_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_6764_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ) ).

% abs_add_abs
tff(fact_6765_abs__add__abs,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ).

% abs_add_abs
tff(fact_6766_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_6767_abs__mult__self__eq,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ) ).

% abs_mult_self_eq
tff(fact_6768_abs__mult__self__eq,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ) ).

% abs_mult_self_eq
tff(fact_6769_abs__mult__self__eq,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ) ).

% abs_mult_self_eq
tff(fact_6770_abs__1,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_6771_abs__1,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_1
tff(fact_6772_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_6773_abs__minus,axiom,
    ! [A: int] : ( aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ) ).

% abs_minus
tff(fact_6774_abs__minus,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ) ).

% abs_minus
tff(fact_6775_abs__minus,axiom,
    ! [A: rat] : ( aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ) ).

% abs_minus
tff(fact_6776_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_6777_abs__minus__cancel,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ) ).

% abs_minus_cancel
tff(fact_6778_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_6779_zdvd1__eq,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),one_one_int)
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_6780_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_6781_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_6782_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_6783_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% abs_le_self_iff
tff(fact_6784_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_6785_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_6786_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = A ) ) ).

% abs_of_nonneg
tff(fact_6787_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_6788_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_6789_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_6790_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_6791_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_6792_abs__neg__numeral,axiom,
    ! [N: num] : ( aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,N) ) ).

% abs_neg_numeral
tff(fact_6793_abs__neg__numeral,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,N) ) ).

% abs_neg_numeral
tff(fact_6794_abs__neg__numeral,axiom,
    ! [N: num] : ( aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = numeral_numeral_rat(N) ) ).

% abs_neg_numeral
tff(fact_6795_abs__neg__one,axiom,
    aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,one_one_int)) = one_one_int ).

% abs_neg_one
tff(fact_6796_abs__neg__one,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_6797_abs__neg__one,axiom,
    aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_6798_abs__power__minus,axiom,
    ! [A: int,N: nat] : ( aa_int_int(abs_abs_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),N)) = aa_int_int(abs_abs_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ).

% abs_power_minus
tff(fact_6799_abs__power__minus,axiom,
    ! [A: code_integer,N: nat] : ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ).

% abs_power_minus
tff(fact_6800_abs__power__minus,axiom,
    ! [A: rat,N: nat] : ( aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N)) = aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),N)) ) ).

% abs_power_minus
tff(fact_6801_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_6802_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_6803_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_6804_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_6805_abs__ge__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) ).

% abs_ge_self
tff(fact_6806_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_6807_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_6808_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% abs_le_D1
tff(fact_6809_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_6810_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_6811_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ) ).

% abs_mult
tff(fact_6812_abs__mult,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ).

% abs_mult
tff(fact_6813_abs__mult,axiom,
    ! [A: int,B: int] : ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ).

% abs_mult
tff(fact_6814_abs__one,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_6815_abs__one,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_one
tff(fact_6816_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_6817_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A)) ) ).

% abs_minus_commute
tff(fact_6818_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)) ) ).

% abs_minus_commute
tff(fact_6819_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_6820_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,X) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( X = Y )
        | ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6821_abs__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y) )
    <=> ( ( X = Y )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6822_abs__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(abs_abs_rat,X) = aa_rat_rat(abs_abs_rat,Y) )
    <=> ( ( X = Y )
        | ( X = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6823_abs__ge__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) ).

% abs_ge_zero
tff(fact_6824_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_6825_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_6826_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = A ) ) ).

% abs_of_pos
tff(fact_6827_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_6828_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_6829_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_6830_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_6831_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_6832_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))) ).

% abs_triangle_ineq
tff(fact_6833_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq
tff(fact_6834_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_6835_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6836_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6837_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_6838_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B))) ).

% abs_triangle_ineq3
tff(fact_6839_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(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))) ).

% abs_triangle_ineq3
tff(fact_6840_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_6841_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B))) ).

% abs_triangle_ineq2
tff(fact_6842_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))) ).

% abs_triangle_ineq2
tff(fact_6843_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_6844_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)),D2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D2)) ) ) ).

% abs_mult_less
tff(fact_6845_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),D2)) ) ) ).

% abs_mult_less
tff(fact_6846_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),D2)
       => 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(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D2)) ) ) ).

% abs_mult_less
tff(fact_6847_abs__ge__minus__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) ).

% abs_ge_minus_self
tff(fact_6848_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_6849_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_6850_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(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_6851_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_6852_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_6853_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(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_6854_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_6855_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_6856_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,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B) ) ) ).

% abs_leI
tff(fact_6857_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_6858_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_6859_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_less_iff
tff(fact_6860_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_less_iff
tff(fact_6861_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_less_iff
tff(fact_6862_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_6863_frac__lt__1,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(archimedean_frac_rat,X)),one_one_rat) ).

% frac_lt_1
tff(fact_6864_frac__1__eq,axiom,
    ! [X: rat] : ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) = aa_rat_rat(archimedean_frac_rat,X) ) ).

% frac_1_eq
tff(fact_6865_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),E3) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_6866_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) ) )
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ) ) ).

% abs_eq_mult
tff(fact_6867_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(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_6868_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_6869_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y)),X) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_6870_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,Y)),X) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_6871_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_6872_abs__minus__le__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_6873_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_6874_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_6875_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(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_6876_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_6877_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_6878_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(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_6879_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_6880_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_6881_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),D2)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),C))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),D2)))) ).

% abs_diff_triangle_ineq
tff(fact_6882_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),D2)))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D2)))) ).

% abs_diff_triangle_ineq
tff(fact_6883_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D2: 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,C),D2)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)))) ).

% abs_diff_triangle_ineq
tff(fact_6884_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))) ).

% abs_triangle_ineq4
tff(fact_6885_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(aa_rat_fun_rat_rat(minus_minus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq4
tff(fact_6886_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_6887_abs__if__raw,axiom,
    ! [X3: int] :
      ( aa_int_int(abs_abs_int,X3) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X3),zero_zero_int),aa_int_int(uminus_uminus_int,X3),X3) ) ).

% abs_if_raw
tff(fact_6888_abs__if__raw,axiom,
    ! [X3: code_integer] :
      ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X3) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X3),X3) ) ).

% abs_if_raw
tff(fact_6889_abs__if__raw,axiom,
    ! [X3: rat] :
      ( aa_rat_rat(abs_abs_rat,X3) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X3),X3) ) ).

% abs_if_raw
tff(fact_6890_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_6891_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_6892_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_6893_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_6894_abs__if,axiom,
    ! [A: code_integer] :
      ( aa_Cod1783974652533396728nteger(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_6895_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_6896_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_6897_abs__integer__code,axiom,
    ! [K: code_integer] :
      ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),K) ) ).

% abs_integer_code
tff(fact_6898_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X))) ).

% abs_add_one_gt_zero
tff(fact_6899_abs__add__one__gt__zero,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(abs_abs_rat,X))) ).

% abs_add_one_gt_zero
tff(fact_6900_abs__add__one__gt__zero,axiom,
    ! [X: 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,X))) ).

% abs_add_one_gt_zero
tff(fact_6901_of__int__leD,axiom,
    ! [N: int,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,N))),X)
     => ( ( N = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),X) ) ) ).

% of_int_leD
tff(fact_6902_of__int__leD,axiom,
    ! [N: int,X: 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,N))),X)
     => ( ( N = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ) ).

% of_int_leD
tff(fact_6903_of__int__leD,axiom,
    ! [N: int,X: 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,N))),X)
     => ( ( N = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),X) ) ) ).

% of_int_leD
tff(fact_6904_of__int__lessD,axiom,
    ! [N: int,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,N))),X)
     => ( ( N = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),X) ) ) ).

% of_int_lessD
tff(fact_6905_of__int__lessD,axiom,
    ! [N: int,X: 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,N))),X)
     => ( ( N = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),X) ) ) ).

% of_int_lessD
tff(fact_6906_of__int__lessD,axiom,
    ! [N: int,X: 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,N))),X)
     => ( ( N = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),X) ) ) ).

% of_int_lessD
tff(fact_6907_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != 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,M),N)),M)
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_6908_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_6909_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_rat )
    <=> ( aa_rat_rat(abs_abs_rat,X) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_6910_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,X),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_6911_frac__eq,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = X )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ) ).

% frac_eq
tff(fact_6912_frac__add,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat)) ) ).

% frac_add
tff(fact_6913_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_6914_abs__square__le__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),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,X)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_6915_abs__square__le__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),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,X)),one_one_int) ) ).

% abs_square_le_1
tff(fact_6916_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_6917_abs__square__less__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_6918_abs__square__less__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int) ) ).

% abs_square_less_1
tff(fact_6919_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,M)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,N))
           => ? [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N)
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_6920_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D2))) ) ).

% incr_lemma
tff(fact_6921_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D2))),Z) ) ).

% decr_lemma
tff(fact_6922_nat__ivt__aux,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,N))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_6923_floor__add,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),one_one_int)) ) ).

% floor_add
tff(fact_6924_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,N))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_6925_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : ( code_divmod_integer(K,L) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(K),L),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(K),L)) ) ).

% divmod_integer_def
tff(fact_6926_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      ( code_bit_cut_integer(K) = $ite(K = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_jy(K),code_divmod_abs(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% bit_cut_integer_code
tff(fact_6927_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( code_nat_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_jz,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% nat_of_integer_code
tff(fact_6928_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_int(code_int_of_integer,K) = $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),
          aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
          $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_ka,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ) ).

% int_of_integer_code
tff(fact_6929_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = $ite(
          K = aa_int_int(uminus_uminus_int,one_one_int),
          aa_int_int(bit_ri7919022796975470100ot_int,L),
          $ite(
            L = aa_int_int(uminus_uminus_int,one_one_int),
            aa_int_int(bit_ri7919022796975470100ot_int,K),
            $ite(
              K = zero_zero_int,
              L,
              $ite(L = zero_zero_int,K,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ) ).

% xor_int_unfold
tff(fact_6930_is__singleton__the__elem,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,A2)
    <=> ( A2 = insert9069300056098147895at_nat(the_el221668144340439132at_nat(A2),bot_bo228742789529271731at_nat) ) ) ).

% is_singleton_the_elem
tff(fact_6931_is__singleton__the__elem,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,A2)
    <=> ( A2 = insert8211810215607154385at_nat(the_el2281957884133575798at_nat(A2),bot_bo2099793752762293965at_nat) ) ) ).

% is_singleton_the_elem
tff(fact_6932_is__singleton__the__elem,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_singleton_o,A2)
    <=> ( A2 = insert_o(aa_set_o_o(the_elem_o,A2),bot_bot_set_o) ) ) ).

% is_singleton_the_elem
tff(fact_6933_is__singleton__the__elem,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A2)
    <=> ( A2 = aa_set_nat_set_nat(insert_nat(aa_set_nat_nat(the_elem_nat,A2)),bot_bot_set_nat) ) ) ).

% is_singleton_the_elem
tff(fact_6934_is__singleton__the__elem,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(is_singleton_int,A2)
    <=> ( A2 = insert_int(the_elem_int(A2),bot_bot_set_int) ) ) ).

% is_singleton_the_elem
tff(fact_6935_integer__of__num_I3_J,axiom,
    ! [N: num] :
      ( code_integer_of_num(aa_num_num(bit1,N)) = $let(
          k: code_integer,
          k:= code_integer_of_num(N),
          aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,k),k)),one_one_Code_integer) ) ) ).

% integer_of_num(3)
tff(fact_6936_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gg,N))) ) ).

% mask_eq_sum_exp_nat
tff(fact_6937_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : ( aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A2)) = groups3906332499630173760nt_rat(aTP_Lamm_kb(F),A2) ) ).

% of_int_sum
tff(fact_6938_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : ( aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_kc(F),A2) ) ).

% of_int_sum
tff(fact_6939_one__integer_Orep__eq,axiom,
    aa_Code_integer_int(code_int_of_integer,one_one_Code_integer) = one_one_int ).

% one_integer.rep_eq
tff(fact_6940_uminus__integer_Orep__eq,axiom,
    ! [X: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)) = aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,X)) ) ).

% uminus_integer.rep_eq
tff(fact_6941_is__singletonI,axiom,
    ! [X: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) ).

% is_singletonI
tff(fact_6942_is__singletonI,axiom,
    ! [X: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ).

% is_singletonI
tff(fact_6943_is__singletonI,axiom,
    ! [X: $o] : aa_set_o_o(is_singleton_o,insert_o((X),bot_bot_set_o)) ).

% is_singletonI
tff(fact_6944_is__singletonI,axiom,
    ! [X: nat] : aa_set_nat_o(is_singleton_nat,aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) ).

% is_singletonI
tff(fact_6945_is__singletonI,axiom,
    ! [X: int] : aa_set_int_o(is_singleton_int,insert_int(X,bot_bot_set_int)) ).

% is_singletonI
tff(fact_6946_bit_Ocompl__zero,axiom,
    aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.compl_zero
tff(fact_6947_bit_Ocompl__zero,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.compl_zero
tff(fact_6948_bit_Ocompl__one,axiom,
    aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% bit.compl_one
tff(fact_6949_bit_Ocompl__one,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% bit.compl_one
tff(fact_6950_bit_Oxor__cancel__right,axiom,
    ! [X: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,X),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ).

% bit.xor_cancel_right
tff(fact_6951_bit_Oxor__cancel__right,axiom,
    ! [X: int] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),aa_int_int(bit_ri7919022796975470100ot_int,X)) = aa_int_int(uminus_uminus_int,one_one_int) ) ).

% bit.xor_cancel_right
tff(fact_6952_bit_Oxor__cancel__left,axiom,
    ! [X: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X)),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ).

% bit.xor_cancel_left
tff(fact_6953_bit_Oxor__cancel__left,axiom,
    ! [X: int] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(bit_ri7919022796975470100ot_int,X)),X) = aa_int_int(uminus_uminus_int,one_one_int) ) ).

% bit.xor_cancel_left
tff(fact_6954_bit_Oxor__one__right,axiom,
    ! [X: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X) ) ).

% bit.xor_one_right
tff(fact_6955_bit_Oxor__one__right,axiom,
    ! [X: int] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,X) ) ).

% bit.xor_one_right
tff(fact_6956_bit_Oxor__one__left,axiom,
    ! [X: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X) ) ).

% bit.xor_one_left
tff(fact_6957_bit_Oxor__one__left,axiom,
    ! [X: int] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(uminus_uminus_int,one_one_int)),X) = aa_int_int(bit_ri7919022796975470100ot_int,X) ) ).

% bit.xor_one_left
tff(fact_6958_minus__not__numeral__eq,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,inc(N)) ) ).

% minus_not_numeral_eq
tff(fact_6959_minus__not__numeral__eq,axiom,
    ! [N: num] : ( aa_int_int(uminus_uminus_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,inc(N)) ) ).

% minus_not_numeral_eq
tff(fact_6960_not__minus__numeral__eq,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = neg_nu5755505904847501662nteger(N,one) ) ).

% not_minus_numeral_eq
tff(fact_6961_not__minus__numeral__eq,axiom,
    ! [N: num] : ( aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = neg_numeral_sub_int(N,one) ) ).

% not_minus_numeral_eq
tff(fact_6962_push__bit__minus__one__eq__not__mask,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_nat_Code_integer(bit_se2119862282449309892nteger,N)) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_6963_push__bit__minus__one__eq__not__mask,axiom,
    ! [N: nat] : ( aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,N)) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_6964_xor__minus__numerals_I2_J,axiom,
    ! [K: int,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),neg_numeral_sub_int(N,one))) ) ).

% xor_minus_numerals(2)
tff(fact_6965_xor__minus__numerals_I1_J,axiom,
    ! [N: num,K: int] : ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),K) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,neg_numeral_sub_int(N,one)),K)) ) ).

% xor_minus_numerals(1)
tff(fact_6966_not__one__eq,axiom,
    aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% not_one_eq
tff(fact_6967_not__one__eq,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% not_one_eq
tff(fact_6968_int__of__integer,axiom,
    ! [X: code_integer] : aa_set_int_o(member_int(aa_Code_integer_int(code_int_of_integer,X)),top_top_set_int) ).

% int_of_integer
tff(fact_6969_int__of__integer__cases,axiom,
    ! [Y: int] :
      ( aa_set_int_o(member_int(Y),top_top_set_int)
     => ~ ! [X2: code_integer] : ( Y != aa_Code_integer_int(code_int_of_integer,X2) ) ) ).

% int_of_integer_cases
tff(fact_6970_int__of__integer__induct,axiom,
    ! [Y: int,P: fun_int_o] :
      ( aa_set_int_o(member_int(Y),top_top_set_int)
     => ( ! [X2: code_integer] : aa_int_o(P,aa_Code_integer_int(code_int_of_integer,X2))
       => aa_int_o(P,Y) ) ) ).

% int_of_integer_induct
tff(fact_6971_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A2: set_nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kd(F),A)),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),A) ) ).

% mod_sum_eq
tff(fact_6972_mod__sum__eq,axiom,
    ! [F: fun_int_int,A: int,A2: set_int] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_ke(F),A),A2)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(F,A2)),A) ) ).

% mod_sum_eq
tff(fact_6973_type__definition__integer,axiom,
    aa_set_int_o(type_d8366093980585677751er_int(code_int_of_integer,code_integer_of_int),top_top_set_int) ).

% type_definition_integer
tff(fact_6974_integer__of__int__inverse,axiom,
    ! [Y: int] :
      ( aa_set_int_o(member_int(Y),top_top_set_int)
     => ( aa_Code_integer_int(code_int_of_integer,aa_int_Code_integer(code_integer_of_int,Y)) = Y ) ) ).

% integer_of_int_inverse
tff(fact_6975_is__singletonI_H,axiom,
    ! [A2: set_se6260736226359567993nt_int] :
      ( ( A2 != bot_bo1488462491386950373nt_int )
     => ( ! [X2: set_Pr958786334691620121nt_int,Y2: set_Pr958786334691620121nt_int] :
            ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),A2)
           => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y2),A2)
             => ( X2 = Y2 ) ) )
       => aa_set6820988702716878619_int_o(is_sin6299389887212142093nt_int,A2) ) ) ).

% is_singletonI'
tff(fact_6976_is__singletonI_H,axiom,
    ! [A2: set_set_nat] :
      ( ( A2 != bot_bot_set_set_nat )
     => ( ! [X2: set_nat,Y2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X2),A2)
           => ( aa_set_set_nat_o(member_set_nat(Y2),A2)
             => ( X2 = Y2 ) ) )
       => aa_set_set_nat_o(is_singleton_set_nat,A2) ) ) ).

% is_singletonI'
tff(fact_6977_is__singletonI_H,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( ( A2 != bot_bo8632255151191677115_nat_o )
     => ( ! [X2: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A2)
           => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A2)
             => ( X2 = Y2 ) ) )
       => aa_set5091322549519198405at_o_o(is_sin624665226112233955_nat_o,A2) ) ) ).

% is_singletonI'
tff(fact_6978_is__singletonI_H,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( A2 != bot_bo2099793752762293965at_nat )
     => ( ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),A2)
           => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y2),A2)
             => ( X2 = Y2 ) ) )
       => aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,A2) ) ) ).

% is_singletonI'
tff(fact_6979_is__singletonI_H,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
     => ( ! [X2: $o,Y2: $o] :
            ( aa_set_o_o(member_o((X2)),A2)
           => ( aa_set_o_o(member_o((Y2)),A2)
             => ( (X2)
              <=> (Y2) ) ) )
       => aa_set_o_o(is_singleton_o,A2) ) ) ).

% is_singletonI'
tff(fact_6980_is__singletonI_H,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [X2: nat,Y2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
           => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y2),A2)
             => ( X2 = Y2 ) ) )
       => aa_set_nat_o(is_singleton_nat,A2) ) ) ).

% is_singletonI'
tff(fact_6981_is__singletonI_H,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
     => ( ! [X2: int,Y2: int] :
            ( aa_set_int_o(member_int(X2),A2)
           => ( aa_set_int_o(member_int(Y2),A2)
             => ( X2 = Y2 ) ) )
       => aa_set_int_o(is_singleton_int,A2) ) ) ).

% is_singletonI'
tff(fact_6982_minus__eq__not__plus__1,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,A)),one_one_Code_integer) ) ).

% minus_eq_not_plus_1
tff(fact_6983_minus__eq__not__plus__1,axiom,
    ! [A: int] : ( aa_int_int(uminus_uminus_int,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),one_one_int) ) ).

% minus_eq_not_plus_1
tff(fact_6984_minus__eq__not__minus__1,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),one_one_Code_integer)) ) ).

% minus_eq_not_minus_1
tff(fact_6985_minus__eq__not__minus__1,axiom,
    ! [A: int] : ( aa_int_int(uminus_uminus_int,A) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)) ) ).

% minus_eq_not_minus_1
tff(fact_6986_not__eq__complement,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,A) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),one_one_Code_integer) ) ).

% not_eq_complement
tff(fact_6987_not__eq__complement,axiom,
    ! [A: int] : ( aa_int_int(bit_ri7919022796975470100ot_int,A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),one_one_int) ) ).

% not_eq_complement
tff(fact_6988_not__int__def,axiom,
    ! [K: int] : ( aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int) ) ).

% not_int_def
tff(fact_6989_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_6990_sum__power__add,axiom,
    ! [X: code_integer,M: nat,I4: set_nat] : ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aTP_Lamm_kf(X),M)),I4) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),M)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(X)),I4)) ) ).

% sum_power_add
tff(fact_6991_sum__power__add,axiom,
    ! [X: rat,M: nat,I4: set_nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_kg(X),M)),I4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),M)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),I4)) ) ).

% sum_power_add
tff(fact_6992_sum__power__add,axiom,
    ! [X: int,M: nat,I4: set_nat] : ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_kh(X),M)),I4) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),M)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),I4)) ) ).

% sum_power_add
tff(fact_6993_minus__numeral__inc__eq,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,inc(N))) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ) ).

% minus_numeral_inc_eq
tff(fact_6994_minus__numeral__inc__eq,axiom,
    ! [N: num] : ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)) ) ).

% minus_numeral_inc_eq
tff(fact_6995_unset__bit__int__def,axiom,
    ! [N: nat,K: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),K) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int))) ) ).

% unset_bit_int_def
tff(fact_6996_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_6997_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : ( code_divmod_abs(J,zero_z3403309356797280102nteger) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,J)) ) ).

% divmod_abs_code(5)
tff(fact_6998_and__not__numerals_I2_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = one_one_int ) ).

% and_not_numerals(2)
tff(fact_6999_and__not__numerals_I4_J,axiom,
    ! [M: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ) ).

% and_not_numerals(4)
tff(fact_7000_not__numeral__Bit0__eq,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N))) ) ).

% not_numeral_Bit0_eq
tff(fact_7001_not__numeral__Bit0__eq,axiom,
    ! [N: num] : ( aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ) ).

% not_numeral_Bit0_eq
tff(fact_7002_minus__numeral__eq__not__sub__one,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,neg_nu5755505904847501662nteger(N,one)) ) ).

% minus_numeral_eq_not_sub_one
tff(fact_7003_minus__numeral__eq__not__sub__one,axiom,
    ! [N: num] : ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) = aa_int_int(bit_ri7919022796975470100ot_int,neg_numeral_sub_int(N,one)) ) ).

% minus_numeral_eq_not_sub_one
tff(fact_7004_not__numeral__BitM__eq,axiom,
    ! [N: num] : ( aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_num_Code_integer(numera6620942414471956472nteger,bitM(N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N))) ) ).

% not_numeral_BitM_eq
tff(fact_7005_not__numeral__BitM__eq,axiom,
    ! [N: num] : ( aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bitM(N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ) ).

% not_numeral_BitM_eq
tff(fact_7006_unset__bit__eq__and__not,axiom,
    ! [N: nat,A: code_integer] : ( aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),one_one_Code_integer))) ) ).

% unset_bit_eq_and_not
tff(fact_7007_unset__bit__eq__and__not,axiom,
    ! [N: nat,A: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),A) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int))) ) ).

% unset_bit_eq_and_not
tff(fact_7008_is__singletonE,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,A2)
     => ~ ! [X2: produc3843707927480180839at_nat] : ( A2 != insert9069300056098147895at_nat(X2,bot_bo228742789529271731at_nat) ) ) ).

% is_singletonE
tff(fact_7009_is__singletonE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,A2)
     => ~ ! [X2: product_prod_nat_nat] : ( A2 != insert8211810215607154385at_nat(X2,bot_bo2099793752762293965at_nat) ) ) ).

% is_singletonE
tff(fact_7010_is__singletonE,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_singleton_o,A2)
     => ~ ! [X2: $o] : ( A2 != insert_o((X2),bot_bot_set_o) ) ) ).

% is_singletonE
tff(fact_7011_is__singletonE,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A2)
     => ~ ! [X2: nat] : ( A2 != aa_set_nat_set_nat(insert_nat(X2),bot_bot_set_nat) ) ) ).

% is_singletonE
tff(fact_7012_is__singletonE,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(is_singleton_int,A2)
     => ~ ! [X2: int] : ( A2 != insert_int(X2,bot_bot_set_int) ) ) ).

% is_singletonE
tff(fact_7013_is__singleton__def,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,A2)
    <=> ? [X4: produc3843707927480180839at_nat] : ( A2 = insert9069300056098147895at_nat(X4,bot_bo228742789529271731at_nat) ) ) ).

% is_singleton_def
tff(fact_7014_is__singleton__def,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,A2)
    <=> ? [X4: product_prod_nat_nat] : ( A2 = insert8211810215607154385at_nat(X4,bot_bo2099793752762293965at_nat) ) ) ).

% is_singleton_def
tff(fact_7015_is__singleton__def,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_singleton_o,A2)
    <=> ? [X4: $o] : ( A2 = insert_o((X4),bot_bot_set_o) ) ) ).

% is_singleton_def
tff(fact_7016_is__singleton__def,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A2)
    <=> ? [X4: nat] : ( A2 = aa_set_nat_set_nat(insert_nat(X4),bot_bot_set_nat) ) ) ).

% is_singleton_def
tff(fact_7017_is__singleton__def,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(is_singleton_int,A2)
    <=> ? [X4: int] : ( A2 = insert_int(X4,bot_bot_set_int) ) ) ).

% is_singleton_def
tff(fact_7018_and__not__numerals_I7_J,axiom,
    ! [M: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ) ).

% and_not_numerals(7)
tff(fact_7019_and__not__numerals_I3_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = zero_zero_int ) ).

% and_not_numerals(3)
tff(fact_7020_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_7021_integer__of__num_I2_J,axiom,
    ! [N: num] :
      ( code_integer_of_num(aa_num_num(bit0,N)) = $let(
          k: code_integer,
          k:= code_integer_of_num(N),
          aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,k),k) ) ) ).

% integer_of_num(2)
tff(fact_7022_minus__exp__eq__not__mask,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_nat_Code_integer(bit_se2119862282449309892nteger,N)) ) ).

% minus_exp_eq_not_mask
tff(fact_7023_minus__exp__eq__not__mask,axiom,
    ! [N: nat] : ( aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,N)) ) ).

% minus_exp_eq_not_mask
tff(fact_7024_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : ( code_divmod_abs(K,L) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L))) ) ).

% divmod_abs_def
tff(fact_7025_mask__eq__sum__exp,axiom,
    ! [N: nat] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gg,N))) ) ).

% mask_eq_sum_exp
tff(fact_7026_mask__eq__sum__exp,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)),one_one_Code_integer) = aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gg,N))) ) ).

% mask_eq_sum_exp
tff(fact_7027_mask__eq__sum__exp,axiom,
    ! [N: nat] : ( minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N),one_one_Code_natural) = groups6325495683096345652atural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gg,N))) ) ).

% mask_eq_sum_exp
tff(fact_7028_mask__eq__sum__exp,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gg,N))) ) ).

% mask_eq_sum_exp
tff(fact_7029_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = 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,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ) ).

% and_not_numerals(8)
tff(fact_7030_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_ki(F),A2)) ).

% sum_abs_ge_zero
tff(fact_7031_sum__abs,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A2))),groups4538972089207619220nt_int(aTP_Lamm_ki(F),A2)) ).

% sum_abs
tff(fact_7032_sum_Oempty,axiom,
    ! [G: fun_o_Code_integer] : ( groups4406642042086082107nteger(G,bot_bot_set_o) = zero_z3403309356797280102nteger ) ).

% sum.empty
tff(fact_7033_sum_Oempty,axiom,
    ! [G: fun_o_rat] : ( groups7872700643590313910_o_rat(G,bot_bot_set_o) = zero_zero_rat ) ).

% sum.empty
tff(fact_7034_sum_Oempty,axiom,
    ! [G: fun_o_nat] : ( groups8507830703676809646_o_nat(G,bot_bot_set_o) = zero_zero_nat ) ).

% sum.empty
tff(fact_7035_sum_Oempty,axiom,
    ! [G: fun_o_int] : ( groups8505340233167759370_o_int(G,bot_bot_set_o) = zero_zero_int ) ).

% sum.empty
tff(fact_7036_sum_Oempty,axiom,
    ! [G: fun_nat_Code_integer] : ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),bot_bot_set_nat) = zero_z3403309356797280102nteger ) ).

% sum.empty
tff(fact_7037_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(G),bot_bot_set_nat) = zero_zero_rat ) ).

% sum.empty
tff(fact_7038_sum_Oempty,axiom,
    ! [G: fun_nat_int] : ( aa_set_nat_int(groups3539618377306564664at_int(G),bot_bot_set_nat) = zero_zero_int ) ).

% sum.empty
tff(fact_7039_sum_Oempty,axiom,
    ! [G: fun_int_Code_integer] : ( groups7873554091576472773nteger(G,bot_bot_set_int) = zero_z3403309356797280102nteger ) ).

% sum.empty
tff(fact_7040_sum_Oempty,axiom,
    ! [G: fun_int_rat] : ( groups3906332499630173760nt_rat(G,bot_bot_set_int) = zero_zero_rat ) ).

% sum.empty
tff(fact_7041_sum_Oempty,axiom,
    ! [G: fun_int_nat] : ( groups4541462559716669496nt_nat(G,bot_bot_set_int) = zero_zero_nat ) ).

% sum.empty
tff(fact_7042_convex__sum__bound__le,axiom,
    ! [I4: set_nat,X: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(X,I2)) )
     => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(X),I4) = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I2)),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_kj(X),A)),I4)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7043_convex__sum__bound__le,axiom,
    ! [I4: set_int,X: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(X,I2)) )
     => ( ( groups7873554091576472773nteger(X,I4) = one_one_Code_integer )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_int_Code_integer(A,I2)),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_kk(X),A),I4)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7044_convex__sum__bound__le,axiom,
    ! [I4: set_nat,X: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(X,I2)) )
     => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(X),I4) = one_one_rat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(minus_minus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kl(X),A)),I4)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7045_convex__sum__bound__le,axiom,
    ! [I4: set_int,X: fun_int_rat,A: fun_int_rat,B: rat,Delta: rat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(X,I2)) )
     => ( ( groups3906332499630173760nt_rat(X,I4) = one_one_rat )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_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(aa_rat_fun_rat_rat(minus_minus_rat,groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_km(X),A),I4)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7046_convex__sum__bound__le,axiom,
    ! [I4: set_nat,X: fun_nat_int,A: fun_nat_int,B: int,Delta: int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(X,I2)) )
     => ( ( aa_set_nat_int(groups3539618377306564664at_int(X),I4) = one_one_int )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I4)
             => 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(A,I2)),B))),Delta) )
         => 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_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_kn(X),A)),I4)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7047_convex__sum__bound__le,axiom,
    ! [I4: set_int,X: fun_int_int,A: fun_int_int,B: int,Delta: int] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(X,I2)) )
     => ( ( groups4538972089207619220nt_int(X,I4) = one_one_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I4)
             => 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(A,I2)),B))),Delta) )
         => 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,groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ko(X),A),I4)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7048_convex__sum__bound__le,axiom,
    ! [I4: set_set_nat,X: fun_se7586849273513198688nteger,A: fun_se7586849273513198688nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(I2),I4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(X,I2)) )
     => ( ( groups9190459664516455967nteger(X,I4) = one_one_Code_integer )
       => ( ! [I2: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(I2),I4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_set871585252387937313nteger(A,I2)),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(groups9190459664516455967nteger(aa_fun4587287018503165694nteger(aTP_Lamm_kp(X),A),I4)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7049_convex__sum__bound__le,axiom,
    ! [I4: set_set_nat,X: fun_set_nat_rat,A: fun_set_nat_rat,B: rat,Delta: rat] :
      ( ! [I2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(I2),I4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_set_nat_rat(X,I2)) )
     => ( ( groups7659867448343625626at_rat(X,I4) = one_one_rat )
       => ( ! [I2: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(I2),I4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_set_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(aa_rat_fun_rat_rat(minus_minus_rat,groups7659867448343625626at_rat(aa_fun8478617152544252148at_rat(aTP_Lamm_kq(X),A),I4)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7050_convex__sum__bound__le,axiom,
    ! [I4: set_set_nat,X: fun_set_nat_int,A: fun_set_nat_int,B: int,Delta: int] :
      ( ! [I2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(I2),I4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_set_nat_int(X,I2)) )
     => ( ( groups8292507037921071086at_int(X,I4) = one_one_int )
       => ( ! [I2: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(I2),I4)
             => 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_set_nat_int(A,I2)),B))),Delta) )
         => 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,groups8292507037921071086at_int(aa_fun3600621996379952540at_int(aTP_Lamm_kr(X),A),I4)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7051_convex__sum__bound__le,axiom,
    ! [I4: set_se6260736226359567993nt_int,X: fun_se1399197888071115401nteger,A: fun_se1399197888071115401nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(I2),I4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_set4224706125354629690nteger(X,I2)) )
     => ( ( groups3738144136874984124nteger(X,I4) = one_one_Code_integer )
       => ( ! [I2: set_Pr958786334691620121nt_int] :
              ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(I2),I4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_set4224706125354629690nteger(A,I2)),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(groups3738144136874984124nteger(aa_fun1473822802349397756nteger(aTP_Lamm_ks(X),A),I4)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7052_abs__sum__abs,axiom,
    ! [F: fun_int_int,A2: set_int] : ( aa_int_int(abs_abs_int,groups4538972089207619220nt_int(aTP_Lamm_ki(F),A2)) = groups4538972089207619220nt_int(aTP_Lamm_ki(F),A2) ) ).

% abs_sum_abs
tff(fact_7053_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fk),A2) = zero_zero_nat ) ).

% sum.neutral_const
tff(fact_7054_sum_Oneutral__const,axiom,
    ! [A2: set_int] : ( groups4538972089207619220nt_int(aTP_Lamm_fl,A2) = zero_zero_int ) ).

% sum.neutral_const
tff(fact_7055_sum__diff1__nat,axiom,
    ! [F: fun_Pr1253225528153422666at_nat,A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( groups3860910324918113789at_nat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups3860910324918113789at_nat(F,A2)),aa_Pro6313114768868464763at_nat(F,A)),groups3860910324918113789at_nat(F,A2)) ) ).

% sum_diff1_nat
tff(fact_7056_sum__diff1__nat,axiom,
    ! [F: fun_se4101287982975402108nt_nat,A2: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int] :
      ( groups185207323561075247nt_nat(F,minus_2612819937483484256nt_int(A2,insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int))) = $ite(aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups185207323561075247nt_nat(F,A2)),aa_set4753215920113914285nt_nat(F,A)),groups185207323561075247nt_nat(F,A2)) ) ).

% sum_diff1_nat
tff(fact_7057_sum__diff1__nat,axiom,
    ! [F: fun_set_nat_nat,A2: set_set_nat,A: set_nat] :
      ( groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = $ite(aa_set_set_nat_o(member_set_nat(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A2)),aa_set_nat_nat(F,A)),groups8294997508430121362at_nat(F,A2)) ) ).

% sum_diff1_nat
tff(fact_7058_sum__diff1__nat,axiom,
    ! [F: fun_fu3504661237974713170_o_nat,A2: set_fu3555266933746557007_nat_o,A: fun_Pr6460944699146269935_nat_o] :
      ( groups2076417071786835717_o_nat(F,minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(A,bot_bo8632255151191677115_nat_o))) = $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups2076417071786835717_o_nat(F,A2)),aa_fun767828674014743683_o_nat(F,A)),groups2076417071786835717_o_nat(F,A2)) ) ).

% sum_diff1_nat
tff(fact_7059_sum__diff1__nat,axiom,
    ! [F: fun_Pr2602117396471556196at_nat,A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( groups977919841031483927at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A2)),aa_Pro4243173770044397973at_nat(F,A)),groups977919841031483927at_nat(F,A2)) ) ).

% sum_diff1_nat
tff(fact_7060_sum__diff1__nat,axiom,
    ! [F: fun_o_nat,A2: set_o,A: $o] :
      ( groups8507830703676809646_o_nat(F,minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8507830703676809646_o_nat(F,A2)),aa_o_nat(F,(A))),groups8507830703676809646_o_nat(F,A2)) ) ).

% sum_diff1_nat
tff(fact_7061_sum__diff1__nat,axiom,
    ! [F: fun_int_nat,A2: set_int,A: int] :
      ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),aa_int_nat(F,A)),groups4541462559716669496nt_nat(F,A2)) ) ).

% sum_diff1_nat
tff(fact_7062_sum__diff1__nat,axiom,
    ! [F: fun_nat_nat,A2: set_nat,A: nat] :
      ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_nat_nat(F,A)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) ) ).

% sum_diff1_nat
tff(fact_7063_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_nat,B2: set_nat,A2: set_nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_kt(G),B2)),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_kv(G),A2)),B2) ) ).

% sum.swap
tff(fact_7064_sum_Oswap,axiom,
    ! [G: fun_int_fun_int_int,B2: set_int,A2: set_int] : ( groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_kw(G),B2),A2) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_ky(G),A2),B2) ) ).

% sum.swap
tff(fact_7065_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_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,aa_set_nat_rat(groups2906978787729119204at_rat(F),K5)),aa_set_nat_rat(groups2906978787729119204at_rat(G),K5)) ) ).

% sum_mono
tff(fact_7066_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_7067_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_7068_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_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,aa_set_nat_int(groups3539618377306564664at_int(F),K5)),aa_set_nat_int(groups3539618377306564664at_int(G),K5)) ) ).

% sum_mono
tff(fact_7069_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I2)),aa_nat_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),K5)),aa_set_nat_nat(groups3542108847815614940at_nat(G),K5)) ) ).

% sum_mono
tff(fact_7070_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_int,G: fun_int_int] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,I2)),aa_int_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4538972089207619220nt_int(F,K5)),groups4538972089207619220nt_int(G,K5)) ) ).

% sum_mono
tff(fact_7071_sum__mono,axiom,
    ! [K5: set_set_nat,F: fun_set_nat_rat,G: fun_set_nat_rat] :
      ( ! [I2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(F,I2)),aa_set_nat_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups7659867448343625626at_rat(F,K5)),groups7659867448343625626at_rat(G,K5)) ) ).

% sum_mono
tff(fact_7072_sum__mono,axiom,
    ! [K5: set_set_nat,F: fun_set_nat_nat,G: fun_set_nat_nat] :
      ( ! [I2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(I2),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(F,I2)),aa_set_nat_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8294997508430121362at_nat(F,K5)),groups8294997508430121362at_nat(G,K5)) ) ).

% sum_mono
tff(fact_7073_sum__mono,axiom,
    ! [K5: set_set_nat,F: fun_set_nat_int,G: fun_set_nat_int] :
      ( ! [I2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(F,I2)),aa_set_nat_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups8292507037921071086at_int(F,K5)),groups8292507037921071086at_int(G,K5)) ) ).

% sum_mono
tff(fact_7074_sum__mono,axiom,
    ! [K5: set_se6260736226359567993nt_int,F: fun_se593059424582005380nt_rat,G: fun_se593059424582005380nt_rat] :
      ( ! [I2: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set4118085860027418549nt_rat(F,I2)),aa_set4118085860027418549nt_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups8773449300329355319nt_rat(F,K5)),groups8773449300329355319nt_rat(G,K5)) ) ).

% sum_mono
tff(fact_7075_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H2: fun_nat_nat,A2: set_nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kz(G),H2)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(H2),A2)) ) ).

% sum.distrib
tff(fact_7076_sum_Odistrib,axiom,
    ! [G: fun_int_int,H2: fun_int_int,A2: set_int] : ( groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_la(G),H2),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A2)),groups4538972089207619220nt_int(H2,A2)) ) ).

% sum.distrib
tff(fact_7077_sum__product,axiom,
    ! [F: fun_nat_nat,A2: set_nat,G: fun_nat_nat,B2: set_nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_lc(F),G),B2)),A2) ) ).

% sum_product
tff(fact_7078_sum__product,axiom,
    ! [F: fun_int_int,A2: set_int,G: fun_int_int,B2: set_int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,B2)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_le(F),G),B2),A2) ) ).

% sum_product
tff(fact_7079_sum__distrib__right,axiom,
    ! [F: fun_nat_nat,A2: set_nat,R2: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),R2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lf(F),R2)),A2) ) ).

% sum_distrib_right
tff(fact_7080_sum__distrib__right,axiom,
    ! [F: fun_int_int,A2: set_int,R2: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A2)),R2) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_lg(F),R2),A2) ) ).

% sum_distrib_right
tff(fact_7081_sum__distrib__left,axiom,
    ! [R2: nat,F: fun_nat_nat,A2: set_nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lh(R2),F)),A2) ) ).

% sum_distrib_left
tff(fact_7082_sum__distrib__left,axiom,
    ! [R2: int,F: fun_int_int,A2: set_int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,R2),groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_li(R2),F),A2) ) ).

% sum_distrib_left
tff(fact_7083_sum__subtractf,axiom,
    ! [F: fun_int_int,G: fun_int_int,A2: set_int] : ( groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_lj(F),G),A2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2)) ) ).

% sum_subtractf
tff(fact_7084_sum__negf,axiom,
    ! [F: fun_int_int,A2: set_int] : ( groups4538972089207619220nt_int(aTP_Lamm_lk(F),A2) = aa_int_int(uminus_uminus_int,groups4538972089207619220nt_int(F,A2)) ) ).

% sum_negf
tff(fact_7085_sum__subtractf__nat,axiom,
    ! [A2: set_se6260736226359567993nt_int,G: fun_se4101287982975402108nt_nat,F: fun_se4101287982975402108nt_nat] :
      ( ! [X2: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set4753215920113914285nt_nat(G,X2)),aa_set4753215920113914285nt_nat(F,X2)) )
     => ( groups185207323561075247nt_nat(aa_fun209864160807355708nt_nat(aTP_Lamm_ll(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups185207323561075247nt_nat(F,A2)),groups185207323561075247nt_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7086_sum__subtractf__nat,axiom,
    ! [A2: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(G,X2)),aa_set_nat_nat(F,X2)) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_lm(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A2)),groups8294997508430121362at_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7087_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X2)),aa_int_nat(F,X2)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ln(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7088_sum__subtractf__nat,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,G: fun_fu3504661237974713170_o_nat,F: fun_fu3504661237974713170_o_nat] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_fun767828674014743683_o_nat(G,X2)),aa_fun767828674014743683_o_nat(F,X2)) )
     => ( groups2076417071786835717_o_nat(aa_fun6312600589078465724_o_nat(aTP_Lamm_lo(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups2076417071786835717_o_nat(F,A2)),groups2076417071786835717_o_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7089_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X2)),aa_nat_nat(F,X2)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lp(G),F)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7090_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      ( code_divmod_integer(K,L) = $ite(
          K = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
          $ite(
            aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
            $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_lq(L)),code_divmod_abs(K,L))),
            $ite(
              L = zero_z3403309356797280102nteger,
              produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
              aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
                $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_lr(L)),code_divmod_abs(K,L)))) ) ) ) ) ).

% divmod_integer_code
tff(fact_7091_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_da),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% Sum_Icc_nat
tff(fact_7092_arith__series__nat,axiom,
    ! [A: nat,D2: nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ls(A),D2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D2)))),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% arith_series_nat
tff(fact_7093_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),N)
     => ( groups4538972089207619220nt_int(aTP_Lamm_db,set_or1266510415728281911st_int(M,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_7094_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_da),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% Sum_Ico_nat
tff(fact_7095_sum__power2,axiom,
    ! [K: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),K)),one_one_nat) ) ).

% sum_power2
tff(fact_7096_gauss__sum__nat,axiom,
    ! [N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_da),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% gauss_sum_nat
tff(fact_7097_apsnd__conv,axiom,
    ! [F: fun_Pr5876161306168176259nteger,X: fun_Pr5098669352780829690e_term,Y: produc8923325533196201883nteger] : ( produc2265439103302406840e_term(F,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X),Y)) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X),aa_Pro4693618923521437820nteger(F,Y)) ) ).

% apsnd_conv
tff(fact_7098_apsnd__conv,axiom,
    ! [F: fun_Pr7091840695141096449et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] : ( aa_Pro5931322043027775230et_nat(produc1991759244683392747_nat_o(F),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X),Y)) = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X),aa_Pro3679490825741336130et_nat(F,Y)) ) ).

% apsnd_conv
tff(fact_7099_apsnd__conv,axiom,
    ! [F: fun_Pr4885567672785483807et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] : ( produc4622684310891945677_nat_o(F,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X),Y)) = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(F,Y)) ) ).

% apsnd_conv
tff(fact_7100_apsnd__conv,axiom,
    ! [F: fun_Pr2813956079162534559et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] : ( produc174852792182663501_nat_o(F,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),Y)) = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X),aa_Pro5541488491759012064et_nat(F,Y)) ) ).

% apsnd_conv
tff(fact_7101_apsnd__conv,axiom,
    ! [F: fun_Pr836988951548913597et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] : ( produc2056317742435807279_nat_o(F,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),Y)) = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro5931322043027775230et_nat(F,Y)) ) ).

% apsnd_conv
tff(fact_7102_apsnd__conv,axiom,
    ! [F: fun_Pr2478283820680522243nt_int,X: fun_Pr7763680713840462443e_term,Y: product_prod_int_int] : ( produc8061333830752525225e_term(F,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X),Y)) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X),aa_Pro6970504127699319548nt_int(F,Y)) ) ).

% apsnd_conv
tff(fact_7103_apsnd__conv,axiom,
    ! [F: fun_Pr2478283820680522243nt_int,X: fun_in5730379987311621695e_term,Y: product_prod_int_int] : ( produc2254893824690362237e_term(F,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X),Y)) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X),aa_Pro6970504127699319548nt_int(F,Y)) ) ).

% apsnd_conv
tff(fact_7104_apsnd__conv,axiom,
    ! [F: fun_Co8499238673658658103nteger,X: code_integer,Y: code_integer] : ( aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),produc1086072967326762835nteger(X,Y)) = produc1086072967326762835nteger(X,aa_Cod1783974652533396728nteger(F,Y)) ) ).

% apsnd_conv
tff(fact_7105_atLeastatMost__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( set_or8904488021354931149Most_o((A),(B)) = bot_bot_set_o )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(B)) ) ).

% atLeastatMost_empty_iff
tff(fact_7106_atLeastatMost__empty__iff,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( set_or1955772592623580779er_nat(A,B) = bot_bo498966703094740906er_nat )
    <=> ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_7107_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( set_or4548717258645045905et_nat(A,B) = bot_bot_set_set_nat )
    <=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_7108_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_7109_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_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_7110_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: 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_7111_atLeastatMost__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( set_or189985376899183464nteger(A,B) = bot_bo3990330152332043303nteger )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_7112_atLeastatMost__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o = set_or8904488021354931149Most_o((A),(B)) )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_7113_atLeastatMost__empty__iff2,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( bot_bo498966703094740906er_nat = set_or1955772592623580779er_nat(A,B) )
    <=> ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_7114_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B) )
    <=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_7115_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_7116_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_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_7117_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_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_7118_atLeastatMost__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger = set_or189985376899183464nteger(A,B) )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_7119_atLeastatMost__empty,axiom,
    ! [B: $o,A: $o] :
      ( aa_o_o(aa_o_fun_o_o(ord_less_o,(B)),(A))
     => ( set_or8904488021354931149Most_o((A),(B)) = bot_bot_set_o ) ) ).

% atLeastatMost_empty
tff(fact_7120_atLeastatMost__empty,axiom,
    ! [B: literal,A: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B),A)
     => ( set_or8436436507035938825iteral(A,B) = bot_bot_set_literal ) ) ).

% atLeastatMost_empty
tff(fact_7121_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_7122_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_7123_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_7124_atLeastatMost__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( set_or189985376899183464nteger(A,B) = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
tff(fact_7125_atLeastLessThan__empty,axiom,
    ! [B: $o,A: $o] :
      ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(B)),(A))
     => ( set_or7139685690850216873Than_o((A),(B)) = bot_bot_set_o ) ) ).

% atLeastLessThan_empty
tff(fact_7126_atLeastLessThan__empty,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A)
     => ( set_or1773934645810362255er_nat(A,B) = bot_bo498966703094740906er_nat ) ) ).

% atLeastLessThan_empty
tff(fact_7127_atLeastLessThan__empty,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( set_or3540276404033026485et_nat(A,B) = bot_bot_set_set_nat ) ) ).

% atLeastLessThan_empty
tff(fact_7128_atLeastLessThan__empty,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( set_or4029947393144176647an_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastLessThan_empty
tff(fact_7129_atLeastLessThan__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
tff(fact_7130_atLeastLessThan__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( set_or8404916559141939852nteger(A,B) = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
tff(fact_7131_atLeastLessThan__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( set_or4662586982721622107an_int(A,B) = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_7132_atLeastLessThan__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( set_or7139685690850216873Than_o((A),(B)) = bot_bot_set_o )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(A)),(B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_7133_atLeastLessThan__empty__iff,axiom,
    ! [A: literal,B: literal] :
      ( ( set_or4828121863581493221iteral(A,B) = bot_bot_set_literal )
    <=> ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7134_atLeastLessThan__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7135_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7136_atLeastLessThan__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = bot_bo3990330152332043303nteger )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7137_atLeastLessThan__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( set_or4662586982721622107an_int(A,B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7138_atLeastLessThan__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o = set_or7139685690850216873Than_o((A),(B)) )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(A)),(B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7139_atLeastLessThan__empty__iff2,axiom,
    ! [A: literal,B: literal] :
      ( ( bot_bot_set_literal = set_or4828121863581493221iteral(A,B) )
    <=> ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7140_atLeastLessThan__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or4029947393144176647an_rat(A,B) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7141_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7142_atLeastLessThan__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger = set_or8404916559141939852nteger(A,B) )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7143_atLeastLessThan__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = set_or4662586982721622107an_int(A,B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7144_atLeastAtMost__singleton,axiom,
    ! [A: $o] : ( set_or8904488021354931149Most_o((A),(A)) = insert_o((A),bot_bot_set_o) ) ).

% atLeastAtMost_singleton
tff(fact_7145_atLeastAtMost__singleton,axiom,
    ! [A: nat] : ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),A) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ) ).

% atLeastAtMost_singleton
tff(fact_7146_atLeastAtMost__singleton,axiom,
    ! [A: int] : ( set_or1266510415728281911st_int(A,A) = insert_int(A,bot_bot_set_int) ) ).

% atLeastAtMost_singleton
tff(fact_7147_atLeastAtMost__singleton,axiom,
    ! [A: code_integer] : ( set_or189985376899183464nteger(A,A) = insert_Code_integer(A,bot_bo3990330152332043303nteger) ) ).

% atLeastAtMost_singleton
tff(fact_7148_atLeastAtMost__singleton__iff,axiom,
    ! [A: $o,B: $o,C: $o] :
      ( ( set_or8904488021354931149Most_o((A),(B)) = insert_o((C),bot_bot_set_o) )
    <=> ( ( (A)
        <=> (B) )
        & ( (B)
        <=> (C) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_7149_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = aa_set_nat_set_nat(insert_nat(C),bot_bot_set_nat) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_7150_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( set_or1266510415728281911st_int(A,B) = insert_int(C,bot_bot_set_int) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_7151_atLeastAtMost__singleton__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( set_or189985376899183464nteger(A,B) = insert_Code_integer(C,bot_bo3990330152332043303nteger) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_7152_atLeastLessThan__singleton,axiom,
    ! [M: nat] : ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,M)) = aa_set_nat_set_nat(insert_nat(M),bot_bot_set_nat) ) ).

% atLeastLessThan_singleton
tff(fact_7153_ivl__disj__un__two_I7_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),U)
       => ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,set_or4029947393144176647an_rat(L,M)),set_or633870826150836451st_rat(M,U)) = set_or633870826150836451st_rat(L,U) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_7154_ivl__disj__un__two_I7_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),U)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_7155_ivl__disj__un__two_I7_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M),U)
       => ( sup_su848401254843788991nteger(set_or8404916559141939852nteger(L,M),set_or189985376899183464nteger(M,U)) = set_or189985376899183464nteger(L,U) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_7156_ivl__disj__un__two_I7_J,axiom,
    ! [L: int,M: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),U)
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,set_or4662586982721622107an_int(L,M)),set_or1266510415728281911st_int(M,U)) = set_or1266510415728281911st_int(L,U) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_7157_ivl__disj__int__two_I7_J,axiom,
    ! [L: $o,M: $o,U: $o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,set_or7139685690850216873Than_o((L),(M))),set_or8904488021354931149Most_o((M),(U))) = bot_bot_set_o ) ).

% ivl_disj_int_two(7)
tff(fact_7158_ivl__disj__int__two_I7_J,axiom,
    ! [L: nat,M: nat,U: nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),U)) = bot_bot_set_nat ) ).

% ivl_disj_int_two(7)
tff(fact_7159_ivl__disj__int__two_I7_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] : ( inf_in1364745209274528805nteger(set_or8404916559141939852nteger(L,M),set_or189985376899183464nteger(M,U)) = bot_bo3990330152332043303nteger ) ).

% ivl_disj_int_two(7)
tff(fact_7160_ivl__disj__int__two_I7_J,axiom,
    ! [L: int,M: int,U: int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_or4662586982721622107an_int(L,M)),set_or1266510415728281911st_int(M,U)) = bot_bot_set_int ) ).

% ivl_disj_int_two(7)
tff(fact_7161_not__UNIV__eq__Icc,axiom,
    ! [L3: rat,H3: rat] : ( top_top_set_rat != set_or633870826150836451st_rat(L3,H3) ) ).

% not_UNIV_eq_Icc
tff(fact_7162_not__UNIV__eq__Icc,axiom,
    ! [L3: nat,H3: nat] : ( top_top_set_nat != aa_nat_set_nat(set_or1269000886237332187st_nat(L3),H3) ) ).

% not_UNIV_eq_Icc
tff(fact_7163_not__UNIV__eq__Icc,axiom,
    ! [L3: int,H3: int] : ( top_top_set_int != set_or1266510415728281911st_int(L3,H3) ) ).

% not_UNIV_eq_Icc
tff(fact_7164_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: literal,M: literal,U: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,L),M)
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,M),U)
       => ( sup_sup_set_literal(set_or8436436507035938825iteral(L,M),set_or4828121863581493221iteral(M,U)) = set_or4828121863581493221iteral(L,U) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_7165_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,M),U)
       => ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,set_or633870826150836451st_rat(L,M)),set_or4029947393144176647an_rat(M,U)) = set_or4029947393144176647an_rat(L,U) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_7166_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),U)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),M)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),U)) = aa_nat_set_nat(set_or4665077453230672383an_nat(L),U) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_7167_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,M),U)
       => ( sup_su848401254843788991nteger(set_or189985376899183464nteger(L,M),set_or8404916559141939852nteger(M,U)) = set_or8404916559141939852nteger(L,U) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_7168_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: int,M: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,M),U)
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,set_or1266510415728281911st_int(L,M)),set_or4662586982721622107an_int(M,U)) = set_or4662586982721622107an_int(L,U) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_7169_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $o,B: $o] : ( set_or7139685690850216873Than_o((A),(B)) = minus_minus_set_o(set_or8904488021354931149Most_o((A),(B)),insert_o((B),bot_bot_set_o)) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_7170_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: nat,B: nat] : ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_7171_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: code_integer,B: code_integer] : ( set_or8404916559141939852nteger(A,B) = minus_2355218937544613996nteger(set_or189985376899183464nteger(A,B),insert_Code_integer(B,bot_bo3990330152332043303nteger)) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_7172_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: int,B: int] : ( set_or4662586982721622107an_int(A,B) = minus_minus_set_int(set_or1266510415728281911st_int(A,B),insert_int(B,bot_bot_set_int)) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_7173_sum_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lt(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lu(A),N)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ) ).

% sum.nested_swap
tff(fact_7174_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: $o,U: $o] :
      ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(L)),(U))
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,set_or7139685690850216873Than_o((L),(U))),insert_o((U),bot_bot_set_o)) = set_or8904488021354931149Most_o((L),(U)) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_7175_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),U)
     => ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,set_or4029947393144176647an_rat(L,U)),insert_rat(U,bot_bot_set_rat)) = set_or633870826150836451st_rat(L,U) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_7176_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),U)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)),aa_set_nat_set_nat(insert_nat(U),bot_bot_set_nat)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_7177_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),U)
     => ( sup_su848401254843788991nteger(set_or8404916559141939852nteger(L,U),insert_Code_integer(U,bot_bo3990330152332043303nteger)) = set_or189985376899183464nteger(L,U) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_7178_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),U)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,set_or4662586982721622107an_int(L,U)),insert_int(U,bot_bot_set_int)) = set_or1266510415728281911st_int(L,U) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_7179_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),U)
       => ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,set_or633870826150836451st_rat(L,M)),set_or633870826150836451st_rat(M,U)) = set_or633870826150836451st_rat(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7180_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),U)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7181_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: int,M: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),U)
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,set_or1266510415728281911st_int(L,M)),set_or1266510415728281911st_int(M,U)) = set_or1266510415728281911st_int(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7182_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M),U)
       => ( sup_su848401254843788991nteger(set_or189985376899183464nteger(L,M),set_or189985376899183464nteger(M,U)) = set_or189985376899183464nteger(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7183_atLeastAtMost__singleton_H,axiom,
    ! [A: $o,B: $o] :
      ( ( (A)
      <=> (B) )
     => ( set_or8904488021354931149Most_o((A),(B)) = insert_o((A),bot_bot_set_o) ) ) ).

% atLeastAtMost_singleton'
tff(fact_7184_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_7185_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( set_or1266510415728281911st_int(A,B) = insert_int(A,bot_bot_set_int) ) ) ).

% atLeastAtMost_singleton'
tff(fact_7186_atLeastAtMost__singleton_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( set_or189985376899183464nteger(A,B) = insert_Code_integer(A,bot_bo3990330152332043303nteger) ) ) ).

% atLeastAtMost_singleton'
tff(fact_7187_not__UNIV__le__Icc,axiom,
    ! [L: rat,H2: rat] : ~ aa_set_rat_o(aa_set229218963967607191_rat_o(ord_less_eq_set_rat,top_top_set_rat),set_or633870826150836451st_rat(L,H2)) ).

% not_UNIV_le_Icc
tff(fact_7188_not__UNIV__le__Icc,axiom,
    ! [L: nat,H2: nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,top_top_set_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(L),H2)) ).

% not_UNIV_le_Icc
tff(fact_7189_not__UNIV__le__Icc,axiom,
    ! [L: int,H2: int] : ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,top_top_set_int),set_or1266510415728281911st_int(L,H2)) ).

% not_UNIV_le_Icc
tff(fact_7190_ivl__disj__un__two_I3_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),U)
       => ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,set_or4029947393144176647an_rat(L,M)),set_or4029947393144176647an_rat(M,U)) = set_or4029947393144176647an_rat(L,U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_7191_ivl__disj__un__two_I3_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),U)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),M)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),U)) = aa_nat_set_nat(set_or4665077453230672383an_nat(L),U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_7192_ivl__disj__un__two_I3_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M),U)
       => ( sup_su848401254843788991nteger(set_or8404916559141939852nteger(L,M),set_or8404916559141939852nteger(M,U)) = set_or8404916559141939852nteger(L,U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_7193_ivl__disj__un__two_I3_J,axiom,
    ! [L: int,M: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),U)
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,set_or4662586982721622107an_int(L,M)),set_or4662586982721622107an_int(M,U)) = set_or4662586982721622107an_int(L,U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_7194_ivl__disj__int__two_I3_J,axiom,
    ! [L: $o,M: $o,U: $o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,set_or7139685690850216873Than_o((L),(M))),set_or7139685690850216873Than_o((M),(U))) = bot_bot_set_o ) ).

% ivl_disj_int_two(3)
tff(fact_7195_ivl__disj__int__two_I3_J,axiom,
    ! [L: nat,M: nat,U: nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),M)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),U)) = bot_bot_set_nat ) ).

% ivl_disj_int_two(3)
tff(fact_7196_ivl__disj__int__two_I3_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] : ( inf_in1364745209274528805nteger(set_or8404916559141939852nteger(L,M),set_or8404916559141939852nteger(M,U)) = bot_bo3990330152332043303nteger ) ).

% ivl_disj_int_two(3)
tff(fact_7197_ivl__disj__int__two_I3_J,axiom,
    ! [L: int,M: int,U: int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_or4662586982721622107an_int(L,M)),set_or4662586982721622107an_int(M,U)) = bot_bot_set_int ) ).

% ivl_disj_int_two(3)
tff(fact_7198_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lv(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_7199_atLeastLessThan0,axiom,
    ! [M: nat] : ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),zero_zero_nat) = bot_bot_set_nat ) ).

% atLeastLessThan0
tff(fact_7200_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lw(G)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_7201_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lw(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7202_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lx(G),K)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_7203_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lx(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7204_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( set_or2784603332384507286at_nat(X,Y) = top_to7629004291339433233at_nat )
    <=> ( ( X = bot_bo2099793752762293965at_nat )
        & ( Y = top_to4669805908274784177at_nat ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_7205_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_list_nat,Y: set_list_nat] :
      ( ( set_or1270514513317581473st_nat(X,Y) = top_to1619013496918823676st_nat )
    <=> ( ( X = bot_bot_set_list_nat )
        & ( Y = top_top_set_list_nat ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_7206_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( set_or266616469461849773_set_o(X,Y) = top_top_set_set_o )
    <=> ( ( X = bot_bot_set_o )
        & ( Y = top_top_set_o ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_7207_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_rat,Y: set_rat] :
      ( ( set_or1040488700251649177et_rat(X,Y) = top_top_set_set_rat )
    <=> ( ( X = bot_bot_set_rat )
        & ( Y = top_top_set_rat ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_7208_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( set_or4548717258645045905et_nat(X,Y) = top_top_set_set_nat )
    <=> ( ( X = bot_bot_set_nat )
        & ( Y = top_top_set_nat ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_7209_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( set_or370866239135849197et_int(X,Y) = top_top_set_set_int )
    <=> ( ( X = bot_bot_set_int )
        & ( Y = top_top_set_int ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_7210_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: $o,Y: $o] :
      ( ( set_or8904488021354931149Most_o((X),(Y)) = top_top_set_o )
    <=> ( ( (X)
        <=> bot_bot_o )
        & ( (Y)
        <=> top_top_o ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_7211_aset_I2_J,axiom,
    ! [D3: int,A2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3)) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(P,X3)
                | aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)) ) ) ) ) ) ).

% aset(2)
tff(fact_7212_aset_I1_J,axiom,
    ! [D3: int,A2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3)) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(P,X3)
                & aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)) ) ) ) ) ) ).

% aset(1)
tff(fact_7213_bset_I2_J,axiom,
    ! [D3: int,B2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3)) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(P,X3)
                | aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)) ) ) ) ) ) ).

% bset(2)
tff(fact_7214_bset_I1_J,axiom,
    ! [D3: int,B2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3)) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(P,X3)
                & aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)) ) ) ) ) ) ).

% bset(1)
tff(fact_7215_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N),aa_set_nat_set_nat(insert_nat(N),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)),bot_bot_set_nat) ) ).

% atLeastLessThanSuc
tff(fact_7216_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(I),J)),aa_nat_set_nat(set_or4665077453230672383an_nat(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_7217_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lv(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_7218_aset_I10_J,axiom,
    ! [D2: int,D3: int,A2: set_int,T2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D3)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T2))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)),T2)) ) ) ) ).

% aset(10)
tff(fact_7219_aset_I9_J,axiom,
    ! [D2: int,D3: int,A2: set_int,T2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D3)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T2))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)),T2)) ) ) ) ).

% aset(9)
tff(fact_7220_bset_I10_J,axiom,
    ! [D2: int,D3: int,B2: set_int,T2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D3)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T2))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)),T2)) ) ) ) ).

% bset(10)
tff(fact_7221_bset_I9_J,axiom,
    ! [D2: int,D3: int,B2: set_int,T2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D3)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T2))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)),T2)) ) ) ) ).

% bset(9)
tff(fact_7222_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N))
     => ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N),set_or1266510415728281911st_int(M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_7223_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),numeral_numeral_nat(K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),pred_numeral(K)),aa_set_nat_set_nat(insert_nat(pred_numeral(K)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),pred_numeral(K))),bot_bot_set_nat) ) ).

% atLeastLessThan_nat_numeral
tff(fact_7224_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ly(F)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,N)),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_7225_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lz(F)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N)),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_7226_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,M)),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ma(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7227_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_mb(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7228_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lw(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7229_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ly(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_7230_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lz(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_7231_sum_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_mc(G),N),M)),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_7232_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_Code_integer,A: nat,B: nat] : ( aa_set871585252387937313nteger(groups7501900531339628137nteger(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_md(F),A,B),zero_z3403309356797280102nteger) ) ).

% sum_atLeastAtMost_code
tff(fact_7233_sum__atLeastAtMost__code,axiom,
    ! [F: fun_na2338080690716861136at_nat,A: nat,B: nat] : ( groups6857163185585827899at_nat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo4497565046347964853at_nat(aTP_Lamm_me(F),A,B,zero_z1048942125864253310at_nat) ) ).

% sum_atLeastAtMost_code
tff(fact_7234_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_mf(F),A,B),zero_zero_rat) ) ).

% sum_atLeastAtMost_code
tff(fact_7235_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_mg(F),A),B),zero_zero_int) ) ).

% sum_atLeastAtMost_code
tff(fact_7236_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_mh(F),A),B),zero_zero_nat) ) ).

% sum_atLeastAtMost_code
tff(fact_7237_periodic__finite__ex,axiom,
    ! [D2: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(P,X2)
          <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) )
       => ( ? [X_12: int] : aa_int_o(P,X_12)
        <=> ? [X4: int] :
              ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D2))
              & aa_int_o(P,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_7238_aset_I7_J,axiom,
    ! [D3: int,A2: set_int,T2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T2),X3)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,T2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)) ) ) ) ).

% aset(7)
tff(fact_7239_aset_I5_J,axiom,
    ! [D3: int,T2: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(T2),A2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),T2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)),T2) ) ) ) ) ).

% aset(5)
tff(fact_7240_aset_I4_J,axiom,
    ! [D3: int,T2: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(T2),A2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X3 != T2 )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3) != T2 ) ) ) ) ) ).

% aset(4)
tff(fact_7241_aset_I3_J,axiom,
    ! [D3: int,T2: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T2),one_one_int)),A2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X3 = T2 )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3) = T2 ) ) ) ) ) ).

% aset(3)
tff(fact_7242_bset_I7_J,axiom,
    ! [D3: int,T2: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(T2),B2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T2),X3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,T2),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)) ) ) ) ) ).

% bset(7)
tff(fact_7243_bset_I5_J,axiom,
    ! [D3: int,B2: set_int,T2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),T2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)),T2) ) ) ) ).

% bset(5)
tff(fact_7244_bset_I4_J,axiom,
    ! [D3: int,T2: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(T2),B2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X3 != T2 )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3) != T2 ) ) ) ) ) ).

% bset(4)
tff(fact_7245_bset_I3_J,axiom,
    ! [D3: int,T2: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T2),one_one_int)),B2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X3 = T2 )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3) = T2 ) ) ) ) ) ).

% bset(3)
tff(fact_7246_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_7247_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_int,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_7248_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_7249_simp__from__to,axiom,
    ! [I: int,J: int] :
      ( set_or1266510415728281911st_int(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),bot_bot_set_int,insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J))) ) ).

% simp_from_to
tff(fact_7250_aset_I8_J,axiom,
    ! [D3: int,A2: set_int,T2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T2),X3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)) ) ) ) ).

% aset(8)
tff(fact_7251_aset_I6_J,axiom,
    ! [D3: int,T2: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T2),one_one_int)),A2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),T2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)),T2) ) ) ) ) ).

% aset(6)
tff(fact_7252_bset_I8_J,axiom,
    ! [D3: int,T2: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T2),one_one_int)),B2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T2),X3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T2),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)) ) ) ) ) ).

% bset(8)
tff(fact_7253_bset_I6_J,axiom,
    ! [D3: int,B2: set_int,T2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),T2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)),T2) ) ) ) ).

% bset(6)
tff(fact_7254_cpmi,axiom,
    ! [D3: int,P: fun_int_o,P4: fun_int_o,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( ? [Z5: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z5)
           => ( aa_int_o(P,X2)
            <=> aa_int_o(P4,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa3: int] :
                  ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3))
                 => ! [Xb2: int] :
                      ( aa_set_int_o(member_int(Xb2),B2)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
             => ( aa_int_o(P,X2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3)) ) )
         => ( ! [X2: int,K2: int] :
                ( aa_int_o(P4,X2)
              <=> aa_int_o(P4,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3))
                    & aa_int_o(P4,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3))
                    & ? [Xa2: int] :
                        ( aa_set_int_o(member_int(Xa2),B2)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa2),X4)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_7255_cppi,axiom,
    ! [D3: int,P: fun_int_o,P4: fun_int_o,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( ? [Z5: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X2)
           => ( aa_int_o(P,X2)
            <=> aa_int_o(P4,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa3: int] :
                  ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3))
                 => ! [Xb2: int] :
                      ( aa_set_int_o(member_int(Xb2),A2)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
             => ( aa_int_o(P,X2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3)) ) )
         => ( ! [X2: int,K2: int] :
                ( aa_int_o(P4,X2)
              <=> aa_int_o(P4,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3))
                    & aa_int_o(P4,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3))
                    & ? [Xa2: int] :
                        ( aa_set_int_o(member_int(Xa2),A2)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa2),X4)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_7256_sum__natinterval__diff,axiom,
    ! [F: fun_nat_Code_integer,M: nat,N: nat] :
      ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_mi(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(F,M)),aa_nat_Code_integer(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),zero_z3403309356797280102nteger) ) ).

% sum_natinterval_diff
tff(fact_7257_sum__natinterval__diff,axiom,
    ! [F: fun_nat_rat,M: nat,N: nat] :
      ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_mj(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,M)),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),zero_zero_rat) ) ).

% sum_natinterval_diff
tff(fact_7258_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,M: nat,N: nat] :
      ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_mk(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),zero_zero_int) ) ).

% sum_natinterval_diff
tff(fact_7259_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ml(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,N)),aa_nat_rat(F,M)) ) ) ).

% sum_telescope''
tff(fact_7260_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_mm(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N)),aa_nat_int(F,M)) ) ) ).

% sum_telescope''
tff(fact_7261_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),X)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),M)),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_7262_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_7263_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_7264_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_mn(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ).

% sum.in_pairs
tff(fact_7265_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_mo(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ).

% sum.in_pairs
tff(fact_7266_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mp(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ).

% sum.in_pairs
tff(fact_7267_sum__gp,axiom,
    ! [X: rat,M: nat,N: nat] :
      ( aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),
          zero_zero_rat,
          $ite(X = 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,N),one_one_nat)),M)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X))) ) ) ).

% sum_gp
tff(fact_7268_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : ( aa_set871585252387937313nteger(groups7501900531339628137nteger(semiri4939895301339042750nteger),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ).

% gauss_sum_from_Suc_0
tff(fact_7269_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : ( groups6325495683096345652atural(semiri3763490453095760265atural,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = divide5121882707175180666atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N),plus_p4538020629002901425atural(aa_nat_Code_natural(semiri3763490453095760265atural,N),one_one_Code_natural)),numera5444537566228673987atural(aa_num_num(bit0,one))) ) ).

% gauss_sum_from_Suc_0
tff(fact_7270_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ).

% gauss_sum_from_Suc_0
tff(fact_7271_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% gauss_sum_from_Suc_0
tff(fact_7272_sum__gp__offset,axiom,
    ! [X: rat,M: nat,N: nat] :
      ( aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = $ite(X = one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),M)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X))) ) ).

% sum_gp_offset
tff(fact_7273_arith__series,axiom,
    ! [A: code_integer,D2: code_integer,N: nat] : ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_mq(A),D2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),D2)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ).

% arith_series
tff(fact_7274_arith__series,axiom,
    ! [A: code_natural,D2: code_natural,N: nat] : ( groups6325495683096345652atural(aa_Cod5320533616963873627atural(aTP_Lamm_mr(A),D2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = divide5121882707175180666atural(times_2397367101498566445atural(plus_p4538020629002901425atural(aa_nat_Code_natural(semiri3763490453095760265atural,N),one_one_Code_natural),plus_p4538020629002901425atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),A),times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N),D2))),numera5444537566228673987atural(aa_num_num(bit0,one))) ) ).

% arith_series
tff(fact_7275_arith__series,axiom,
    ! [A: int,D2: int,N: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ms(A),D2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ).

% arith_series
tff(fact_7276_arith__series,axiom,
    ! [A: nat,D2: nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mt(A),D2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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,N)),D2)))),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% arith_series
tff(fact_7277_gauss__sum,axiom,
    ! [N: nat] : ( aa_set871585252387937313nteger(groups7501900531339628137nteger(semiri4939895301339042750nteger),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ).

% gauss_sum
tff(fact_7278_gauss__sum,axiom,
    ! [N: nat] : ( groups6325495683096345652atural(semiri3763490453095760265atural,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = divide5121882707175180666atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N),plus_p4538020629002901425atural(aa_nat_Code_natural(semiri3763490453095760265atural,N),one_one_Code_natural)),numera5444537566228673987atural(aa_num_num(bit0,one))) ) ).

% gauss_sum
tff(fact_7279_gauss__sum,axiom,
    ! [N: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ).

% gauss_sum
tff(fact_7280_gauss__sum,axiom,
    ! [N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% gauss_sum
tff(fact_7281_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(semiri4939895301339042750nteger),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer)) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_7282_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : ( times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),groups6325495683096345652atural(semiri3763490453095760265atural,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N),plus_p4538020629002901425atural(aa_nat_Code_natural(semiri3763490453095760265atural,N),one_one_Code_natural)) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_7283_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_7284_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(semiri681578069525770553at_rat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_7285_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_7286_double__arith__series,axiom,
    ! [A: code_integer,D2: code_integer,N: nat] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_mq(A),D2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),D2))) ) ).

% double_arith_series
tff(fact_7287_double__arith__series,axiom,
    ! [A: code_natural,D2: code_natural,N: nat] : ( times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),groups6325495683096345652atural(aa_Cod5320533616963873627atural(aTP_Lamm_mr(A),D2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = times_2397367101498566445atural(plus_p4538020629002901425atural(aa_nat_Code_natural(semiri3763490453095760265atural,N),one_one_Code_natural),plus_p4538020629002901425atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),A),times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N),D2))) ) ).

% double_arith_series
tff(fact_7288_double__arith__series,axiom,
    ! [A: int,D2: int,N: nat] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ms(A),D2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D2))) ) ).

% double_arith_series
tff(fact_7289_double__arith__series,axiom,
    ! [A: rat,D2: rat,N: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_mu(A),D2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),D2))) ) ).

% double_arith_series
tff(fact_7290_double__arith__series,axiom,
    ! [A: nat,D2: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mt(A),D2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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,N)),D2))) ) ).

% double_arith_series
tff(fact_7291_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_7292_negative__zle,axiom,
    ! [N: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zle
tff(fact_7293_of__nat__mult,axiom,
    ! [M: nat,N: nat] : ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ) ).

% of_nat_mult
tff(fact_7294_of__nat__mult,axiom,
    ! [M: nat,N: nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% of_nat_mult
tff(fact_7295_of__nat__mult,axiom,
    ! [M: nat,N: nat] : ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ).

% of_nat_mult
tff(fact_7296_of__nat__mult,axiom,
    ! [M: nat,N: nat] : ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ).

% of_nat_mult
tff(fact_7297_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_7298_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_7299_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_7300_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_7301_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7302_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7303_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7304_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7305_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,N) = one_one_Code_integer )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7306_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,N) = one_one_int )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7307_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,N) = one_one_nat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7308_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,N) = one_one_rat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7309_negative__zless,axiom,
    ! [N: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zless
tff(fact_7310_of__nat__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : ( aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_mv(F),A2) ) ).

% of_nat_sum
tff(fact_7311_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_mw(F)),A2) ) ).

% of_nat_sum
tff(fact_7312_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : ( aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_mx(F)),A2) ) ).

% of_nat_sum
tff(fact_7313_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : ( aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_my(F)),A2) ) ).

% of_nat_sum
tff(fact_7314_of__nat__Suc,axiom,
    ! [M: nat] : ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_nat_Code_integer(semiri4939895301339042750nteger,M)) ) ).

% of_nat_Suc
tff(fact_7315_of__nat__Suc,axiom,
    ! [M: nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ) ).

% of_nat_Suc
tff(fact_7316_of__nat__Suc,axiom,
    ! [M: nat] : ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M)) ) ).

% of_nat_Suc
tff(fact_7317_of__nat__Suc,axiom,
    ! [M: nat] : ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ) ).

% of_nat_Suc
tff(fact_7318_mult__of__nat__commute,axiom,
    ! [X: nat,Y: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(semiri4939895301339042750nteger,X)) ) ).

% mult_of_nat_commute
tff(fact_7319_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X)) ) ).

% mult_of_nat_commute
tff(fact_7320_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X)) ) ).

% mult_of_nat_commute
tff(fact_7321_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(semiri681578069525770553at_rat,X)) ) ).

% mult_of_nat_commute
tff(fact_7322_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( set_or8404916559141939852nteger(L,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),one_one_Code_integer)) = set_or189985376899183464nteger(L,U) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_7323_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N3: nat] : ( Z != aa_nat_int(semiri1314217659103216013at_int,N3) )
     => ~ ! [N3: nat] : ( Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) ) ) ).

% int_cases2
tff(fact_7324_div__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N: nat] : ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ) ).

% div_mult2_eq'
tff(fact_7325_div__mult2__eq_H,axiom,
    ! [A: code_natural,M: nat,N: nat] : ( divide5121882707175180666atural(A,times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,M),aa_nat_Code_natural(semiri3763490453095760265atural,N))) = divide5121882707175180666atural(divide5121882707175180666atural(A,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N)) ) ).

% div_mult2_eq'
tff(fact_7326_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = 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(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% div_mult2_eq'
tff(fact_7327_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ).

% div_mult2_eq'
tff(fact_7328_int__cases,axiom,
    ! [Z: int] :
      ( ! [N3: nat] : ( Z != aa_nat_int(semiri1314217659103216013at_int,N3) )
     => ~ ! [N3: nat] : ( Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ) ).

% int_cases
tff(fact_7329_int__of__nat__induct,axiom,
    ! [P: fun_int_o,Z: int] :
      ( ! [N3: nat] : aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,N3))
     => ( ! [N3: nat] : aa_int_o(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))))
       => aa_int_o(P,Z) ) ) ).

% int_of_nat_induct
tff(fact_7330_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_7331_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))) ).

% not_int_zless_negative
tff(fact_7332_nat__less__as__int,axiom,
    ! [X3: nat,Xa3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xa3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa3)) ) ).

% nat_less_as_int
tff(fact_7333_nat__leq__as__int,axiom,
    ! [X3: nat,Xa3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa3)) ) ).

% nat_leq_as_int
tff(fact_7334_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ? [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N3)),X)) ) ).

% ex_less_of_nat_mult
tff(fact_7335_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : ( set_or4662586982721622107an_int(L,aa_int_int(aa_int_fun_int_int(plus_plus_int,U),one_one_int)) = set_or1266510415728281911st_int(L,U) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_7336_int__cases4,axiom,
    ! [M: int] :
      ( ! [N3: nat] : ( M != aa_nat_int(semiri1314217659103216013at_int,N3) )
     => ~ ! [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3)
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) ) ) ) ).

% int_cases4
tff(fact_7337_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_7338_int__Suc,axiom,
    ! [N: nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ) ).

% int_Suc
tff(fact_7339_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_7340_negative__zle__0,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),zero_zero_int) ).

% negative_zle_0
tff(fact_7341_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ~ ! [N3: nat] : ( K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) ) ) ).

% nonpos_int_cases
tff(fact_7342_int__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : ( aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_mv(F),A2) ) ).

% int_sum
tff(fact_7343_int__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_mw(F)),A2) ) ).

% int_sum
tff(fact_7344_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N: nat] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ) ).

% mod_mult2_eq'
tff(fact_7345_mod__mult2__eq_H,axiom,
    ! [A: code_natural,M: nat,N: nat] : ( modulo8411746178871703098atural(A,times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,M),aa_nat_Code_natural(semiri3763490453095760265atural,N))) = plus_p4538020629002901425atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,M),modulo8411746178871703098atural(divide5121882707175180666atural(A,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N))),modulo8411746178871703098atural(A,aa_nat_Code_natural(semiri3763490453095760265atural,M))) ) ).

% mod_mult2_eq'
tff(fact_7346_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: 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,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = 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,M)),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,M))),aa_nat_int(semiri1314217659103216013at_int,N)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,M))) ) ).

% mod_mult2_eq'
tff(fact_7347_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ) ).

% mod_mult2_eq'
tff(fact_7348_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N3: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3) )
       => ~ ! [N3: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3) ) ) ) ).

% int_cases3
tff(fact_7349_not__zle__0__negative,axiom,
    ! [N: 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,N)))) ).

% not_zle_0_negative
tff(fact_7350_negD,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int)
     => ? [N3: nat] : ( X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ) ).

% negD
tff(fact_7351_negative__zless__0,axiom,
    ! [N: 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,N)))),zero_zero_int) ).

% negative_zless_0
tff(fact_7352_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
     => ~ ! [N3: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N3)))),E2) ) ).

% nat_approx_posE
tff(fact_7353_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N3: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3) ) ) ).

% neg_int_cases
tff(fact_7354_double__gauss__sum,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(semiri4939895301339042750nteger),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer)) ) ).

% double_gauss_sum
tff(fact_7355_double__gauss__sum,axiom,
    ! [N: nat] : ( times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),groups6325495683096345652atural(semiri3763490453095760265atural,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N),plus_p4538020629002901425atural(aa_nat_Code_natural(semiri3763490453095760265atural,N),one_one_Code_natural)) ) ).

% double_gauss_sum
tff(fact_7356_double__gauss__sum,axiom,
    ! [N: nat] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ) ).

% double_gauss_sum
tff(fact_7357_double__gauss__sum,axiom,
    ! [N: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(semiri681578069525770553at_rat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ) ).

% double_gauss_sum
tff(fact_7358_double__gauss__sum,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ) ).

% double_gauss_sum
tff(fact_7359_pochhammer__double,axiom,
    ! [Z: rat,N: nat] : ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one))))),N)) ) ).

% pochhammer_double
tff(fact_7360_of__nat__code,axiom,
    ! [N: nat] : ( aa_nat_Code_integer(semiri4939895301339042750nteger,N) = aa_Cod1783974652533396728nteger(semiri4055485073559036834nteger(aTP_Lamm_mz,N),zero_z3403309356797280102nteger) ) ).

% of_nat_code
tff(fact_7361_of__nat__code,axiom,
    ! [N: nat] : ( aa_nat_int(semiri1314217659103216013at_int,N) = aa_int_int(aa_nat_fun_int_int(semiri8420488043553186161ux_int(aTP_Lamm_na),N),zero_zero_int) ) ).

% of_nat_code
tff(fact_7362_of__nat__code,axiom,
    ! [N: nat] : ( aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_nb),N),zero_zero_nat) ) ).

% of_nat_code
tff(fact_7363_of__nat__code,axiom,
    ! [N: nat] : ( aa_nat_rat(semiri681578069525770553at_rat,N) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_nc,N),zero_zero_rat) ) ).

% of_nat_code
tff(fact_7364_pochhammer__code,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N) = $ite(N = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_nd(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_Code_integer)) ) ).

% pochhammer_code
tff(fact_7365_pochhammer__code,axiom,
    ! [A: int,N: nat] :
      ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N) = $ite(N = zero_zero_nat,one_one_int,aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_ne(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_int)) ) ).

% pochhammer_code
tff(fact_7366_pochhammer__code,axiom,
    ! [A: rat,N: nat] :
      ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = $ite(N = zero_zero_nat,one_one_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_nf(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_rat)) ) ).

% pochhammer_code
tff(fact_7367_pochhammer__code,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = $ite(N = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_ng(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_nat)) ) ).

% pochhammer_code
tff(fact_7368_sum__gp0,axiom,
    ! [X: rat,N: nat] :
      ( aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_atMost_nat,N)) = $ite(X = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X))) ) ).

% sum_gp0
tff(fact_7369_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_nh(R2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M))),numeral_numeral_rat(aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R2),aa_nat_nat(suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_7370_case__nat__add__eq__if,axiom,
    ! [A: $o,F: fun_nat_o,V: num,N: nat] :
      ( aa_nat_o(case_nat_o((A),F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(V)),N))
    <=> aa_nat_o(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,pred_numeral(V)),N)) ) ).

% case_nat_add_eq_if
tff(fact_7371_case__nat__add__eq__if,axiom,
    ! [A: nat,F: fun_nat_nat,V: num,N: nat] : ( aa_nat_nat(case_nat_nat(A,F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(V)),N)) = aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,pred_numeral(V)),N)) ) ).

% case_nat_add_eq_if
tff(fact_7372_case__nat__add__eq__if,axiom,
    ! [A: option_num,F: fun_nat_option_num,V: num,N: nat] : ( aa_nat_option_num(case_nat_option_num(A,F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(V)),N)) = aa_nat_option_num(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,pred_numeral(V)),N)) ) ).

% case_nat_add_eq_if
tff(fact_7373_gbinomial__0_I1_J,axiom,
    ! [A: code_integer] : ( aa_nat_Code_integer(gbinom8545251970709558553nteger(A),zero_zero_nat) = one_one_Code_integer ) ).

% gbinomial_0(1)
tff(fact_7374_gbinomial__0_I1_J,axiom,
    ! [A: rat] : ( aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ) ).

% gbinomial_0(1)
tff(fact_7375_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_7376_gbinomial__0_I1_J,axiom,
    ! [A: int] : ( aa_nat_int(aa_int_fun_nat_int(gbinomial_int,A),zero_zero_nat) = one_one_int ) ).

% gbinomial_0(1)
tff(fact_7377_pochhammer__0,axiom,
    ! [A: code_integer] : ( aa_nat_Code_integer(comm_s8582702949713902594nteger(A),zero_zero_nat) = one_one_Code_integer ) ).

% pochhammer_0
tff(fact_7378_pochhammer__0,axiom,
    ! [A: rat] : ( aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ) ).

% pochhammer_0
tff(fact_7379_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_7380_pochhammer__0,axiom,
    ! [A: int] : ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),zero_zero_nat) = one_one_int ) ).

% pochhammer_0
tff(fact_7381_case__nat__numeral,axiom,
    ! [A: $o,F: fun_nat_o,V: num] :
      ( aa_nat_o(case_nat_o((A),F),numeral_numeral_nat(V))
    <=> aa_nat_o(F,pred_numeral(V)) ) ).

% case_nat_numeral
tff(fact_7382_case__nat__numeral,axiom,
    ! [A: nat,F: fun_nat_nat,V: num] : ( aa_nat_nat(case_nat_nat(A,F),numeral_numeral_nat(V)) = aa_nat_nat(F,pred_numeral(V)) ) ).

% case_nat_numeral
tff(fact_7383_case__nat__numeral,axiom,
    ! [A: option_num,F: fun_nat_option_num,V: num] : ( aa_nat_option_num(case_nat_option_num(A,F),numeral_numeral_nat(V)) = aa_nat_option_num(F,pred_numeral(V)) ) ).

% case_nat_numeral
tff(fact_7384_atMost__0,axiom,
    aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat) ).

% atMost_0
tff(fact_7385_nat_Ocase__distrib,axiom,
    ! [H2: fun_o_o,F1: $o,F22: fun_nat_o,Nat: nat] :
      ( aa_o_o(H2,aa_nat_o(case_nat_o((F1),F22),Nat))
    <=> aa_nat_o(case_nat_o(aa_o_o(H2,(F1)),aa_fun5937193402907943164_nat_o(aTP_Lamm_ni(H2),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_7386_nat_Ocase__distrib,axiom,
    ! [H2: fun_o_nat,F1: $o,F22: fun_nat_o,Nat: nat] : ( aa_o_nat(H2,aa_nat_o(case_nat_o((F1),F22),Nat)) = aa_nat_nat(case_nat_nat(aa_o_nat(H2,(F1)),aa_fun4204860950123276802at_nat(aTP_Lamm_nj(H2),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_7387_nat_Ocase__distrib,axiom,
    ! [H2: fun_o_option_num,F1: $o,F22: fun_nat_o,Nat: nat] : ( aa_o_option_num(H2,aa_nat_o(case_nat_o((F1),F22),Nat)) = aa_nat_option_num(case_nat_option_num(aa_o_option_num(H2,(F1)),aa_fun6990899178391151900on_num(aTP_Lamm_nk(H2),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_7388_nat_Ocase__distrib,axiom,
    ! [H2: fun_nat_o,F1: nat,F22: fun_nat_nat,Nat: nat] :
      ( aa_nat_o(H2,aa_nat_nat(case_nat_nat(F1,F22),Nat))
    <=> aa_nat_o(case_nat_o(aa_nat_o(H2,F1),aa_fun8673969331427483654_nat_o(aTP_Lamm_nl(H2),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_7389_nat_Ocase__distrib,axiom,
    ! [H2: fun_nat_nat,F1: nat,F22: fun_nat_nat,Nat: nat] : ( aa_nat_nat(H2,aa_nat_nat(case_nat_nat(F1,F22),Nat)) = aa_nat_nat(case_nat_nat(aa_nat_nat(H2,F1),aa_fun5791449513213591928at_nat(aTP_Lamm_nm(H2),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_7390_nat_Ocase__distrib,axiom,
    ! [H2: fun_nat_option_num,F1: nat,F22: fun_nat_nat,Nat: nat] : ( aa_nat_option_num(H2,aa_nat_nat(case_nat_nat(F1,F22),Nat)) = aa_nat_option_num(case_nat_option_num(aa_nat_option_num(H2,F1),aa_fun6517009837249254290on_num(aTP_Lamm_nn(H2),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_7391_nat_Ocase__distrib,axiom,
    ! [H2: fun_option_num_o,F1: option_num,F22: fun_nat_option_num,Nat: nat] :
      ( aa_option_num_o(H2,aa_nat_option_num(case_nat_option_num(F1,F22),Nat))
    <=> aa_nat_o(case_nat_o(aa_option_num_o(H2,F1),aa_fun1675964276245824108_nat_o(aTP_Lamm_no(H2),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_7392_nat_Ocase__distrib,axiom,
    ! [H2: fun_option_num_nat,F1: option_num,F22: fun_nat_option_num,Nat: nat] : ( aa_option_num_nat(H2,aa_nat_option_num(case_nat_option_num(F1,F22),Nat)) = aa_nat_nat(case_nat_nat(aa_option_num_nat(H2,F1),aa_fun7387201323300553874at_nat(aTP_Lamm_np(H2),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_7393_nat_Ocase__distrib,axiom,
    ! [H2: fun_op8989173174992673105on_num,F1: option_num,F22: fun_nat_option_num,Nat: nat] : ( aa_opt5369159329878751634on_num(H2,aa_nat_option_num(case_nat_option_num(F1,F22),Nat)) = aa_nat_option_num(case_nat_option_num(aa_opt5369159329878751634on_num(H2,F1),aa_fun7716400287237121964on_num(aTP_Lamm_nq(H2),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_7394_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: $o] : ( bot_bot_set_o != set_ord_atMost_o((H2)) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_7395_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: int] : ( bot_bot_set_int != set_ord_atMost_int(H2) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_7396_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: nat] : ( bot_bot_set_nat != aa_nat_set_nat(set_ord_atMost_nat,H2) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_7397_not__UNIV__eq__Iic,axiom,
    ! [H3: rat] : ( top_top_set_rat != set_ord_atMost_rat(H3) ) ).

% not_UNIV_eq_Iic
tff(fact_7398_not__UNIV__eq__Iic,axiom,
    ! [H3: int] : ( top_top_set_int != set_ord_atMost_int(H3) ) ).

% not_UNIV_eq_Iic
tff(fact_7399_not__UNIV__eq__Iic,axiom,
    ! [H3: nat] : ( top_top_set_nat != aa_nat_set_nat(set_ord_atMost_nat,H3) ) ).

% not_UNIV_eq_Iic
tff(fact_7400_atMost__def,axiom,
    ! [U: set_Pr958786334691620121nt_int] : ( set_or2459421552957432928nt_int(U) = collec5210948495886036740nt_int(aa_set4993530631398190431_int_o(aTP_Lamm_nr,U)) ) ).

% atMost_def
tff(fact_7401_atMost__def,axiom,
    ! [U: fun_Pr6460944699146269935_nat_o] : ( set_or403216468232133430_nat_o(U) = collec6407654713642994138_nat_o(aa_fun3380769697021226975at_o_o(aTP_Lamm_ns,U)) ) ).

% atMost_def
tff(fact_7402_atMost__def,axiom,
    ! [U: filter_nat] : ( set_or9144418160755794905er_nat(U) = collect_filter_nat(aa_fil403049915117619187_nat_o(aTP_Lamm_gb,U)) ) ).

% atMost_def
tff(fact_7403_atMost__def,axiom,
    ! [U: set_nat] : ( set_or4236626031148496127et_nat(U) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_gd,U)) ) ).

% atMost_def
tff(fact_7404_atMost__def,axiom,
    ! [U: rat] : ( set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_nt,U)) ) ).

% atMost_def
tff(fact_7405_atMost__def,axiom,
    ! [U: int] : ( set_ord_atMost_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_nu,U)) ) ).

% atMost_def
tff(fact_7406_atMost__def,axiom,
    ! [U: nat] : ( aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gf,U)) ) ).

% atMost_def
tff(fact_7407_gbinomial__parallel__sum,axiom,
    ! [A: rat,N: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_nv(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,N))),one_one_rat)),N) ) ).

% gbinomial_parallel_sum
tff(fact_7408_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] : ( aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) ) ).

% gbinomial_Suc_Suc
tff(fact_7409_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_di),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_7410_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_bj),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_7411_not__UNIV__le__Iic,axiom,
    ! [H2: rat] : ~ aa_set_rat_o(aa_set229218963967607191_rat_o(ord_less_eq_set_rat,top_top_set_rat),set_ord_atMost_rat(H2)) ).

% not_UNIV_le_Iic
tff(fact_7412_not__UNIV__le__Iic,axiom,
    ! [H2: int] : ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,top_top_set_int),set_ord_atMost_int(H2)) ).

% not_UNIV_le_Iic
tff(fact_7413_not__UNIV__le__Iic,axiom,
    ! [H2: nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,top_top_set_nat),aa_nat_set_nat(set_ord_atMost_nat,H2)) ).

% not_UNIV_le_Iic
tff(fact_7414_atMost__eq__UNIV__iff,axiom,
    ! [X: set_list_nat] :
      ( ( set_or2492388921469580815st_nat(X) = top_to1619013496918823676st_nat )
    <=> ( X = top_top_set_list_nat ) ) ).

% atMost_eq_UNIV_iff
tff(fact_7415_atMost__eq__UNIV__iff,axiom,
    ! [X: set_o] :
      ( ( set_ord_atMost_set_o(X) = top_top_set_set_o )
    <=> ( X = top_top_set_o ) ) ).

% atMost_eq_UNIV_iff
tff(fact_7416_atMost__eq__UNIV__iff,axiom,
    ! [X: set_rat] :
      ( ( set_or728397472755099399et_rat(X) = top_top_set_set_rat )
    <=> ( X = top_top_set_rat ) ) ).

% atMost_eq_UNIV_iff
tff(fact_7417_atMost__eq__UNIV__iff,axiom,
    ! [X: set_nat] :
      ( ( set_or4236626031148496127et_nat(X) = top_top_set_set_nat )
    <=> ( X = top_top_set_nat ) ) ).

% atMost_eq_UNIV_iff
tff(fact_7418_atMost__eq__UNIV__iff,axiom,
    ! [X: set_int] :
      ( ( set_or58775011639299419et_int(X) = top_top_set_set_int )
    <=> ( X = top_top_set_int ) ) ).

% atMost_eq_UNIV_iff
tff(fact_7419_atMost__eq__UNIV__iff,axiom,
    ! [X: $o] :
      ( ( set_ord_atMost_o((X)) = top_top_set_o )
    <=> ( (X)
      <=> top_top_o ) ) ).

% atMost_eq_UNIV_iff
tff(fact_7420_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_nw(A)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),M)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),M)) ) ).

% gbinomial_sum_lower_neg
tff(fact_7421_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] : ( aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),K)) ) ).

% gbinomial_addition_formula
tff(fact_7422_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),K)) ) ).

% gbinomial_absorb_comp
tff(fact_7423_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ) ).

% gbinomial_mult_1
tff(fact_7424_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ) ).

% gbinomial_mult_1'
tff(fact_7425_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: rat,X: rat,Y: rat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_nx(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ny(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_7426_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( aa_nat_Code_integer(comm_s8582702949713902594nteger(zero_z3403309356797280102nteger),N) = $ite(N = zero_zero_nat,one_one_Code_integer,zero_z3403309356797280102nteger) ) ).

% pochhammer_0_left
tff(fact_7427_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = $ite(N = zero_zero_nat,one_one_rat,zero_zero_rat) ) ).

% pochhammer_0_left
tff(fact_7428_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = $ite(N = zero_zero_nat,one_one_nat,zero_zero_nat) ) ).

% pochhammer_0_left
tff(fact_7429_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,zero_zero_int),N) = $ite(N = zero_zero_nat,one_one_int,zero_zero_int) ) ).

% pochhammer_0_left
tff(fact_7430_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: rat,X: rat,Y: rat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_nx(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_nz(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_7431_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_oa(M)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),M) ) ).

% gbinomial_sum_nat_pow2
tff(fact_7432_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ) ).

% Suc_times_gbinomial
tff(fact_7433_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),K)) ) ).

% gbinomial_absorption
tff(fact_7434_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,M)),N) ) ).

% less_eq_nat.simps(2)
tff(fact_7435_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),M)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7436_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ma(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ) ).

% sum.atMost_Suc_shift
tff(fact_7437_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_mb(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ) ).

% sum.atMost_Suc_shift
tff(fact_7438_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = 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_lw(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ) ).

% sum.atMost_Suc_shift
tff(fact_7439_sum__telescope,axiom,
    ! [F: fun_nat_rat,I: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ob(F)),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_rat_rat(aa_rat_fun_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_7440_sum__telescope,axiom,
    ! [F: fun_nat_int,I: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_oc(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_7441_pochhammer__rec,axiom,
    ! [A: code_integer,N: nat] : ( aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),N)) ) ).

% pochhammer_rec
tff(fact_7442_pochhammer__rec,axiom,
    ! [A: rat,N: nat] : ( aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),N)) ) ).

% pochhammer_rec
tff(fact_7443_pochhammer__rec,axiom,
    ! [A: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),N)) ) ).

% pochhammer_rec
tff(fact_7444_pochhammer__rec,axiom,
    ! [A: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),N)) ) ).

% pochhammer_rec
tff(fact_7445_pochhammer__Suc,axiom,
    ! [A: code_integer,N: nat] : ( aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) ) ).

% pochhammer_Suc
tff(fact_7446_pochhammer__Suc,axiom,
    ! [A: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% pochhammer_Suc
tff(fact_7447_pochhammer__Suc,axiom,
    ! [A: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,N))) ) ).

% pochhammer_Suc
tff(fact_7448_pochhammer__Suc,axiom,
    ! [A: rat,N: nat] : ( aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(A),N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,N))) ) ).

% pochhammer_Suc
tff(fact_7449_pochhammer__rec_H,axiom,
    ! [Z: code_integer,N: nat] : ( aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,N))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N)) ) ).

% pochhammer_rec'
tff(fact_7450_pochhammer__rec_H,axiom,
    ! [Z: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),N)) ) ).

% pochhammer_rec'
tff(fact_7451_pochhammer__rec_H,axiom,
    ! [Z: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)) ) ).

% pochhammer_rec'
tff(fact_7452_pochhammer__rec_H,axiom,
    ! [Z: rat,N: nat] : ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,N))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)) ) ).

% pochhammer_rec'
tff(fact_7453_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7454_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7455_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7456_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7457_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7458_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7459_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
    <=> ? [K4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K4),N)
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K4)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7460_gbinomial__rec,axiom,
    ! [A: rat,K: nat] : ( aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ) ).

% gbinomial_rec
tff(fact_7461_gbinomial__factors,axiom,
    ! [A: rat,K: nat] : ( aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))),aa_nat_rat(gbinomial_rat(A),K)) ) ).

% gbinomial_factors
tff(fact_7462_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7463_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7464_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7465_pochhammer__product_H,axiom,
    ! [Z: code_integer,N: nat,M: nat] : ( aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,N))),M)) ) ).

% pochhammer_product'
tff(fact_7466_pochhammer__product_H,axiom,
    ! [Z: int,N: nat,M: nat] : ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),N)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),M)) ) ).

% pochhammer_product'
tff(fact_7467_pochhammer__product_H,axiom,
    ! [Z: nat,N: nat,M: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),M)) ) ).

% pochhammer_product'
tff(fact_7468_pochhammer__product_H,axiom,
    ! [Z: rat,N: nat,M: nat] : ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,N))),M)) ) ).

% pochhammer_product'
tff(fact_7469_gbinomial__negated__upper,axiom,
    ! [A: rat,K: nat] : ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A)),one_one_rat)),K)) ) ).

% gbinomial_negated_upper
tff(fact_7470_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),one_one_rat)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),N)) ) ).

% gbinomial_index_swap
tff(fact_7471_gbinomial__r__part__sum,axiom,
    ! [M: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,M))),one_one_rat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)) ) ).

% gbinomial_r_part_sum
tff(fact_7472_diff__Suc,axiom,
    ! [M: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_da),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ).

% diff_Suc
tff(fact_7473_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_nh(A)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),one_one_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,M),one_one_nat))) ) ).

% gbinomial_partial_row_sum
tff(fact_7474_Nitpick_Ocase__nat__unfold,axiom,
    ! [X: $o,F: fun_nat_o,N: nat] :
      ( aa_nat_o(case_nat_o((X),F),N)
    <=> $ite(N = zero_zero_nat,(X),aa_nat_o(F,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% Nitpick.case_nat_unfold
tff(fact_7475_Nitpick_Ocase__nat__unfold,axiom,
    ! [X: nat,F: fun_nat_nat,N: nat] :
      ( aa_nat_nat(case_nat_nat(X,F),N) = $ite(N = zero_zero_nat,X,aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% Nitpick.case_nat_unfold
tff(fact_7476_Nitpick_Ocase__nat__unfold,axiom,
    ! [X: option_num,F: fun_nat_option_num,N: nat] :
      ( aa_nat_option_num(case_nat_option_num(X,F),N) = $ite(N = zero_zero_nat,X,aa_nat_option_num(F,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% Nitpick.case_nat_unfold
tff(fact_7477_gbinomial__minus,axiom,
    ! [A: rat,K: nat] : ( aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ) ).

% gbinomial_minus
tff(fact_7478_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : ( aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atMost_nat,N)),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat)) ) ).

% atLeast1_atMost_eq_remove0
tff(fact_7479_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7480_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : ( groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_od(N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_of(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% sum.triangle_reindex_eq
tff(fact_7481_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),M)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_7482_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),M)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_7483_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),M)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_7484_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_7485_sum__gp__basic,axiom,
    ! [X: code_integer,N: nat] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),X)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_nat_nat(suc,N))) ) ).

% sum_gp_basic
tff(fact_7486_sum__gp__basic,axiom,
    ! [X: rat,N: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))) ) ).

% sum_gp_basic
tff(fact_7487_sum__gp__basic,axiom,
    ! [X: int,N: 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),X)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_nat_nat(suc,N))) ) ).

% sum_gp_basic
tff(fact_7488_sum__power__shift,axiom,
    ! [M: nat,N: nat,X: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),M)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_7489_sum__power__shift,axiom,
    ! [M: nat,N: nat,X: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),M)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_7490_sum__power__shift,axiom,
    ! [M: nat,N: nat,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),M)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_7491_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_og(K)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ) ).

% gbinomial_sum_up_index
tff(fact_7492_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(R2),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),one_one_Code_integer)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_7493_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,R2)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R2)),one_one_int)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_7494_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,R2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R2)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,R2)),one_one_rat)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_7495_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7496_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N: nat] : ( aa_set_nat_rat(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,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_mn(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% sum.in_pairs_0
tff(fact_7497_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: nat] : ( aa_set_nat_int(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,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_mo(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% sum.in_pairs_0
tff(fact_7498_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N: 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,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mp(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% sum.in_pairs_0
tff(fact_7499_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] : ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),one_one_Code_integer)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K)) ) ).

% pochhammer_minus'
tff(fact_7500_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] : ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,B)),K)) ) ).

% pochhammer_minus'
tff(fact_7501_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] : ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K)) ) ).

% pochhammer_minus'
tff(fact_7502_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] : ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),one_one_Code_integer)),K)) ) ).

% pochhammer_minus
tff(fact_7503_pochhammer__minus,axiom,
    ! [B: int,K: nat] : ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,B)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ) ).

% pochhammer_minus
tff(fact_7504_pochhammer__minus,axiom,
    ! [B: rat,K: nat] : ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ) ).

% pochhammer_minus
tff(fact_7505_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_Code_integer,H2: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_oh(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_oi(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7506_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_na2338080690716861136at_nat,H2: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( groups6857163185585827899at_nat(aa_fun4141531684813184444at_nat(aa_fun8230936523094545239at_nat(aTP_Lamm_oj(K),G),H2),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups6857163185585827899at_nat(aa_fun4141531684813184444at_nat(aa_fun8230936523094545239at_nat(aTP_Lamm_ok(K),G),H2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7507_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_rat,H2: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ol(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_om(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7508_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_int,H2: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_on(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_oo(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7509_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_nat,H2: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_op(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_oq(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7510_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_or(Z)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),one_one_nat))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_7511_choose__odd__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_os(N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_7512_choose__odd__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ot,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_7513_choose__odd__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ou(N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_7514_choose__even__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_ov(N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_7515_choose__even__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ow,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_7516_choose__even__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ox(N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_7517_gbinomial__code,axiom,
    ! [A: rat,K: nat] :
      ( aa_nat_rat(gbinomial_rat(A),K) = $ite(K = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_oy(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_rat)),aa_nat_rat(semiri773545260158071498ct_rat,K))) ) ).

% gbinomial_code
tff(fact_7518_fact__double,axiom,
    ! [N: nat] : ( aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one)))),N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ) ).

% fact_double
tff(fact_7519_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_oz(N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_7520_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_pa,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_7521_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_pb(N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_7522_atMost__UNIV__triv,axiom,
    set_or2492388921469580815st_nat(top_top_set_list_nat) = top_to1619013496918823676st_nat ).

% atMost_UNIV_triv
tff(fact_7523_atMost__UNIV__triv,axiom,
    set_ord_atMost_set_o(top_top_set_o) = top_top_set_set_o ).

% atMost_UNIV_triv
tff(fact_7524_atMost__UNIV__triv,axiom,
    set_or728397472755099399et_rat(top_top_set_rat) = top_top_set_set_rat ).

% atMost_UNIV_triv
tff(fact_7525_atMost__UNIV__triv,axiom,
    set_or4236626031148496127et_nat(top_top_set_nat) = top_top_set_set_nat ).

% atMost_UNIV_triv
tff(fact_7526_atMost__UNIV__triv,axiom,
    set_or58775011639299419et_int(top_top_set_int) = top_top_set_set_int ).

% atMost_UNIV_triv
tff(fact_7527_binomial__n__n,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),N) = one_one_nat ) ).

% binomial_n_n
tff(fact_7528_prod_Oneutral__const,axiom,
    ! [A2: set_nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_pc),A2) = one_one_nat ) ).

% prod.neutral_const
tff(fact_7529_prod_Oneutral__const,axiom,
    ! [A2: set_nat] : ( aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_pd),A2) = one_one_int ) ).

% prod.neutral_const
tff(fact_7530_prod_Oneutral__const,axiom,
    ! [A2: set_int] : ( groups1705073143266064639nt_int(aTP_Lamm_pe,A2) = one_one_int ) ).

% prod.neutral_const
tff(fact_7531_of__nat__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : ( aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_mv(F),A2) ) ).

% of_nat_prod
tff(fact_7532_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : ( aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_mx(F)),A2) ) ).

% of_nat_prod
tff(fact_7533_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : ( aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_my(F)),A2) ) ).

% of_nat_prod
tff(fact_7534_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_mw(F)),A2) ) ).

% of_nat_prod
tff(fact_7535_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : ( aa_int_rat(ring_1_of_int_rat,aa_set_nat_int(groups705719431365010083at_int(F),A2)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_pf(F)),A2) ) ).

% of_int_prod
tff(fact_7536_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : ( aa_int_int(ring_1_of_int_int,aa_set_nat_int(groups705719431365010083at_int(F),A2)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_pg(F)),A2) ) ).

% of_int_prod
tff(fact_7537_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : ( aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(F,A2)) = groups1072433553688619179nt_rat(aTP_Lamm_kb(F),A2) ) ).

% of_int_prod
tff(fact_7538_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : ( aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_kc(F),A2) ) ).

% of_int_prod
tff(fact_7539_prod_Oempty,axiom,
    ! [G: fun_o_Code_integer] : ( groups7694694392188491536nteger(G,bot_bot_set_o) = one_one_Code_integer ) ).

% prod.empty
tff(fact_7540_prod_Oempty,axiom,
    ! [G: fun_o_assn] : ( groups5301882518646026715o_assn(G,bot_bot_set_o) = one_one_assn ) ).

% prod.empty
tff(fact_7541_prod_Oempty,axiom,
    ! [G: fun_o_rat] : ( groups2869687844427037835_o_rat(G,bot_bot_set_o) = one_one_rat ) ).

% prod.empty
tff(fact_7542_prod_Oempty,axiom,
    ! [G: fun_o_nat] : ( groups3504817904513533571_o_nat(G,bot_bot_set_o) = one_one_nat ) ).

% prod.empty
tff(fact_7543_prod_Oempty,axiom,
    ! [G: fun_o_int] : ( groups3502327434004483295_o_int(G,bot_bot_set_o) = one_one_int ) ).

% prod.empty
tff(fact_7544_prod_Oempty,axiom,
    ! [G: fun_nat_Code_integer] : ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),bot_bot_set_nat) = one_one_Code_integer ) ).

% prod.empty
tff(fact_7545_prod_Oempty,axiom,
    ! [G: fun_nat_assn] : ( groups6906906614972039071t_assn(G,bot_bot_set_nat) = one_one_assn ) ).

% prod.empty
tff(fact_7546_prod_Oempty,axiom,
    ! [G: fun_nat_rat] : ( aa_set_nat_rat(groups73079841787564623at_rat(G),bot_bot_set_nat) = one_one_rat ) ).

% prod.empty
tff(fact_7547_prod_Oempty,axiom,
    ! [G: fun_int_Code_integer] : ( groups3827104343326376752nteger(G,bot_bot_set_int) = one_one_Code_integer ) ).

% prod.empty
tff(fact_7548_prod_Oempty,axiom,
    ! [G: fun_int_assn] : ( groups7882442080178216443t_assn(G,bot_bot_set_int) = one_one_assn ) ).

% prod.empty
tff(fact_7549_fact__0,axiom,
    aa_nat_Code_integer(semiri3624122377584611663nteger,zero_zero_nat) = one_one_Code_integer ).

% fact_0
tff(fact_7550_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_7551_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_7552_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_7553_fact__1,axiom,
    aa_nat_Code_integer(semiri3624122377584611663nteger,one_one_nat) = one_one_Code_integer ).

% fact_1
tff(fact_7554_fact__1,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).

% fact_1
tff(fact_7555_fact__1,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).

% fact_1
tff(fact_7556_fact__1,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).

% fact_1
tff(fact_7557_binomial__n__0,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),zero_zero_nat) = one_one_nat ) ).

% binomial_n_0
tff(fact_7558_fact__Suc__0,axiom,
    aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,zero_zero_nat)) = one_one_Code_integer ).

% fact_Suc_0
tff(fact_7559_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_7560_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_7561_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_7562_fact__Suc,axiom,
    ! [N: nat] : ( aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,N))),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ) ).

% fact_Suc
tff(fact_7563_fact__Suc,axiom,
    ! [N: nat] : ( aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))),aa_nat_int(semiri1406184849735516958ct_int,N)) ) ).

% fact_Suc
tff(fact_7564_fact__Suc,axiom,
    ! [N: nat] : ( aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ) ).

% fact_Suc
tff(fact_7565_fact__Suc,axiom,
    ! [N: nat] : ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N))),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ) ).

% fact_Suc
tff(fact_7566_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_Code_integer(G,aa_nat_nat(suc,N))) ) ).

% prod.atMost_Suc
tff(fact_7567_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_assn,N: nat] : ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_assn(G,aa_nat_nat(suc,N))) ) ).

% prod.atMost_Suc
tff(fact_7568_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ).

% prod.atMost_Suc
tff(fact_7569_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = 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,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ).

% prod.atMost_Suc
tff(fact_7570_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ).

% prod.atMost_Suc
tff(fact_7571_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N: nat] :
      ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_Code_integer(G,aa_nat_nat(suc,N)))) ) ).

% prod.cl_ivl_Suc
tff(fact_7572_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_assn,M: nat,N: nat] :
      ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_assn(G,aa_nat_nat(suc,N)))) ) ).

% prod.cl_ivl_Suc
tff(fact_7573_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] :
      ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N)))) ) ).

% prod.cl_ivl_Suc
tff(fact_7574_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] :
      ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N)))) ) ).

% prod.cl_ivl_Suc
tff(fact_7575_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] :
      ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N)))) ) ).

% prod.cl_ivl_Suc
tff(fact_7576_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N: nat] :
      ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_Code_integer(G,N))) ) ).

% prod.op_ivl_Suc
tff(fact_7577_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_assn,M: nat,N: nat] :
      ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_assn(G,N))) ) ).

% prod.op_ivl_Suc
tff(fact_7578_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] :
      ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_rat(G,N))) ) ).

% prod.op_ivl_Suc
tff(fact_7579_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] :
      ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_nat(G,N))) ) ).

% prod.op_ivl_Suc
tff(fact_7580_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] :
      ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_int(G,N))) ) ).

% prod.op_ivl_Suc
tff(fact_7581_prod_Oswap,axiom,
    ! [G: fun_nat_fun_nat_nat,B2: set_nat,A2: set_nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_ph(G),B2)),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_pi(G),A2)),B2) ) ).

% prod.swap
tff(fact_7582_prod_Oswap,axiom,
    ! [G: fun_nat_fun_nat_int,B2: set_nat,A2: set_nat] : ( aa_set_nat_int(groups705719431365010083at_int(aa_set2359242532610849469at_int(aTP_Lamm_pj(G),B2)),A2) = aa_set_nat_int(groups705719431365010083at_int(aa_set2359242532610849469at_int(aTP_Lamm_pl(G),A2)),B2) ) ).

% prod.swap
tff(fact_7583_prod_Oswap,axiom,
    ! [G: fun_nat_fun_int_int,B2: set_int,A2: set_nat] : ( aa_set_nat_int(groups705719431365010083at_int(aa_set6310687161026558745at_int(aTP_Lamm_pm(G),B2)),A2) = groups1705073143266064639nt_int(aa_set3358596244511904025nt_int(aTP_Lamm_po(G),A2),B2) ) ).

% prod.swap
tff(fact_7584_prod_Oswap,axiom,
    ! [G: fun_int_fun_nat_int,B2: set_nat,A2: set_int] : ( groups1705073143266064639nt_int(aa_set3358596244511904025nt_int(aTP_Lamm_pp(G),B2),A2) = aa_set_nat_int(groups705719431365010083at_int(aa_set6310687161026558745at_int(aTP_Lamm_pr(G),A2)),B2) ) ).

% prod.swap
tff(fact_7585_prod_Oswap,axiom,
    ! [G: fun_int_fun_int_int,B2: set_int,A2: set_int] : ( groups1705073143266064639nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_ps(G),B2),A2) = groups1705073143266064639nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_pt(G),A2),B2) ) ).

% prod.swap
tff(fact_7586_prod_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
         => ( aa_nat_nat(G,X2) = one_one_nat ) )
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A2) = one_one_nat ) ) ).

% prod.neutral
tff(fact_7587_prod_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
         => ( aa_nat_int(G,X2) = one_one_int ) )
     => ( aa_set_nat_int(groups705719431365010083at_int(G),A2) = one_one_int ) ) ).

% prod.neutral
tff(fact_7588_prod_Oneutral,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => ( aa_int_int(G,X2) = one_one_int ) )
     => ( groups1705073143266064639nt_int(G,A2) = one_one_int ) ) ).

% prod.neutral
tff(fact_7589_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_Code_integer,A2: set_nat] :
      ( ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2) != one_one_Code_integer )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),A2)
           => ( aa_nat_Code_integer(G,A4) = one_one_Code_integer ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7590_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_Code_integer,A2: set_int] :
      ( ( groups3827104343326376752nteger(G,A2) != one_one_Code_integer )
     => ~ ! [A4: int] :
            ( aa_set_int_o(member_int(A4),A2)
           => ( aa_int_Code_integer(G,A4) = one_one_Code_integer ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7591_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_assn,A2: set_nat] :
      ( ( groups6906906614972039071t_assn(G,A2) != one_one_assn )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),A2)
           => ( aa_nat_assn(G,A4) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7592_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_assn,A2: set_int] :
      ( ( groups7882442080178216443t_assn(G,A2) != one_one_assn )
     => ~ ! [A4: int] :
            ( aa_set_int_o(member_int(A4),A2)
           => ( aa_int_assn(G,A4) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7593_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A2: set_nat] :
      ( ( aa_set_nat_rat(groups73079841787564623at_rat(G),A2) != one_one_rat )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),A2)
           => ( aa_nat_rat(G,A4) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7594_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A2: set_int] :
      ( ( groups1072433553688619179nt_rat(G,A2) != one_one_rat )
     => ~ ! [A4: int] :
            ( aa_set_int_o(member_int(A4),A2)
           => ( aa_int_rat(G,A4) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7595_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_nat,A2: set_int] :
      ( ( groups1707563613775114915nt_nat(G,A2) != one_one_nat )
     => ~ ! [A4: int] :
            ( aa_set_int_o(member_int(A4),A2)
           => ( aa_int_nat(G,A4) = one_one_nat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7596_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_nat,A2: set_nat] :
      ( ( aa_set_nat_nat(groups708209901874060359at_nat(G),A2) != one_one_nat )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),A2)
           => ( aa_nat_nat(G,A4) = one_one_nat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7597_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_int,A2: set_nat] :
      ( ( aa_set_nat_int(groups705719431365010083at_int(G),A2) != one_one_int )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),A2)
           => ( aa_nat_int(G,A4) = one_one_int ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7598_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_int,A2: set_int] :
      ( ( groups1705073143266064639nt_int(G,A2) != one_one_int )
     => ~ ! [A4: int] :
            ( aa_set_int_o(member_int(A4),A2)
           => ( aa_int_int(G,A4) = one_one_int ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7599_choose__one,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),one_one_nat) = N ) ).

% choose_one
tff(fact_7600_prod_Odistrib,axiom,
    ! [G: fun_nat_nat,H2: fun_nat_nat,A2: set_nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pu(G),H2)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),A2)),aa_set_nat_nat(groups708209901874060359at_nat(H2),A2)) ) ).

% prod.distrib
tff(fact_7601_prod_Odistrib,axiom,
    ! [G: fun_nat_int,H2: fun_nat_int,A2: set_nat] : ( aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_pv(G),H2)),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),A2)),aa_set_nat_int(groups705719431365010083at_int(H2),A2)) ) ).

% prod.distrib
tff(fact_7602_prod_Odistrib,axiom,
    ! [G: fun_int_int,H2: fun_int_int,A2: set_int] : ( groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_pw(G),H2),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,A2)),groups1705073143266064639nt_int(H2,A2)) ) ).

% prod.distrib
tff(fact_7603_prod__power__distrib,axiom,
    ! [F: fun_nat_nat,A2: set_nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),N) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_px(F),N)),A2) ) ).

% prod_power_distrib
tff(fact_7604_prod__power__distrib,axiom,
    ! [F: fun_nat_int,A2: set_nat,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_set_nat_int(groups705719431365010083at_int(F),A2)),N) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_py(F),N)),A2) ) ).

% prod_power_distrib
tff(fact_7605_prod__power__distrib,axiom,
    ! [F: fun_int_int,A2: set_int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,groups1705073143266064639nt_int(F,A2)),N) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_pz(F),N),A2) ) ).

% prod_power_distrib
tff(fact_7606_mod__prod__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A2: set_nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kd(F),A)),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),A) ) ).

% mod_prod_eq
tff(fact_7607_mod__prod__eq,axiom,
    ! [F: fun_nat_int,A: int,A2: set_nat] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_qa(F),A)),A2)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_set_nat_int(groups705719431365010083at_int(F),A2)),A) ) ).

% mod_prod_eq
tff(fact_7608_mod__prod__eq,axiom,
    ! [F: fun_int_int,A: int,A2: set_int] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_ke(F),A),A2)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(F,A2)),A) ) ).

% mod_prod_eq
tff(fact_7609_fact__prod,axiom,
    ! [N: nat] : ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_da),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N))) ) ).

% fact_prod
tff(fact_7610_fact__prod,axiom,
    ! [N: nat] : ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_da),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N))) ) ).

% fact_prod
tff(fact_7611_fact__prod,axiom,
    ! [N: nat] : ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_da),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N))) ) ).

% fact_prod
tff(fact_7612_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_Code_integer] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(F,X2)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A2)) ) ).

% prod_ge_1
tff(fact_7613_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_Code_integer] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(F,X2)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),groups3827104343326376752nteger(F,A2)) ) ).

% prod_ge_1
tff(fact_7614_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_set_nat_rat(groups73079841787564623at_rat(F),A2)) ) ).

% prod_ge_1
tff(fact_7615_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_7616_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_int_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups1707563613775114915nt_nat(F,A2)) ) ).

% prod_ge_1
tff(fact_7617_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) ) ).

% prod_ge_1
tff(fact_7618_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_set_nat_int(groups705719431365010083at_int(F),A2)) ) ).

% prod_ge_1
tff(fact_7619_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),groups1705073143266064639nt_int(F,A2)) ) ).

% prod_ge_1
tff(fact_7620_prod__ge__1,axiom,
    ! [A2: set_set_nat,F: fun_se7586849273513198688nteger] :
      ( ! [X2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X2),A2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_set871585252387937313nteger(F,X2)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),groups2761809935038513290nteger(F,A2)) ) ).

% prod_ge_1
tff(fact_7621_prod__ge__1,axiom,
    ! [A2: set_set_nat,F: fun_set_nat_rat] :
      ( ! [X2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_set_nat_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups3613417700093529605at_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_7622_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_da),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_7623_fact__prod__rev,axiom,
    ! [N: nat] : ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,N)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ) ).

% fact_prod_rev
tff(fact_7624_fact__prod__rev,axiom,
    ! [N: nat] : ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,N)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ) ).

% fact_prod_rev
tff(fact_7625_fact__prod__rev,axiom,
    ! [N: nat] : ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,N)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ) ).

% fact_prod_rev
tff(fact_7626_fact__ge__1,axiom,
    ! [N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).

% fact_ge_1
tff(fact_7627_fact__ge__1,axiom,
    ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% fact_ge_1
tff(fact_7628_fact__ge__1,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,N)) ).

% fact_ge_1
tff(fact_7629_fact__ge__1,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).

% fact_ge_1
tff(fact_7630_pochhammer__fact,axiom,
    ! [N: nat] : ( aa_nat_Code_integer(semiri3624122377584611663nteger,N) = aa_nat_Code_integer(comm_s8582702949713902594nteger(one_one_Code_integer),N) ) ).

% pochhammer_fact
tff(fact_7631_pochhammer__fact,axiom,
    ! [N: nat] : ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),N) ) ).

% pochhammer_fact
tff(fact_7632_pochhammer__fact,axiom,
    ! [N: nat] : ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,one_one_int),N) ) ).

% pochhammer_fact
tff(fact_7633_pochhammer__fact,axiom,
    ! [N: nat] : ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),N) ) ).

% pochhammer_fact
tff(fact_7634_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_lw(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_7635_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_mb(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_7636_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_lw(G)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_7637_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_mb(G)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_7638_power__sum,axiom,
    ! [C: nat,F: fun_nat_nat,A2: set_nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qb(C),F)),A2) ) ).

% power_sum
tff(fact_7639_power__sum,axiom,
    ! [C: int,F: fun_nat_nat,A2: set_nat] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_int(groups705719431365010083at_int(aa_fun1613598493704395220at_int(aTP_Lamm_qc(C),F)),A2) ) ).

% power_sum
tff(fact_7640_power__sum,axiom,
    ! [C: int,F: fun_int_nat,A2: set_int] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,C),groups4541462559716669496nt_nat(F,A2)) = groups1705073143266064639nt_int(aa_fun2328901131235339916nt_int(aTP_Lamm_qd(C),F),A2) ) ).

% power_sum
tff(fact_7641_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lx(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_7642_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qe(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_7643_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lx(G),K)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_7644_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qe(G),K)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_7645_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_binomial
tff(fact_7646_binomial__fact,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_fact
tff(fact_7647_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_da),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) ) ).

% fact_div_fact
tff(fact_7648_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_Code_integer] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X2))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X2)),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A2)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_7649_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_Code_integer] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X2))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,X2)),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3827104343326376752nteger(F,A2)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_7650_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups73079841787564623at_rat(F),A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_7651_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_7652_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A2)),one_one_nat) ) ).

% prod_le_1
tff(fact_7653_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),one_one_nat) ) ).

% prod_le_1
tff(fact_7654_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),one_one_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups705719431365010083at_int(F),A2)),one_one_int) ) ).

% prod_le_1
tff(fact_7655_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X2)),one_one_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1705073143266064639nt_int(F,A2)),one_one_int) ) ).

% prod_le_1
tff(fact_7656_prod__le__1,axiom,
    ! [A2: set_set_nat,F: fun_se7586849273513198688nteger] :
      ( ! [X2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X2),A2)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(F,X2))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(F,X2)),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups2761809935038513290nteger(F,A2)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_7657_prod__le__1,axiom,
    ! [A2: set_set_nat,F: fun_set_nat_rat] :
      ( ! [X2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X2),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_set_nat_rat(F,X2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(F,X2)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3613417700093529605at_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_7658_fact__split,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_Code_integer(semiri3624122377584611663nteger,N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),N)))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_split
tff(fact_7659_fact__split,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),N)))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_split
tff(fact_7660_fact__split,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),N)))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_split
tff(fact_7661_fact__split,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),N)))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_split
tff(fact_7662_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P3: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),P3)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),P3))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_7663_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P3: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),P3)
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(N),P3))) = groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_7664_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P3: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),P3)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),P3))) = aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_7665_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P3: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),P3)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),P3))) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_7666_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P3: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),P3)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),P3))) = aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_7667_binomial__altdef__of__nat,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_qf(K),N)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% binomial_altdef_of_nat
tff(fact_7668_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ).

% binomial_absorb_comp
tff(fact_7669_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ) ) ).

% dvd_fact
tff(fact_7670_gbinomial__mult__fact_H,axiom,
    ! [A: rat,K: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_qg(A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_7671_gbinomial__mult__fact,axiom,
    ! [K: nat,A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(gbinomial_rat(A),K)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_qg(A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_7672_gbinomial__prod__rev,axiom,
    ! [A: code_integer,K: nat] : ( aa_nat_Code_integer(gbinom8545251970709558553nteger(A),K) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_qh(A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K))),aa_nat_Code_integer(semiri3624122377584611663nteger,K)) ) ).

% gbinomial_prod_rev
tff(fact_7673_gbinomial__prod__rev,axiom,
    ! [A: code_natural,K: nat] : ( aa_nat_Code_natural(gbinom7368847122466276068atural(A),K) = divide5121882707175180666atural(groups2279045934846249631atural(aTP_Lamm_qi(A),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)),aa_nat_Code_natural(semiri2447717529341329178atural,K)) ) ).

% gbinomial_prod_rev
tff(fact_7674_gbinomial__prod__rev,axiom,
    ! [A: rat,K: nat] : ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_qg(A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K))),aa_nat_rat(semiri773545260158071498ct_rat,K)) ) ).

% gbinomial_prod_rev
tff(fact_7675_gbinomial__prod__rev,axiom,
    ! [A: nat,K: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qj,A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K))),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ) ).

% gbinomial_prod_rev
tff(fact_7676_gbinomial__prod__rev,axiom,
    ! [A: int,K: nat] : ( aa_nat_int(aa_int_fun_nat_int(gbinomial_int,A),K) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_qk,A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K))),aa_nat_int(semiri1406184849735516958ct_int,K)) ) ).

% gbinomial_prod_rev
tff(fact_7677_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,N))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N))) ).

% fact_fact_dvd_fact
tff(fact_7678_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,N))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N))) ).

% fact_fact_dvd_fact
tff(fact_7679_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: 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,K)),aa_nat_int(semiri1406184849735516958ct_int,N))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N))) ).

% fact_fact_dvd_fact
tff(fact_7680_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,N))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N))) ).

% fact_fact_dvd_fact
tff(fact_7681_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lv(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_7682_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,N: nat,M: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ql(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_7683_gbinomial__Suc,axiom,
    ! [A: code_integer,K: nat] : ( aa_nat_Code_integer(gbinom8545251970709558553nteger(A),aa_nat_nat(suc,K)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_qh(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,K))) ) ).

% gbinomial_Suc
tff(fact_7684_gbinomial__Suc,axiom,
    ! [A: code_natural,K: nat] : ( aa_nat_Code_natural(gbinom7368847122466276068atural(A),aa_nat_nat(suc,K)) = divide5121882707175180666atural(groups2279045934846249631atural(aTP_Lamm_qi(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)),aa_nat_Code_natural(semiri2447717529341329178atural,aa_nat_nat(suc,K))) ) ).

% gbinomial_Suc
tff(fact_7685_gbinomial__Suc,axiom,
    ! [A: rat,K: nat] : ( aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_qg(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,K))) ) ).

% gbinomial_Suc
tff(fact_7686_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qj,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K))) ) ).

% gbinomial_Suc
tff(fact_7687_gbinomial__Suc,axiom,
    ! [A: int,K: nat] : ( aa_nat_int(aa_int_fun_nat_int(gbinomial_int,A),aa_nat_nat(suc,K)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_qk,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K))) ) ).

% gbinomial_Suc
tff(fact_7688_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qm,M)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)) ) ).

% sum_choose_upper
tff(fact_7689_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_Code_integer(G,aa_nat_nat(suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_7690_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_assn,N: nat] : ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_assn(G,aa_nat_nat(suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_7691_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_7692_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_7693_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_7694_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_Code_integer(G,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_7695_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_assn,N: nat] : ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_assn(G,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_7696_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_rat(G,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_7697_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_nat(G,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_7698_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_int(G,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_7699_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,M)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_7700_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,M)),groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_7701_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,M)),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_7702_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_7703_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_7704_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,aa_nat_nat(suc,N))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7705_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,aa_nat_nat(suc,N))),groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7706_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,aa_nat_nat(suc,N))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7707_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,N))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7708_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,N))),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7709_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,M)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7710_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,M)),groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7711_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,M)),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7712_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7713_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7714_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_Code_integer(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_7715_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_assn(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_7716_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_rat(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_7717_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_nat(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_7718_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_int(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_7719_prod_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,N)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) ) ) ).

% prod.last_plus
tff(fact_7720_prod_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,N)),groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) ) ) ).

% prod.last_plus
tff(fact_7721_prod_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,N)),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) ) ) ).

% prod.last_plus
tff(fact_7722_prod_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,N)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) ) ) ).

% prod.last_plus
tff(fact_7723_prod_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,N)),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) ) ) ).

% prod.last_plus
tff(fact_7724_binomial__absorption,axiom,
    ! [K: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ).

% binomial_absorption
tff(fact_7725_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_Code_integer(G,aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,M)),aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_qn(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7726_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_assn(G,aa_nat_nat(suc,N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,M)),groups6906906614972039071t_assn(aTP_Lamm_qo(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7727_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,M)),aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_ma(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7728_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_lw(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7729_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_mb(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7730_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,zero_zero_nat)),aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_qn(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ) ).

% prod.atMost_Suc_shift
tff(fact_7731_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_assn,N: nat] : ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,zero_zero_nat)),groups6906906614972039071t_assn(aTP_Lamm_qo(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ) ).

% prod.atMost_Suc_shift
tff(fact_7732_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_ma(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ) ).

% prod.atMost_Suc_shift
tff(fact_7733_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = 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_lw(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ) ).

% prod.atMost_Suc_shift
tff(fact_7734_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_mb(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ) ).

% prod.atMost_Suc_shift
tff(fact_7735_prod_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_mc(G),N),M)),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_7736_prod_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_int,N: nat,M: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_qp(G),N),M)),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_7737_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ) ).

% choose_dvd
tff(fact_7738_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ) ).

% choose_dvd
tff(fact_7739_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => 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,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_int(semiri1406184849735516958ct_int,N)) ) ).

% choose_dvd
tff(fact_7740_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ) ).

% choose_dvd
tff(fact_7741_prod_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_qq(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qr(A),N)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ) ).

% prod.nested_swap
tff(fact_7742_prod_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_int,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_qs(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qt(A),N)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ) ).

% prod.nested_swap
tff(fact_7743_fact__numeral,axiom,
    ! [K: num] : ( aa_nat_int(semiri1406184849735516958ct_int,numeral_numeral_nat(K)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,K)),aa_nat_int(semiri1406184849735516958ct_int,pred_numeral(K))) ) ).

% fact_numeral
tff(fact_7744_fact__numeral,axiom,
    ! [K: num] : ( aa_nat_Code_integer(semiri3624122377584611663nteger,numeral_numeral_nat(K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,pred_numeral(K))) ) ).

% fact_numeral
tff(fact_7745_fact__numeral,axiom,
    ! [K: num] : ( aa_nat_rat(semiri773545260158071498ct_rat,numeral_numeral_nat(K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(K)),aa_nat_rat(semiri773545260158071498ct_rat,pred_numeral(K))) ) ).

% fact_numeral
tff(fact_7746_fact__numeral,axiom,
    ! [K: num] : ( aa_nat_Code_natural(semiri2447717529341329178atural,numeral_numeral_nat(K)) = times_2397367101498566445atural(numera5444537566228673987atural(K),aa_nat_Code_natural(semiri2447717529341329178atural,pred_numeral(K))) ) ).

% fact_numeral
tff(fact_7747_fact__numeral,axiom,
    ! [K: num] : ( aa_nat_nat(semiri1408675320244567234ct_nat,numeral_numeral_nat(K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(K)),aa_nat_nat(semiri1408675320244567234ct_nat,pred_numeral(K))) ) ).

% fact_numeral
tff(fact_7748_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_Code_integer,A: nat,B: nat] : ( aa_set871585252387937313nteger(groups3455450783089532116nteger(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_qu(F),A,B),one_one_Code_integer) ) ).

% prod_atLeastAtMost_code
tff(fact_7749_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_assn,A: nat,B: nat] : ( groups6906906614972039071t_assn(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo1959793692361082170t_assn(aTP_Lamm_qv(F),A,B,one_one_assn) ) ).

% prod_atLeastAtMost_code
tff(fact_7750_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : ( aa_set_nat_rat(groups73079841787564623at_rat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_qw(F),A,B),one_one_rat) ) ).

% prod_atLeastAtMost_code
tff(fact_7751_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_qx(F),A),B),one_one_nat) ) ).

% prod_atLeastAtMost_code
tff(fact_7752_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : ( aa_set_nat_int(groups705719431365010083at_int(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_qy(F),A),B),one_one_int) ) ).

% prod_atLeastAtMost_code
tff(fact_7753_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qz,R2)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),N))),N) ) ).

% sum_choose_lower
tff(fact_7754_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ra,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),M) ) ).

% choose_rising_sum(2)
tff(fact_7755_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ra,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) ).

% choose_rising_sum(1)
tff(fact_7756_binomial__code,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K),
          zero_zero_nat,
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),one_one_nat)),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ) ).

% binomial_code
tff(fact_7757_prod_Ohead__if,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N: nat] :
      ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_Code_integer(G,N))) ) ).

% prod.head_if
tff(fact_7758_prod_Ohead__if,axiom,
    ! [G: fun_nat_assn,M: nat,N: nat] :
      ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_assn(G,N))) ) ).

% prod.head_if
tff(fact_7759_prod_Ohead__if,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] :
      ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_rat(G,N))) ) ).

% prod.head_if
tff(fact_7760_prod_Ohead__if,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] :
      ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_nat(G,N))) ) ).

% prod.head_if
tff(fact_7761_prod_Ohead__if,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] :
      ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_int(G,N))) ) ).

% prod.head_if
tff(fact_7762_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_Code_integer,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_7763_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_assn,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_7764_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_7765_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_7766_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_int,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_7767_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_7768_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_7769_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lv(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_7770_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_int,N: nat,M: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ql(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_7771_pochhammer__prod,axiom,
    ! [A: rat,N: nat] : ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_rb(A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ) ).

% pochhammer_prod
tff(fact_7772_pochhammer__prod,axiom,
    ! [A: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rc,A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ) ).

% pochhammer_prod
tff(fact_7773_pochhammer__prod,axiom,
    ! [A: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_rd,A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ) ).

% pochhammer_prod
tff(fact_7774_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_re(M),N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_7775_vandermonde,axiom,
    ! [M: nat,N: nat,R2: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_rf(M),N),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),R2) ) ).

% vandermonde
tff(fact_7776_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_7777_pochhammer__Suc__prod,axiom,
    ! [A: rat,N: nat] : ( aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_rb(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ) ).

% pochhammer_Suc_prod
tff(fact_7778_pochhammer__Suc__prod,axiom,
    ! [A: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rc,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ) ).

% pochhammer_Suc_prod
tff(fact_7779_pochhammer__Suc__prod,axiom,
    ! [A: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,N)) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_rd,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ) ).

% pochhammer_Suc_prod
tff(fact_7780_pochhammer__prod__rev,axiom,
    ! [A: rat,N: nat] : ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_rg(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ) ).

% pochhammer_prod_rev
tff(fact_7781_pochhammer__prod__rev,axiom,
    ! [A: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rh(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ) ).

% pochhammer_prod_rev
tff(fact_7782_pochhammer__prod__rev,axiom,
    ! [A: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ri(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ) ).

% pochhammer_prod_rev
tff(fact_7783_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( aa_nat_Code_integer(semiri3624122377584611663nteger,M) = $ite(M = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ) ).

% fact_num_eq_if
tff(fact_7784_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( aa_nat_int(semiri1406184849735516958ct_int,M) = $ite(M = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ) ).

% fact_num_eq_if
tff(fact_7785_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( aa_nat_rat(semiri773545260158071498ct_rat,M) = $ite(M = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ) ).

% fact_num_eq_if
tff(fact_7786_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = $ite(M = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ) ).

% fact_num_eq_if
tff(fact_7787_fact__reduce,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_Code_integer(semiri3624122377584611663nteger,N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7788_fact__reduce,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7789_fact__reduce,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7790_fact__reduce,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7791_fact__code,axiom,
    ! [N: nat] : ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N),one_one_nat)) ) ).

% fact_code
tff(fact_7792_fact__code,axiom,
    ! [N: nat] : ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N),one_one_nat)) ) ).

% fact_code
tff(fact_7793_fact__code,axiom,
    ! [N: nat] : ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N),one_one_nat)) ) ).

% fact_code
tff(fact_7794_pochhammer__same,axiom,
    ! [N: nat] : ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ) ).

% pochhammer_same
tff(fact_7795_pochhammer__same,axiom,
    ! [N: nat] : ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(semiri1406184849735516958ct_int,N)) ) ).

% pochhammer_same
tff(fact_7796_pochhammer__same,axiom,
    ! [N: nat] : ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(semiri773545260158071498ct_rat,N)) ) ).

% pochhammer_same
tff(fact_7797_choose__row__sum,axiom,
    ! [N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N) ) ).

% choose_row_sum
tff(fact_7798_binomial,axiom,
    ! [A: nat,B: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_rj(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% binomial
tff(fact_7799_choose__two,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% choose_two
tff(fact_7800_prod_Oin__pairs,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N: nat] : ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_rk(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ).

% prod.in_pairs
tff(fact_7801_prod_Oin__pairs,axiom,
    ! [G: fun_nat_assn,M: nat,N: nat] : ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = groups6906906614972039071t_assn(aTP_Lamm_rl(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ).

% prod.in_pairs
tff(fact_7802_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] : ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_rm(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ).

% prod.in_pairs
tff(fact_7803_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_rn(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ).

% prod.in_pairs
tff(fact_7804_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ro(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ).

% prod.in_pairs
tff(fact_7805_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_rk(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% prod.in_pairs_0
tff(fact_7806_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_assn,N: nat] : ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = groups6906906614972039071t_assn(aTP_Lamm_rl(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% prod.in_pairs_0
tff(fact_7807_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N: nat] : ( aa_set_nat_rat(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,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_rm(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% prod.in_pairs_0
tff(fact_7808_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N: 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,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_rn(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% prod.in_pairs_0
tff(fact_7809_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: nat] : ( aa_set_nat_int(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,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ro(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% prod.in_pairs_0
tff(fact_7810_gbinomial__altdef__of__nat,axiom,
    ! [A: rat,K: nat] : ( aa_nat_rat(gbinomial_rat(A),K) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_rp(A),K)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_7811_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N: nat] : ( aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_rg(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_7812_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rh(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_7813_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,N)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ri(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_7814_binomial__ring,axiom,
    ! [A: code_integer,B: code_integer,N: nat] : ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),N) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_rq(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% binomial_ring
tff(fact_7815_binomial__ring,axiom,
    ! [A: int,B: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rr(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% binomial_ring
tff(fact_7816_binomial__ring,axiom,
    ! [A: rat,B: rat,N: nat] : ( aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),N) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rs(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% binomial_ring
tff(fact_7817_binomial__ring,axiom,
    ! [A: nat,B: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_rj(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% binomial_ring
tff(fact_7818_pochhammer__binomial__sum,axiom,
    ! [A: code_integer,B: code_integer,N: nat] : ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),N) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_rt(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% pochhammer_binomial_sum
tff(fact_7819_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N: nat] : ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ru(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% pochhammer_binomial_sum
tff(fact_7820_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N: nat] : ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),N) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rv(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% pochhammer_binomial_sum
tff(fact_7821_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K: nat] : ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ) ).

% gbinomial_pochhammer'
tff(fact_7822_choose__square__sum,axiom,
    ! [N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rw,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),N) ) ).

% choose_square_sum
tff(fact_7823_gbinomial__pochhammer,axiom,
    ! [A: rat,K: nat] : ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,A)),K))),aa_nat_rat(semiri773545260158071498ct_rat,K)) ) ).

% gbinomial_pochhammer
tff(fact_7824_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_Code_integer,H2: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_rx(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_oi(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7825_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_assn,H2: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_ry(K),G),H2),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_rz(K),G),H2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7826_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_rat,H2: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_sa(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_om(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7827_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_nat,H2: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sb(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_oq(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7828_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_int,H2: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_sc(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_oo(K),G),H2)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7829_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_sd(N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_7830_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_se,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_7831_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_sf(N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_7832_binomial__r__part__sum,axiom,
    ! [M: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)) ) ).

% binomial_r_part_sum
tff(fact_7833_choose__linear__sum,axiom,
    ! [N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sg,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% choose_linear_sum
tff(fact_7834_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : ( groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_sh(N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_of(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ).

% sum.triangle_reindex
tff(fact_7835_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,N: nat] :
      ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = $ite(
          ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N = zero_zero_nat ) ),
          zero_zero_int,
          $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N),M)))))) ) ) ).

% divide_int_unfold
tff(fact_7836_sum__zero__power_H,axiom,
    ! [C: fun_nat_rat,D2: fun_nat_rat,A2: set_nat] :
      ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_si(C),D2)),A2) = $ite(
          ( aa_set_nat_o(finite_finite_nat,A2)
          & aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),A2) ),
          aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(C,zero_zero_nat)),aa_nat_rat(D2,zero_zero_nat)),
          zero_zero_rat ) ) ).

% sum_zero_power'
tff(fact_7837_bezw__0,axiom,
    ! [X: nat] : ( bezw(X,zero_zero_nat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ) ).

% bezw_0
tff(fact_7838_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,pred_numeral(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ) ).

% drop_bit_numeral_minus_bit1
tff(fact_7839_sgn__1,axiom,
    aa_int_int(sgn_sgn_int,one_one_int) = one_one_int ).

% sgn_1
tff(fact_7840_sgn__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% sgn_1
tff(fact_7841_sgn__1,axiom,
    aa_rat_rat(sgn_sgn_rat,one_one_rat) = one_one_rat ).

% sgn_1
tff(fact_7842_sgn__minus,axiom,
    ! [A: int] : ( aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,A)) ) ).

% sgn_minus
tff(fact_7843_sgn__minus,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) ) ).

% sgn_minus
tff(fact_7844_sgn__minus,axiom,
    ! [A: rat] : ( aa_rat_rat(sgn_sgn_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(sgn_sgn_rat,A)) ) ).

% sgn_minus
tff(fact_7845_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_Code_integer] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2) = one_one_Code_integer ) ) ).

% prod.infinite
tff(fact_7846_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_Code_integer] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups3827104343326376752nteger(G,A2) = one_one_Code_integer ) ) ).

% prod.infinite
tff(fact_7847_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_assn] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups6906906614972039071t_assn(G,A2) = one_one_assn ) ) ).

% prod.infinite
tff(fact_7848_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_assn] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups7882442080178216443t_assn(G,A2) = one_one_assn ) ) ).

% prod.infinite
tff(fact_7849_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7850_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups1072433553688619179nt_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7851_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups1707563613775114915nt_nat(G,A2) = one_one_nat ) ) ).

% prod.infinite
tff(fact_7852_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A2) = one_one_nat ) ) ).

% prod.infinite
tff(fact_7853_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),A2) = one_one_int ) ) ).

% prod.infinite
tff(fact_7854_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups1705073143266064639nt_int(G,A2) = one_one_int ) ) ).

% prod.infinite
tff(fact_7855_divide__sgn,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(sgn_sgn_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(sgn_sgn_rat,B)) ) ).

% divide_sgn
tff(fact_7856_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_7857_drop__bit__minus__one,axiom,
    ! [N: nat] : ( aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ) ).

% drop_bit_minus_one
tff(fact_7858_sum_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_sj(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta'
tff(fact_7859_sum_Odelta_H,axiom,
    ! [S: set_int,A: int,B: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_sk(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta'
tff(fact_7860_sum_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_sl(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_7861_sum_Odelta_H,axiom,
    ! [S: set_int,A: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_sm(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_7862_sum_Odelta_H,axiom,
    ! [S: set_int,A: int,B: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_sn(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_7863_sum_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_so(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_int(B,A),zero_zero_int) ) ) ).

% sum.delta'
tff(fact_7864_sum_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_sp(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_7865_sum_Odelta_H,axiom,
    ! [S: set_int,A: int,B: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_sq(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_int(B,A),zero_zero_int) ) ) ).

% sum.delta'
tff(fact_7866_sum_Odelta_H,axiom,
    ! [S: set_set_nat,A: set_nat,B: fun_se7586849273513198688nteger] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,S)
     => ( groups9190459664516455967nteger(aa_fun4587287018503165694nteger(aTP_Lamm_sr(A),B),S) = $ite(aa_set_set_nat_o(member_set_nat(A),S),aa_set871585252387937313nteger(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta'
tff(fact_7867_sum_Odelta_H,axiom,
    ! [S: set_set_nat,A: set_nat,B: fun_set_nat_rat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,S)
     => ( groups7659867448343625626at_rat(aa_fun8478617152544252148at_rat(aTP_Lamm_ss(A),B),S) = $ite(aa_set_set_nat_o(member_set_nat(A),S),aa_set_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_7868_sum_Odelta,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_st(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta
tff(fact_7869_sum_Odelta,axiom,
    ! [S: set_int,A: int,B: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_su(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta
tff(fact_7870_sum_Odelta,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_sv(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_7871_sum_Odelta,axiom,
    ! [S: set_int,A: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_sw(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_7872_sum_Odelta,axiom,
    ! [S: set_int,A: int,B: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_sx(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_7873_sum_Odelta,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_sy(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_int(B,A),zero_zero_int) ) ) ).

% sum.delta
tff(fact_7874_sum_Odelta,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_sz(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_7875_sum_Odelta,axiom,
    ! [S: set_int,A: int,B: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ta(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_int(B,A),zero_zero_int) ) ) ).

% sum.delta
tff(fact_7876_sum_Odelta,axiom,
    ! [S: set_set_nat,A: set_nat,B: fun_se7586849273513198688nteger] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,S)
     => ( groups9190459664516455967nteger(aa_fun4587287018503165694nteger(aTP_Lamm_tb(A),B),S) = $ite(aa_set_set_nat_o(member_set_nat(A),S),aa_set871585252387937313nteger(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta
tff(fact_7877_sum_Odelta,axiom,
    ! [S: set_set_nat,A: set_nat,B: fun_set_nat_rat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,S)
     => ( groups7659867448343625626at_rat(aa_fun8478617152544252148at_rat(aTP_Lamm_tc(A),B),S) = $ite(aa_set_set_nat_o(member_set_nat(A),S),aa_set_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_7878_prod__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups1707563613775114915nt_nat(F,A2) = one_one_nat )
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),A2)
           => ( aa_int_nat(F,X4) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_7879_prod__eq__1__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( ( groups4077766827762148844at_nat(F,A2) = one_one_nat )
      <=> ! [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A2)
           => ( aa_Pro4243173770044397973at_nat(F,X4) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_7880_prod__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_set_nat_nat(groups708209901874060359at_nat(F),A2) = one_one_nat )
      <=> ! [X4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A2)
           => ( aa_nat_nat(F,X4) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_7881_prod_Odelta,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aTP_Lamm_td(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_Code_integer(B,A),one_one_Code_integer) ) ) ).

% prod.delta
tff(fact_7882_prod_Odelta,axiom,
    ! [S: set_int,A: int,B: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups3827104343326376752nteger(aa_fun5304498221708291146nteger(aTP_Lamm_te(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_Code_integer(B,A),one_one_Code_integer) ) ) ).

% prod.delta
tff(fact_7883_prod_Odelta,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aTP_Lamm_tf(A),B),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_assn(B,A),one_one_assn) ) ) ).

% prod.delta
tff(fact_7884_prod_Odelta,axiom,
    ! [S: set_int,A: int,B: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aTP_Lamm_tg(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_assn(B,A),one_one_assn) ) ) ).

% prod.delta
tff(fact_7885_prod_Odelta,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_th(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_7886_prod_Odelta,axiom,
    ! [S: set_int,A: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ti(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_7887_prod_Odelta,axiom,
    ! [S: set_int,A: int,B: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_tj(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_nat(B,A),one_one_nat) ) ) ).

% prod.delta
tff(fact_7888_prod_Odelta,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_tk(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_nat(B,A),one_one_nat) ) ) ).

% prod.delta
tff(fact_7889_prod_Odelta,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_tl(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_int(B,A),one_one_int) ) ) ).

% prod.delta
tff(fact_7890_prod_Odelta,axiom,
    ! [S: set_int,A: int,B: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_tm(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_int(B,A),one_one_int) ) ) ).

% prod.delta
tff(fact_7891_prod_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aTP_Lamm_tn(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_Code_integer(B,A),one_one_Code_integer) ) ) ).

% prod.delta'
tff(fact_7892_prod_Odelta_H,axiom,
    ! [S: set_int,A: int,B: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups3827104343326376752nteger(aa_fun5304498221708291146nteger(aTP_Lamm_to(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_Code_integer(B,A),one_one_Code_integer) ) ) ).

% prod.delta'
tff(fact_7893_prod_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aTP_Lamm_tp(A),B),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_assn(B,A),one_one_assn) ) ) ).

% prod.delta'
tff(fact_7894_prod_Odelta_H,axiom,
    ! [S: set_int,A: int,B: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aTP_Lamm_tq(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_assn(B,A),one_one_assn) ) ) ).

% prod.delta'
tff(fact_7895_prod_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_tr(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_7896_prod_Odelta_H,axiom,
    ! [S: set_int,A: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ts(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_7897_prod_Odelta_H,axiom,
    ! [S: set_int,A: int,B: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_tt(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_nat(B,A),one_one_nat) ) ) ).

% prod.delta'
tff(fact_7898_prod_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_tu(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_nat(B,A),one_one_nat) ) ) ).

% prod.delta'
tff(fact_7899_prod_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_tv(A),B)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_int(B,A),one_one_int) ) ) ).

% prod.delta'
tff(fact_7900_prod_Odelta_H,axiom,
    ! [S: set_int,A: int,B: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_tw(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_int(B,A),one_one_int) ) ) ).

% prod.delta'
tff(fact_7901_sgn__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer ) ) ).

% sgn_pos
tff(fact_7902_sgn__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) ) ).

% sgn_pos
tff(fact_7903_sgn__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(sgn_sgn_int,A) = one_one_int ) ) ).

% sgn_pos
tff(fact_7904_prod_Oinsert,axiom,
    ! [A2: set_o,X: $o,G: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((X)),A2)
       => ( groups7694694392188491536nteger(G,insert_o((X),A2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(G,(X))),groups7694694392188491536nteger(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7905_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2)
       => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(insert_nat(X),A2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,X)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2)) ) ) ) ).

% prod.insert
tff(fact_7906_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(X),A2)
       => ( groups3827104343326376752nteger(G,insert_int(X,A2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(G,X)),groups3827104343326376752nteger(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7907_prod_Oinsert,axiom,
    ! [A2: set_o,X: $o,G: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((X)),A2)
       => ( groups5301882518646026715o_assn(G,insert_o((X),A2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(G,(X))),groups5301882518646026715o_assn(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7908_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2)
       => ( groups6906906614972039071t_assn(G,aa_set_nat_set_nat(insert_nat(X),A2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,X)),groups6906906614972039071t_assn(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7909_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(X),A2)
       => ( groups7882442080178216443t_assn(G,insert_int(X,A2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(G,X)),groups7882442080178216443t_assn(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7910_prod_Oinsert,axiom,
    ! [A2: set_o,X: $o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((X)),A2)
       => ( groups2869687844427037835_o_rat(G,insert_o((X),A2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(G,(X))),groups2869687844427037835_o_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7911_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2)
       => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(insert_nat(X),A2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,X)),aa_set_nat_rat(groups73079841787564623at_rat(G),A2)) ) ) ) ).

% prod.insert
tff(fact_7912_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(X),A2)
       => ( groups1072433553688619179nt_rat(G,insert_int(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G,X)),groups1072433553688619179nt_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7913_prod_Oinsert,axiom,
    ! [A2: set_o,X: $o,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((X)),A2)
       => ( groups3504817904513533571_o_nat(G,insert_o((X),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(G,(X))),groups3504817904513533571_o_nat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7914_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
tff(fact_7915_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = one_one_rat ) ) ).

% abs_sgn_eq_1
tff(fact_7916_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ).

% abs_sgn_eq_1
tff(fact_7917_single__Diff__lessThan,axiom,
    ! [K: $o] : ( minus_minus_set_o(insert_o((K),bot_bot_set_o),set_ord_lessThan_o((K))) = insert_o((K),bot_bot_set_o) ) ).

% single_Diff_lessThan
tff(fact_7918_single__Diff__lessThan,axiom,
    ! [K: int] : ( minus_minus_set_int(insert_int(K,bot_bot_set_int),set_ord_lessThan_int(K)) = insert_int(K,bot_bot_set_int) ) ).

% single_Diff_lessThan
tff(fact_7919_single__Diff__lessThan,axiom,
    ! [K: nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(insert_nat(K),bot_bot_set_nat)),aa_nat_set_nat(set_ord_lessThan_nat,K)) = aa_set_nat_set_nat(insert_nat(K),bot_bot_set_nat) ) ).

% single_Diff_lessThan
tff(fact_7920_sgn__mult__self__eq,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = aa_o_Code_integer(zero_n356916108424825756nteger,A != zero_z3403309356797280102nteger) ) ).

% sgn_mult_self_eq
tff(fact_7921_sgn__mult__self__eq,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,A)) = aa_o_rat(zero_n2052037380579107095ol_rat,A != zero_zero_rat) ) ).

% sgn_mult_self_eq
tff(fact_7922_sgn__mult__self__eq,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ) ).

% sgn_mult_self_eq
tff(fact_7923_drop__bit__of__1,axiom,
    ! [N: nat] : ( aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(N),one_one_Code_integer) = aa_o_Code_integer(zero_n356916108424825756nteger,N = zero_zero_nat) ) ).

% drop_bit_of_1
tff(fact_7924_drop__bit__of__1,axiom,
    ! [N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,N = zero_zero_nat) ) ).

% drop_bit_of_1
tff(fact_7925_drop__bit__of__1,axiom,
    ! [N: nat] : ( aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,N = zero_zero_nat) ) ).

% drop_bit_of_1
tff(fact_7926_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_Code_integer(G,N)) ) ).

% prod.lessThan_Suc
tff(fact_7927_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_assn,N: nat] : ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_assn(G,N)) ) ).

% prod.lessThan_Suc
tff(fact_7928_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_rat(G,N)) ) ).

% prod.lessThan_Suc
tff(fact_7929_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = 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,N))),aa_nat_nat(G,N)) ) ).

% prod.lessThan_Suc
tff(fact_7930_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G,N)) ) ).

% prod.lessThan_Suc
tff(fact_7931_sgn__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% sgn_neg
tff(fact_7932_sgn__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% sgn_neg
tff(fact_7933_sgn__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% sgn_neg
tff(fact_7934_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ) ).

% drop_bit_Suc_minus_bit0
tff(fact_7935_sum__mult__of__bool__eq,axiom,
    ! [A2: set_int,F: fun_int_Code_integer,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7873554091576472773nteger(aa_fun6019004165459918095nteger(aTP_Lamm_tx(F),P),A2) = groups7873554091576472773nteger(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7936_sum__mult__of__bool__eq,axiom,
    ! [A2: set_nat,F: fun_nat_Code_integer,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun3969837373441324047nteger(aTP_Lamm_ty(F),P)),A2) = aa_set871585252387937313nteger(groups7501900531339628137nteger(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7937_sum__mult__of__bool__eq,axiom,
    ! [A2: set_int,F: fun_int_rat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_tz(F),P),A2) = groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7938_sum__mult__of__bool__eq,axiom,
    ! [A2: set_nat,F: fun_nat_rat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_ua(F),P)),A2) = aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7939_sum__mult__of__bool__eq,axiom,
    ! [A2: set_nat,F: fun_nat_int,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun27009930614080094at_int(aTP_Lamm_ub(F),P)),A2) = aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7940_sum__mult__of__bool__eq,axiom,
    ! [A2: set_int,F: fun_int_nat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_uc(F),P),A2) = groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7941_sum__mult__of__bool__eq,axiom,
    ! [A2: set_nat,F: fun_nat_nat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_ud(F),P)),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7942_sum__mult__of__bool__eq,axiom,
    ! [A2: set_int,F: fun_int_int,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4538972089207619220nt_int(aa_fun8822586851003501406nt_int(aTP_Lamm_ue(F),P),A2) = groups4538972089207619220nt_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7943_sum__mult__of__bool__eq,axiom,
    ! [A2: set_set_nat,F: fun_se7586849273513198688nteger,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups9190459664516455967nteger(aa_fun7102796863437819535nteger(aTP_Lamm_uf(F),P),A2) = groups9190459664516455967nteger(F,inf_inf_set_set_nat(A2,collect_set_nat(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7944_sum__mult__of__bool__eq,axiom,
    ! [A2: set_set_nat,F: fun_set_nat_rat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups7659867448343625626at_rat(aa_fun7116522257906611850at_rat(aTP_Lamm_ug(F),P),A2) = groups7659867448343625626at_rat(F,inf_inf_set_set_nat(A2,collect_set_nat(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7945_sum__of__bool__mult__eq,axiom,
    ! [A2: set_int,P: fun_int_o,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_uh(P),F),A2) = groups7873554091576472773nteger(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7946_sum__of__bool__mult__eq,axiom,
    ! [A2: set_nat,P: fun_nat_o,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_ui(P),F)),A2) = aa_set871585252387937313nteger(groups7501900531339628137nteger(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7947_sum__of__bool__mult__eq,axiom,
    ! [A2: set_int,P: fun_int_o,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_uj(P),F),A2) = groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7948_sum__of__bool__mult__eq,axiom,
    ! [A2: set_nat,P: fun_nat_o,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_uk(P),F)),A2) = aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7949_sum__of__bool__mult__eq,axiom,
    ! [A2: set_nat,P: fun_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_ul(P),F)),A2) = aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7950_sum__of__bool__mult__eq,axiom,
    ! [A2: set_int,P: fun_int_o,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_um(P),F),A2) = groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7951_sum__of__bool__mult__eq,axiom,
    ! [A2: set_nat,P: fun_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_un(P),F)),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7952_sum__of__bool__mult__eq,axiom,
    ! [A2: set_int,P: fun_int_o,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_uo(P),F),A2) = groups4538972089207619220nt_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7953_sum__of__bool__mult__eq,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_o,F: fun_se7586849273513198688nteger] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups9190459664516455967nteger(aa_fun4587287018503165694nteger(aTP_Lamm_up(P),F),A2) = groups9190459664516455967nteger(F,inf_inf_set_set_nat(A2,collect_set_nat(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7954_sum__of__bool__mult__eq,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_o,F: fun_set_nat_rat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups7659867448343625626at_rat(aa_fun8478617152544252148at_rat(aTP_Lamm_uq(P),F),A2) = groups7659867448343625626at_rat(F,inf_inf_set_set_nat(A2,collect_set_nat(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7955_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,pred_numeral(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ) ).

% drop_bit_numeral_minus_bit0
tff(fact_7956_sum__zero__power,axiom,
    ! [C: fun_nat_rat,A2: set_nat] :
      ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ur(C)),A2) = $ite(
          ( aa_set_nat_o(finite_finite_nat,A2)
          & aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),A2) ),
          aa_nat_rat(C,zero_zero_nat),
          zero_zero_rat ) ) ).

% sum_zero_power
tff(fact_7957_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ) ).

% drop_bit_Suc_minus_bit1
tff(fact_7958_int__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : ( aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_mv(F),A2) ) ).

% int_prod
tff(fact_7959_int__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_mw(F)),A2) ) ).

% int_prod
tff(fact_7960_finite__if__eq__beyond__finite,axiom,
    ! [S: set_Pr958786334691620121nt_int,S4: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(finite2998713641127702882nt_int,S)
     => aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aa_set4993530631398190431_int_o(aTP_Lamm_us(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_7961_finite__if__eq__beyond__finite,axiom,
    ! [S: set_int,S4: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => finite6197958912794628473et_int(collect_set_int(aa_set5425576183772390127_int_o(aTP_Lamm_ut(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_7962_finite__if__eq__beyond__finite,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
     => finite9047747110432174090at_nat(collec5514110066124741708at_nat(aa_set7675215768510388063_nat_o(aTP_Lamm_uu(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_7963_finite__if__eq__beyond__finite,axiom,
    ! [S: set_nat,S4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_uv(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_7964_sgn__mult,axiom,
    ! [A: code_integer,B: code_integer] : ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B)) ) ).

% sgn_mult
tff(fact_7965_sgn__mult,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(sgn_sgn_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,B)) ) ).

% sgn_mult
tff(fact_7966_sgn__mult,axiom,
    ! [A: int,B: int] : ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B)) ) ).

% sgn_mult
tff(fact_7967_lessThan__non__empty,axiom,
    ! [X: int] : ( set_ord_lessThan_int(X) != bot_bot_set_int ) ).

% lessThan_non_empty
tff(fact_7968_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,I: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_uw(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_7969_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),aa_nat_nat(F,N3))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ux(F),U))) ) ).

% finite_less_ub
tff(fact_7970_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_uz(B2),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_vc(A2),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7971_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_int,G: fun_nat_fun_int_int,R: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_ve(B2),G),R)),A2) = groups4538972089207619220nt_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_vg(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7972_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_int,G: fun_nat_fun_int_nat,R: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8666973529875159599at_nat(aa_fun3399039426249668090at_nat(aTP_Lamm_vh(B2),G),R)),A2) = groups4541462559716669496nt_nat(aa_fun442955204921438347nt_nat(aa_fun5585961000942689878nt_nat(aTP_Lamm_vj(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7973_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_fun_nat_nat,R: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_vl(B2),G),R)),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_vn(A2),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7974_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G: fun_int_fun_nat_int,R: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups4538972089207619220nt_int(aa_fun472946340410393319nt_int(aa_fun6062826860762256214nt_int(aTP_Lamm_vo(B2),G),R),A2) = aa_set_nat_int(groups3539618377306564664at_int(aa_fun8696964665364114571at_int(aa_fun3875905286069234426at_int(aTP_Lamm_vp(A2),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7975_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_fun_int_int,R: fun_int_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups4538972089207619220nt_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_vr(B2),G),R),A2) = groups4538972089207619220nt_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_vt(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7976_sum_Oswap__restrict,axiom,
    ! [A2: set_set_nat,B2: set_nat,G: fun_se1179353343636554400at_nat,R: fun_se382663106265794036_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups8294997508430121362at_nat(aa_fun6238879321244136459at_nat(aa_fun1036415571424436988at_nat(aTP_Lamm_vv(B2),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun1153650177998607445at_nat(aa_fun5868473082678171206at_nat(aTP_Lamm_vy(A2),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7977_sum_Oswap__restrict,axiom,
    ! [A2: set_set_nat,B2: set_int,G: fun_se876191773397849432nt_int,R: fun_se7360890175450123928_int_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups8292507037921071086at_int(aa_fun5327250345464651019at_int(aa_fun3157859007690329412at_int(aTP_Lamm_wa(B2),G),R),A2) = groups4538972089207619220nt_int(aa_fun7007818992012977073nt_int(aa_fun7130641361752856810nt_int(aTP_Lamm_wd(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7978_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_set_nat,G: fun_na106483476154090400at_nat,R: fun_na8563630686737515764_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7187098123162309461at_nat(aa_fun585184524491904070at_nat(aTP_Lamm_wf(B2),G),R)),A2) = groups8294997508430121362at_nat(aa_fun255155394271802123at_nat(aa_fun6284103964554420476at_nat(aTP_Lamm_wi(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7979_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_set_nat,G: fun_in6555721293707029592at_int,R: fun_in7136481858371106712_nat_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
       => ( groups4538972089207619220nt_int(aa_fun1461796484577889457nt_int(aa_fun1283830802835827434nt_int(aTP_Lamm_wk(B2),G),R),A2) = groups8292507037921071086at_int(aa_fun5805668902077597195at_int(aa_fun5576367408028688196at_int(aTP_Lamm_wn(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7980_prod_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups1707563613775114915nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_wo(B2),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_wp(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7981_prod_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_int,G: fun_nat_fun_int_nat,R: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun8666973529875159599at_nat(aa_fun3399039426249668090at_nat(aTP_Lamm_wq(B2),G),R)),A2) = groups1707563613775114915nt_nat(aa_fun442955204921438347nt_nat(aa_fun5585961000942689878nt_nat(aTP_Lamm_wr(A2),G),R),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7982_prod_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_fun_nat_nat,R: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_ws(B2),G),R)),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_wt(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7983_prod_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_fun_nat_int,R: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun4679828425598226919at_int(aa_fun5734951013428723122at_int(aTP_Lamm_wu(B2),G),R)),A2) = aa_set_nat_int(groups705719431365010083at_int(aa_fun4679828425598226919at_int(aa_fun5734951013428723122at_int(aTP_Lamm_wv(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7984_prod_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_int,G: fun_nat_fun_int_int,R: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_ww(B2),G),R)),A2) = groups1705073143266064639nt_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_wx(A2),G),R),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7985_prod_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G: fun_int_fun_nat_int,R: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups1705073143266064639nt_int(aa_fun472946340410393319nt_int(aa_fun6062826860762256214nt_int(aTP_Lamm_wy(B2),G),R),A2) = aa_set_nat_int(groups705719431365010083at_int(aa_fun8696964665364114571at_int(aa_fun3875905286069234426at_int(aTP_Lamm_wz(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7986_prod_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_fun_int_int,R: fun_int_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups1705073143266064639nt_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_xa(B2),G),R),A2) = groups1705073143266064639nt_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_xb(A2),G),R),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7987_prod_Oswap__restrict,axiom,
    ! [A2: set_set_nat,B2: set_nat,G: fun_se1179353343636554400at_nat,R: fun_se382663106265794036_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups4248547760180025341at_nat(aa_fun6238879321244136459at_nat(aa_fun1036415571424436988at_nat(aTP_Lamm_xc(B2),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun1153650177998607445at_nat(aa_fun5868473082678171206at_nat(aTP_Lamm_xd(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7988_prod_Oswap__restrict,axiom,
    ! [A2: set_set_nat,B2: set_nat,G: fun_se7912642235559603452at_int,R: fun_se382663106265794036_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups4246057289670975065at_int(aa_fun2061028301734939751at_int(aa_fun8609273137425597436at_int(aTP_Lamm_xe(B2),G),R),A2) = aa_set_nat_int(groups705719431365010083at_int(aa_fun6199171195344186545at_int(aa_fun2727794920753907526at_int(aTP_Lamm_xg(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7989_prod_Oswap__restrict,axiom,
    ! [A2: set_set_nat,B2: set_int,G: fun_se876191773397849432nt_int,R: fun_se7360890175450123928_int_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups4246057289670975065at_int(aa_fun5327250345464651019at_int(aa_fun3157859007690329412at_int(aTP_Lamm_xh(B2),G),R),A2) = groups1705073143266064639nt_int(aa_fun7007818992012977073nt_int(aa_fun7130641361752856810nt_int(aTP_Lamm_xi(A2),G),R),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7990_lessThan__def,axiom,
    ! [U: set_Pr958786334691620121nt_int] : ( set_or4940836740269066044nt_int(U) = collec5210948495886036740nt_int(aa_set4993530631398190431_int_o(aTP_Lamm_xj,U)) ) ).

% lessThan_def
tff(fact_7991_lessThan__def,axiom,
    ! [U: set_nat] : ( set_or890127255671739683et_nat(U) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_ge,U)) ) ).

% lessThan_def
tff(fact_7992_lessThan__def,axiom,
    ! [U: fun_Pr6460944699146269935_nat_o] : ( set_or152332918318897682_nat_o(U) = collec6407654713642994138_nat_o(aa_fun3380769697021226975at_o_o(aTP_Lamm_xk,U)) ) ).

% lessThan_def
tff(fact_7993_lessThan__def,axiom,
    ! [U: literal] : ( set_or2436086697161274615iteral(U) = collect_literal(aa_lit5986169069484365087eral_o(aTP_Lamm_xl,U)) ) ).

% lessThan_def
tff(fact_7994_lessThan__def,axiom,
    ! [U: rat] : ( set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_xm,U)) ) ).

% lessThan_def
tff(fact_7995_lessThan__def,axiom,
    ! [U: int] : ( set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_xn,U)) ) ).

% lessThan_def
tff(fact_7996_lessThan__def,axiom,
    ! [U: nat] : ( aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gg,U)) ) ).

% lessThan_def
tff(fact_7997_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_7998_sgn__not__eq__imp,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) != aa_int_int(sgn_sgn_int,A) )
     => ( ( aa_int_int(sgn_sgn_int,A) != zero_zero_int )
       => ( ( aa_int_int(sgn_sgn_int,B) != zero_zero_int )
         => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_7999_sgn__not__eq__imp,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) != aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) != zero_z3403309356797280102nteger )
       => ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) != zero_z3403309356797280102nteger )
         => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_8000_sgn__not__eq__imp,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B) != aa_rat_rat(sgn_sgn_rat,A) )
     => ( ( aa_rat_rat(sgn_sgn_rat,A) != zero_zero_rat )
       => ( ( aa_rat_rat(sgn_sgn_rat,B) != zero_zero_rat )
         => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(sgn_sgn_rat,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_8001_sgn__minus__1,axiom,
    aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% sgn_minus_1
tff(fact_8002_sgn__minus__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% sgn_minus_1
tff(fact_8003_sgn__minus__1,axiom,
    aa_rat_rat(sgn_sgn_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% sgn_minus_1
tff(fact_8004_mult__sgn__abs,axiom,
    ! [X: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)) = X ) ).

% mult_sgn_abs
tff(fact_8005_mult__sgn__abs,axiom,
    ! [X: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(sgn_sgn_rat,X)),aa_rat_rat(abs_abs_rat,X)) = X ) ).

% mult_sgn_abs
tff(fact_8006_mult__sgn__abs,axiom,
    ! [X: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,X)),aa_int_int(abs_abs_int,X)) = X ) ).

% mult_sgn_abs
tff(fact_8007_sgn__mult__abs,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = A ) ).

% sgn_mult_abs
tff(fact_8008_sgn__mult__abs,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(abs_abs_rat,A)) = A ) ).

% sgn_mult_abs
tff(fact_8009_sgn__mult__abs,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(abs_abs_int,A)) = A ) ).

% sgn_mult_abs
tff(fact_8010_abs__mult__sgn,axiom,
    ! [A: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = A ) ).

% abs_mult_sgn
tff(fact_8011_abs__mult__sgn,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(sgn_sgn_rat,A)) = A ) ).

% abs_mult_sgn
tff(fact_8012_abs__mult__sgn,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(sgn_sgn_int,A)) = A ) ).

% abs_mult_sgn
tff(fact_8013_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: code_integer] : ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_8014_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: rat] : ( aa_rat_rat(abs_abs_rat,K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),aa_rat_rat(sgn_sgn_rat,K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_8015_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: int] : ( aa_int_int(abs_abs_int,K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_8016_Iio__eq__empty__iff,axiom,
    ! [N: $o] :
      ( ( set_ord_lessThan_o((N)) = bot_bot_set_o )
    <=> ( (N)
      <=> bot_bot_o ) ) ).

% Iio_eq_empty_iff
tff(fact_8017_Iio__eq__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
tff(fact_8018_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: fun_nat_Code_integer,Y: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_xo(I4),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_xo(I4),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aa_fun2745573882733218024_nat_o(aTP_Lamm_xp(I4),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8019_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_int,X: fun_int_Code_integer,Y: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_xq(I4),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_xq(I4),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aa_fun8818272713621706452_int_o(aTP_Lamm_xr(I4),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8020_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_xs(I4),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_xs(I4),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_xt(I4),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8021_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_int,X: fun_int_rat,Y: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_xu(I4),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_xu(I4),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_xv(I4),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8022_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: fun_nat_nat,Y: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_xw(I4),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_xw(I4),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aa_fun6348600740168882050_nat_o(aTP_Lamm_xx(I4),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8023_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_int,X: fun_int_nat,Y: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_xy(I4),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_xy(I4),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aa_fun8703205213358249326_int_o(aTP_Lamm_xz(I4),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8024_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: fun_nat_int,Y: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun2180201864171524266_nat_o(aTP_Lamm_ya(I4),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun2180201864171524266_nat_o(aTP_Lamm_ya(I4),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun2180201864171524266_nat_o(aa_fun8566290670789036234_nat_o(aTP_Lamm_yb(I4),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8025_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_int,X: fun_int_int,Y: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun8155382727227203058_int_o(aTP_Lamm_yc(I4),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun8155382727227203058_int_o(aTP_Lamm_yc(I4),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun8155382727227203058_int_o(aa_fun1697523107123627702_int_o(aTP_Lamm_yd(I4),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8026_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_set_nat,X: fun_se7586849273513198688nteger,Y: fun_se7586849273513198688nteger] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun5627232717963835469_nat_o(aTP_Lamm_ye(I4),X)))
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun5627232717963835469_nat_o(aTP_Lamm_ye(I4),Y)))
       => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun5627232717963835469_nat_o(aa_fun4072113896377786438_nat_o(aTP_Lamm_yf(I4),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8027_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_set_nat,X: fun_set_nat_rat,Y: fun_set_nat_rat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun3672760304936660370_nat_o(aTP_Lamm_yg(I4),X)))
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun3672760304936660370_nat_o(aTP_Lamm_yg(I4),Y)))
       => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun3672760304936660370_nat_o(aa_fun4148231721690063568_nat_o(aTP_Lamm_yh(I4),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8028_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: fun_nat_Code_integer,Y: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_yi(I4),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_yi(I4),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aa_fun2745573882733218024_nat_o(aTP_Lamm_yj(I4),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8029_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_int,X: fun_int_Code_integer,Y: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_yk(I4),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_yk(I4),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aa_fun8818272713621706452_int_o(aTP_Lamm_yl(I4),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8030_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: fun_nat_assn,Y: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7183233715851567790_nat_o(aTP_Lamm_ym(I4),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7183233715851567790_nat_o(aTP_Lamm_ym(I4),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7183233715851567790_nat_o(aa_fun8595640264602358482_nat_o(aTP_Lamm_yn(I4),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8031_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_int,X: fun_int_assn,Y: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun202081646129719286_int_o(aTP_Lamm_yo(I4),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun202081646129719286_int_o(aTP_Lamm_yo(I4),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun202081646129719286_int_o(aa_fun871127997271750846_int_o(aTP_Lamm_yp(I4),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8032_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_yq(I4),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_yq(I4),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_yr(I4),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8033_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_int,X: fun_int_rat,Y: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ys(I4),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ys(I4),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_yt(I4),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8034_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: fun_nat_nat,Y: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_yu(I4),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_yu(I4),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aa_fun6348600740168882050_nat_o(aTP_Lamm_yv(I4),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8035_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_int,X: fun_int_nat,Y: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_yw(I4),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_yw(I4),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aa_fun8703205213358249326_int_o(aTP_Lamm_yx(I4),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8036_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: fun_nat_int,Y: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun2180201864171524266_nat_o(aTP_Lamm_yy(I4),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun2180201864171524266_nat_o(aTP_Lamm_yy(I4),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun2180201864171524266_nat_o(aa_fun8566290670789036234_nat_o(aTP_Lamm_yz(I4),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8037_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_int,X: fun_int_int,Y: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun8155382727227203058_int_o(aTP_Lamm_za(I4),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun8155382727227203058_int_o(aTP_Lamm_za(I4),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun8155382727227203058_int_o(aa_fun1697523107123627702_int_o(aTP_Lamm_zb(I4),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8038_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_Code_integer,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gl(A2),P))) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun3969837373441324047nteger(aTP_Lamm_zc(G),P)),A2) ) ) ).

% sum.inter_filter
tff(fact_8039_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_Code_integer,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7873554091576472773nteger(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gj(A2),P))) = groups7873554091576472773nteger(aa_fun6019004165459918095nteger(aTP_Lamm_zd(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8040_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_rat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gl(A2),P))) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_ze(G),P)),A2) ) ) ).

% sum.inter_filter
tff(fact_8041_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_rat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gj(A2),P))) = groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_zf(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8042_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_nat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gj(A2),P))) = groups4541462559716669496nt_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_zg(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8043_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_int,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gl(A2),P))) = aa_set_nat_int(groups3539618377306564664at_int(aa_fun27009930614080094at_int(aTP_Lamm_zh(G),P)),A2) ) ) ).

% sum.inter_filter
tff(fact_8044_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_nat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gl(A2),P))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_zi(G),P)),A2) ) ) ).

% sum.inter_filter
tff(fact_8045_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_int,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4538972089207619220nt_int(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gj(A2),P))) = groups4538972089207619220nt_int(aa_fun8822586851003501406nt_int(aTP_Lamm_zj(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8046_sum_Ointer__filter,axiom,
    ! [A2: set_set_nat,G: fun_se7586849273513198688nteger,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups9190459664516455967nteger(G,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_gi(A2),P))) = groups9190459664516455967nteger(aa_fun7102796863437819535nteger(aTP_Lamm_zk(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8047_sum_Ointer__filter,axiom,
    ! [A2: set_set_nat,G: fun_set_nat_rat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups7659867448343625626at_rat(G,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_gi(A2),P))) = groups7659867448343625626at_rat(aa_fun7116522257906611850at_rat(aTP_Lamm_zl(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8048_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_Code_integer,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gl(A2),P))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun3969837373441324047nteger(aTP_Lamm_zm(G),P)),A2) ) ) ).

% prod.inter_filter
tff(fact_8049_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_Code_integer,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3827104343326376752nteger(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gj(A2),P))) = groups3827104343326376752nteger(aa_fun6019004165459918095nteger(aTP_Lamm_zn(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8050_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_assn,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups6906906614972039071t_assn(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gl(A2),P))) = groups6906906614972039071t_assn(aa_fun3990007403578212698t_assn(aTP_Lamm_zo(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8051_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_assn,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7882442080178216443t_assn(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gj(A2),P))) = groups7882442080178216443t_assn(aa_fun9028805181350005338t_assn(aTP_Lamm_zp(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8052_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_rat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gl(A2),P))) = aa_set_nat_rat(groups73079841787564623at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_zq(G),P)),A2) ) ) ).

% prod.inter_filter
tff(fact_8053_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_rat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1072433553688619179nt_rat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gj(A2),P))) = groups1072433553688619179nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_zr(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8054_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_nat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1707563613775114915nt_nat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gj(A2),P))) = groups1707563613775114915nt_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_zs(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8055_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_nat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gl(A2),P))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_zt(G),P)),A2) ) ) ).

% prod.inter_filter
tff(fact_8056_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_int,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gl(A2),P))) = aa_set_nat_int(groups705719431365010083at_int(aa_fun27009930614080094at_int(aTP_Lamm_zu(G),P)),A2) ) ) ).

% prod.inter_filter
tff(fact_8057_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_int,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1705073143266064639nt_int(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_gj(A2),P))) = groups1705073143266064639nt_int(aa_fun8822586851003501406nt_int(aTP_Lamm_zv(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8058_sum__strict__mono,axiom,
    ! [A2: set_o,F: fun_o_rat,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(F,(X2))),aa_o_rat(G,(X2))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups7872700643590313910_o_rat(F,A2)),groups7872700643590313910_o_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8059_sum__strict__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),A2)),aa_set_nat_rat(groups2906978787729119204at_rat(G),A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8060_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8061_sum__strict__mono,axiom,
    ! [A2: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(F,(X2))),aa_o_nat(G,(X2))) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups8507830703676809646_o_nat(F,A2)),groups8507830703676809646_o_nat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8062_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(G,X2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8063_sum__strict__mono,axiom,
    ! [A2: set_o,F: fun_o_int,G: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(F,(X2))),aa_o_int(G,(X2))) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups8505340233167759370_o_int(F,A2)),groups8505340233167759370_o_int(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8064_sum__strict__mono,axiom,
    ! [A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X2)),aa_nat_int(G,X2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),A2)),aa_set_nat_int(groups3539618377306564664at_int(G),A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8065_sum__strict__mono,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X2)),aa_nat_nat(G,X2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8066_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X2)),aa_int_int(G,X2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8067_sum__strict__mono,axiom,
    ! [A2: set_set_nat,F: fun_set_nat_rat,G: fun_set_nat_rat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( ! [X2: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(X2),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_set_nat_rat(F,X2)),aa_set_nat_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups7659867448343625626at_rat(F,A2)),groups7659867448343625626at_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8068_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% sgn_1_pos
tff(fact_8069_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat )
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% sgn_1_pos
tff(fact_8070_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = one_one_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_1_pos
tff(fact_8071_prod_Orelated,axiom,
    ! [R: fun_Co2081214437782444048eger_o,S: set_nat,H2: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,one_one_Code_integer),one_one_Code_integer)
     => ( ! [X12: code_integer,Y12: code_integer,X23: code_integer,Y23: code_integer] :
            ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,X12),X23)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,Y12),Y23) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X12),Y12)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,aa_nat_Code_integer(H2,X2)),aa_nat_Code_integer(G,X2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,aa_set871585252387937313nteger(groups3455450783089532116nteger(H2),S)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),S)) ) ) ) ) ).

% prod.related
tff(fact_8072_prod_Orelated,axiom,
    ! [R: fun_Co2081214437782444048eger_o,S: set_int,H2: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,one_one_Code_integer),one_one_Code_integer)
     => ( ! [X12: code_integer,Y12: code_integer,X23: code_integer,Y23: code_integer] :
            ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,X12),X23)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,Y12),Y23) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X12),Y12)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,aa_int_Code_integer(H2,X2)),aa_int_Code_integer(G,X2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,groups3827104343326376752nteger(H2,S)),groups3827104343326376752nteger(G,S)) ) ) ) ) ).

% prod.related
tff(fact_8073_prod_Orelated,axiom,
    ! [R: fun_assn_fun_assn_o,S: set_nat,H2: fun_nat_assn,G: fun_nat_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(R,one_one_assn),one_one_assn)
     => ( ! [X12: assn,Y12: assn,X23: assn,Y23: assn] :
            ( ( aa_assn_o(aa_assn_fun_assn_o(R,X12),X23)
              & aa_assn_o(aa_assn_fun_assn_o(R,Y12),Y23) )
           => aa_assn_o(aa_assn_fun_assn_o(R,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X12),Y12)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => aa_assn_o(aa_assn_fun_assn_o(R,aa_nat_assn(H2,X2)),aa_nat_assn(G,X2)) )
           => aa_assn_o(aa_assn_fun_assn_o(R,groups6906906614972039071t_assn(H2,S)),groups6906906614972039071t_assn(G,S)) ) ) ) ) ).

% prod.related
tff(fact_8074_prod_Orelated,axiom,
    ! [R: fun_assn_fun_assn_o,S: set_int,H2: fun_int_assn,G: fun_int_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(R,one_one_assn),one_one_assn)
     => ( ! [X12: assn,Y12: assn,X23: assn,Y23: assn] :
            ( ( aa_assn_o(aa_assn_fun_assn_o(R,X12),X23)
              & aa_assn_o(aa_assn_fun_assn_o(R,Y12),Y23) )
           => aa_assn_o(aa_assn_fun_assn_o(R,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X12),Y12)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => aa_assn_o(aa_assn_fun_assn_o(R,aa_int_assn(H2,X2)),aa_int_assn(G,X2)) )
           => aa_assn_o(aa_assn_fun_assn_o(R,groups7882442080178216443t_assn(H2,S)),groups7882442080178216443t_assn(G,S)) ) ) ) ) ).

% prod.related
tff(fact_8075_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S: set_nat,H2: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X12: rat,Y12: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X12),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y12),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X12),Y12)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H2,X2)),aa_nat_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_set_nat_rat(groups73079841787564623at_rat(H2),S)),aa_set_nat_rat(groups73079841787564623at_rat(G),S)) ) ) ) ) ).

% prod.related
tff(fact_8076_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S: set_int,H2: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X12: rat,Y12: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X12),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y12),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X12),Y12)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_int_rat(H2,X2)),aa_int_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups1072433553688619179nt_rat(H2,S)),groups1072433553688619179nt_rat(G,S)) ) ) ) ) ).

% prod.related
tff(fact_8077_prod_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S: set_int,H2: fun_int_nat,G: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,one_one_nat),one_one_nat)
     => ( ! [X12: nat,Y12: nat,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X12),X23)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y12),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X12),Y12)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_int_nat(H2,X2)),aa_int_nat(G,X2)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups1707563613775114915nt_nat(H2,S)),groups1707563613775114915nt_nat(G,S)) ) ) ) ) ).

% prod.related
tff(fact_8078_prod_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S: set_nat,H2: fun_nat_nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,one_one_nat),one_one_nat)
     => ( ! [X12: nat,Y12: nat,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X12),X23)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y12),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X12),Y12)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(H2,X2)),aa_nat_nat(G,X2)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_set_nat_nat(groups708209901874060359at_nat(H2),S)),aa_set_nat_nat(groups708209901874060359at_nat(G),S)) ) ) ) ) ).

% prod.related
tff(fact_8079_prod_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S: set_nat,H2: fun_nat_int,G: fun_nat_int] :
      ( aa_int_o(aa_int_fun_int_o(R,one_one_int),one_one_int)
     => ( ! [X12: int,Y12: int,X23: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X12),X23)
              & aa_int_o(aa_int_fun_int_o(R,Y12),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(times_times_int,X12),Y12)),aa_int_int(aa_int_fun_int_int(times_times_int,X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => aa_int_o(aa_int_fun_int_o(R,aa_nat_int(H2,X2)),aa_nat_int(G,X2)) )
           => aa_int_o(aa_int_fun_int_o(R,aa_set_nat_int(groups705719431365010083at_int(H2),S)),aa_set_nat_int(groups705719431365010083at_int(G),S)) ) ) ) ) ).

% prod.related
tff(fact_8080_prod_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S: set_int,H2: fun_int_int,G: fun_int_int] :
      ( aa_int_o(aa_int_fun_int_o(R,one_one_int),one_one_int)
     => ( ! [X12: int,Y12: int,X23: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X12),X23)
              & aa_int_o(aa_int_fun_int_o(R,Y12),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(times_times_int,X12),Y12)),aa_int_int(aa_int_fun_int_int(times_times_int,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => aa_int_o(aa_int_fun_int_o(R,aa_int_int(H2,X2)),aa_int_int(G,X2)) )
           => aa_int_o(aa_int_fun_int_o(R,groups1705073143266064639nt_int(H2,S)),groups1705073143266064639nt_int(G,S)) ) ) ) ) ).

% prod.related
tff(fact_8081_prod_Oinsert__if,axiom,
    ! [A2: set_o,G: fun_o_Code_integer,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups7694694392188491536nteger(G,insert_o((X),A2)) = $ite(aa_set_o_o(member_o((X)),A2),groups7694694392188491536nteger(G,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(G,(X))),groups7694694392188491536nteger(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8082_prod_Oinsert__if,axiom,
    ! [A2: set_nat,G: fun_nat_Code_integer,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(insert_nat(X),A2)) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,X)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2))) ) ) ).

% prod.insert_if
tff(fact_8083_prod_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_Code_integer,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3827104343326376752nteger(G,insert_int(X,A2)) = $ite(aa_set_int_o(member_int(X),A2),groups3827104343326376752nteger(G,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(G,X)),groups3827104343326376752nteger(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8084_prod_Oinsert__if,axiom,
    ! [A2: set_o,G: fun_o_assn,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups5301882518646026715o_assn(G,insert_o((X),A2)) = $ite(aa_set_o_o(member_o((X)),A2),groups5301882518646026715o_assn(G,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(G,(X))),groups5301882518646026715o_assn(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8085_prod_Oinsert__if,axiom,
    ! [A2: set_nat,G: fun_nat_assn,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups6906906614972039071t_assn(G,aa_set_nat_set_nat(insert_nat(X),A2)) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2),groups6906906614972039071t_assn(G,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,X)),groups6906906614972039071t_assn(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8086_prod_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_assn,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7882442080178216443t_assn(G,insert_int(X,A2)) = $ite(aa_set_int_o(member_int(X),A2),groups7882442080178216443t_assn(G,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(G,X)),groups7882442080178216443t_assn(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8087_prod_Oinsert__if,axiom,
    ! [A2: set_o,G: fun_o_rat,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups2869687844427037835_o_rat(G,insert_o((X),A2)) = $ite(aa_set_o_o(member_o((X)),A2),groups2869687844427037835_o_rat(G,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(G,(X))),groups2869687844427037835_o_rat(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8088_prod_Oinsert__if,axiom,
    ! [A2: set_nat,G: fun_nat_rat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(insert_nat(X),A2)) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2),aa_set_nat_rat(groups73079841787564623at_rat(G),A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,X)),aa_set_nat_rat(groups73079841787564623at_rat(G),A2))) ) ) ).

% prod.insert_if
tff(fact_8089_prod_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_rat,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1072433553688619179nt_rat(G,insert_int(X,A2)) = $ite(aa_set_int_o(member_int(X),A2),groups1072433553688619179nt_rat(G,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G,X)),groups1072433553688619179nt_rat(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8090_prod_Oinsert__if,axiom,
    ! [A2: set_o,G: fun_o_nat,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups3504817904513533571_o_nat(G,insert_o((X),A2)) = $ite(aa_set_o_o(member_o((X)),A2),groups3504817904513533571_o_nat(G,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(G,(X))),groups3504817904513533571_o_nat(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8091_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S3: set_int,T3: set_int,S: set_int,I: fun_int_int,J: fun_int_int,T: set_int,G: fun_int_Code_integer,H2: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),minus_minus_set_int(S,S3))
             => ( aa_int_int(I,aa_int_int(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(S,S3))
               => aa_set_int_o(member_int(aa_int_int(J,A4)),minus_minus_set_int(T,T3)) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(T,T3))
                 => ( aa_int_int(J,aa_int_int(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( aa_set_int_o(member_int(B4),minus_minus_set_int(T,T3))
                   => aa_set_int_o(member_int(aa_int_int(I,B4)),minus_minus_set_int(S,S3)) )
               => ( ! [A4: int] :
                      ( aa_set_int_o(member_int(A4),S3)
                     => ( aa_int_Code_integer(G,A4) = one_one_Code_integer ) )
                 => ( ! [B4: int] :
                        ( aa_set_int_o(member_int(B4),T3)
                       => ( aa_int_Code_integer(H2,B4) = one_one_Code_integer ) )
                   => ( ! [A4: int] :
                          ( aa_set_int_o(member_int(A4),S)
                         => ( aa_int_Code_integer(H2,aa_int_int(J,A4)) = aa_int_Code_integer(G,A4) ) )
                     => ( groups3827104343326376752nteger(G,S) = groups3827104343326376752nteger(H2,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8092_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S3: set_int,T3: set_int,S: set_int,I: fun_int_int,J: fun_int_int,T: set_int,G: fun_int_assn,H2: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),minus_minus_set_int(S,S3))
             => ( aa_int_int(I,aa_int_int(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(S,S3))
               => aa_set_int_o(member_int(aa_int_int(J,A4)),minus_minus_set_int(T,T3)) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(T,T3))
                 => ( aa_int_int(J,aa_int_int(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( aa_set_int_o(member_int(B4),minus_minus_set_int(T,T3))
                   => aa_set_int_o(member_int(aa_int_int(I,B4)),minus_minus_set_int(S,S3)) )
               => ( ! [A4: int] :
                      ( aa_set_int_o(member_int(A4),S3)
                     => ( aa_int_assn(G,A4) = one_one_assn ) )
                 => ( ! [B4: int] :
                        ( aa_set_int_o(member_int(B4),T3)
                       => ( aa_int_assn(H2,B4) = one_one_assn ) )
                   => ( ! [A4: int] :
                          ( aa_set_int_o(member_int(A4),S)
                         => ( aa_int_assn(H2,aa_int_int(J,A4)) = aa_int_assn(G,A4) ) )
                     => ( groups7882442080178216443t_assn(G,S) = groups7882442080178216443t_assn(H2,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8093_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S3: set_int,T3: set_int,S: set_int,I: fun_int_int,J: fun_int_int,T: set_int,G: fun_int_rat,H2: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),minus_minus_set_int(S,S3))
             => ( aa_int_int(I,aa_int_int(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(S,S3))
               => aa_set_int_o(member_int(aa_int_int(J,A4)),minus_minus_set_int(T,T3)) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(T,T3))
                 => ( aa_int_int(J,aa_int_int(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( aa_set_int_o(member_int(B4),minus_minus_set_int(T,T3))
                   => aa_set_int_o(member_int(aa_int_int(I,B4)),minus_minus_set_int(S,S3)) )
               => ( ! [A4: int] :
                      ( aa_set_int_o(member_int(A4),S3)
                     => ( aa_int_rat(G,A4) = one_one_rat ) )
                 => ( ! [B4: int] :
                        ( aa_set_int_o(member_int(B4),T3)
                       => ( aa_int_rat(H2,B4) = one_one_rat ) )
                   => ( ! [A4: int] :
                          ( aa_set_int_o(member_int(A4),S)
                         => ( aa_int_rat(H2,aa_int_int(J,A4)) = aa_int_rat(G,A4) ) )
                     => ( groups1072433553688619179nt_rat(G,S) = groups1072433553688619179nt_rat(H2,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8094_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S3: set_int,T3: set_int,S: set_int,I: fun_int_int,J: fun_int_int,T: set_int,G: fun_int_nat,H2: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),minus_minus_set_int(S,S3))
             => ( aa_int_int(I,aa_int_int(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(S,S3))
               => aa_set_int_o(member_int(aa_int_int(J,A4)),minus_minus_set_int(T,T3)) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(T,T3))
                 => ( aa_int_int(J,aa_int_int(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( aa_set_int_o(member_int(B4),minus_minus_set_int(T,T3))
                   => aa_set_int_o(member_int(aa_int_int(I,B4)),minus_minus_set_int(S,S3)) )
               => ( ! [A4: int] :
                      ( aa_set_int_o(member_int(A4),S3)
                     => ( aa_int_nat(G,A4) = one_one_nat ) )
                 => ( ! [B4: int] :
                        ( aa_set_int_o(member_int(B4),T3)
                       => ( aa_int_nat(H2,B4) = one_one_nat ) )
                   => ( ! [A4: int] :
                          ( aa_set_int_o(member_int(A4),S)
                         => ( aa_int_nat(H2,aa_int_int(J,A4)) = aa_int_nat(G,A4) ) )
                     => ( groups1707563613775114915nt_nat(G,S) = groups1707563613775114915nt_nat(H2,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8095_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S3: set_int,T3: set_nat,S: set_int,I: fun_nat_int,J: fun_int_nat,T: set_nat,G: fun_int_Code_integer,H2: fun_nat_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_nat_o(finite_finite_nat,T3)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),minus_minus_set_int(S,S3))
             => ( aa_nat_int(I,aa_int_nat(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(S,S3))
               => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(J,A4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),T3)) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),T3))
                 => ( aa_int_nat(J,aa_nat_int(I,B4)) = B4 ) )
             => ( ! [B4: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),T3))
                   => aa_set_int_o(member_int(aa_nat_int(I,B4)),minus_minus_set_int(S,S3)) )
               => ( ! [A4: int] :
                      ( aa_set_int_o(member_int(A4),S3)
                     => ( aa_int_Code_integer(G,A4) = one_one_Code_integer ) )
                 => ( ! [B4: nat] :
                        ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),T3)
                       => ( aa_nat_Code_integer(H2,B4) = one_one_Code_integer ) )
                   => ( ! [A4: int] :
                          ( aa_set_int_o(member_int(A4),S)
                         => ( aa_nat_Code_integer(H2,aa_int_nat(J,A4)) = aa_int_Code_integer(G,A4) ) )
                     => ( groups3827104343326376752nteger(G,S) = aa_set871585252387937313nteger(groups3455450783089532116nteger(H2),T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8096_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S3: set_int,T3: set_nat,S: set_int,I: fun_nat_int,J: fun_int_nat,T: set_nat,G: fun_int_assn,H2: fun_nat_assn] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_nat_o(finite_finite_nat,T3)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),minus_minus_set_int(S,S3))
             => ( aa_nat_int(I,aa_int_nat(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(S,S3))
               => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(J,A4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),T3)) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),T3))
                 => ( aa_int_nat(J,aa_nat_int(I,B4)) = B4 ) )
             => ( ! [B4: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),T3))
                   => aa_set_int_o(member_int(aa_nat_int(I,B4)),minus_minus_set_int(S,S3)) )
               => ( ! [A4: int] :
                      ( aa_set_int_o(member_int(A4),S3)
                     => ( aa_int_assn(G,A4) = one_one_assn ) )
                 => ( ! [B4: nat] :
                        ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),T3)
                       => ( aa_nat_assn(H2,B4) = one_one_assn ) )
                   => ( ! [A4: int] :
                          ( aa_set_int_o(member_int(A4),S)
                         => ( aa_nat_assn(H2,aa_int_nat(J,A4)) = aa_int_assn(G,A4) ) )
                     => ( groups7882442080178216443t_assn(G,S) = groups6906906614972039071t_assn(H2,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8097_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S3: set_int,T3: set_nat,S: set_int,I: fun_nat_int,J: fun_int_nat,T: set_nat,G: fun_int_rat,H2: fun_nat_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_nat_o(finite_finite_nat,T3)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),minus_minus_set_int(S,S3))
             => ( aa_nat_int(I,aa_int_nat(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(S,S3))
               => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(J,A4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),T3)) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),T3))
                 => ( aa_int_nat(J,aa_nat_int(I,B4)) = B4 ) )
             => ( ! [B4: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),T3))
                   => aa_set_int_o(member_int(aa_nat_int(I,B4)),minus_minus_set_int(S,S3)) )
               => ( ! [A4: int] :
                      ( aa_set_int_o(member_int(A4),S3)
                     => ( aa_int_rat(G,A4) = one_one_rat ) )
                 => ( ! [B4: nat] :
                        ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),T3)
                       => ( aa_nat_rat(H2,B4) = one_one_rat ) )
                   => ( ! [A4: int] :
                          ( aa_set_int_o(member_int(A4),S)
                         => ( aa_nat_rat(H2,aa_int_nat(J,A4)) = aa_int_rat(G,A4) ) )
                     => ( groups1072433553688619179nt_rat(G,S) = aa_set_nat_rat(groups73079841787564623at_rat(H2),T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8098_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S3: set_nat,T3: set_int,S: set_nat,I: fun_int_nat,J: fun_nat_int,T: set_int,G: fun_nat_Code_integer,H2: fun_int_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),S3))
             => ( aa_int_nat(I,aa_nat_int(J,A4)) = A4 ) )
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),S3))
               => aa_set_int_o(member_int(aa_nat_int(J,A4)),minus_minus_set_int(T,T3)) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(T,T3))
                 => ( aa_nat_int(J,aa_int_nat(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( aa_set_int_o(member_int(B4),minus_minus_set_int(T,T3))
                   => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(I,B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),S3)) )
               => ( ! [A4: nat] :
                      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),S3)
                     => ( aa_nat_Code_integer(G,A4) = one_one_Code_integer ) )
                 => ( ! [B4: int] :
                        ( aa_set_int_o(member_int(B4),T3)
                       => ( aa_int_Code_integer(H2,B4) = one_one_Code_integer ) )
                   => ( ! [A4: nat] :
                          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),S)
                         => ( aa_int_Code_integer(H2,aa_nat_int(J,A4)) = aa_nat_Code_integer(G,A4) ) )
                     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),S) = groups3827104343326376752nteger(H2,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8099_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S3: set_nat,T3: set_int,S: set_nat,I: fun_int_nat,J: fun_nat_int,T: set_int,G: fun_nat_assn,H2: fun_int_assn] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),S3))
             => ( aa_int_nat(I,aa_nat_int(J,A4)) = A4 ) )
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),S3))
               => aa_set_int_o(member_int(aa_nat_int(J,A4)),minus_minus_set_int(T,T3)) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(T,T3))
                 => ( aa_nat_int(J,aa_int_nat(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( aa_set_int_o(member_int(B4),minus_minus_set_int(T,T3))
                   => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(I,B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),S3)) )
               => ( ! [A4: nat] :
                      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),S3)
                     => ( aa_nat_assn(G,A4) = one_one_assn ) )
                 => ( ! [B4: int] :
                        ( aa_set_int_o(member_int(B4),T3)
                       => ( aa_int_assn(H2,B4) = one_one_assn ) )
                   => ( ! [A4: nat] :
                          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),S)
                         => ( aa_int_assn(H2,aa_nat_int(J,A4)) = aa_nat_assn(G,A4) ) )
                     => ( groups6906906614972039071t_assn(G,S) = groups7882442080178216443t_assn(H2,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8100_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S3: set_nat,T3: set_int,S: set_nat,I: fun_int_nat,J: fun_nat_int,T: set_int,G: fun_nat_rat,H2: fun_int_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),S3))
             => ( aa_int_nat(I,aa_nat_int(J,A4)) = A4 ) )
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),S3))
               => aa_set_int_o(member_int(aa_nat_int(J,A4)),minus_minus_set_int(T,T3)) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(T,T3))
                 => ( aa_nat_int(J,aa_int_nat(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( aa_set_int_o(member_int(B4),minus_minus_set_int(T,T3))
                   => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(I,B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),S3)) )
               => ( ! [A4: nat] :
                      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),S3)
                     => ( aa_nat_rat(G,A4) = one_one_rat ) )
                 => ( ! [B4: int] :
                        ( aa_set_int_o(member_int(B4),T3)
                       => ( aa_int_rat(H2,B4) = one_one_rat ) )
                   => ( ! [A4: nat] :
                          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),S)
                         => ( aa_int_rat(H2,aa_nat_int(J,A4)) = aa_nat_rat(G,A4) ) )
                     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),S) = groups1072433553688619179nt_rat(H2,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8101_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = $ite(A = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,one_one_Code_integer) ) ).

% abs_sgn_eq
tff(fact_8102_abs__sgn__eq,axiom,
    ! [A: rat] :
      ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ) ).

% abs_sgn_eq
tff(fact_8103_abs__sgn__eq,axiom,
    ! [A: int] :
      ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = $ite(A = zero_zero_int,zero_zero_int,one_one_int) ) ).

% abs_sgn_eq
tff(fact_8104_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_zw(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ).

% sum.nat_diff_reindex
tff(fact_8105_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_zw(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ).

% prod.nat_diff_reindex
tff(fact_8106_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_zx(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ).

% prod.nat_diff_reindex
tff(fact_8107_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups4541462559716669496nt_nat(F,A2) = one_one_nat )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A2)
            & ( aa_int_nat(F,X4) = one_one_nat )
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A2)
               => ( ( X4 != Xa2 )
                 => ( aa_int_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8108_sum__eq__1__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( ( groups977919841031483927at_nat(F,A2) = one_one_nat )
      <=> ? [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A2)
            & ( aa_Pro4243173770044397973at_nat(F,X4) = one_one_nat )
            & ! [Xa2: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa2),A2)
               => ( ( X4 != Xa2 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8109_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = one_one_nat )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A2)
            & ( aa_nat_nat(F,X4) = one_one_nat )
            & ! [Xa2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa2),A2)
               => ( ( X4 != Xa2 )
                 => ( aa_nat_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8110_prod__int__eq,axiom,
    ! [I: nat,J: nat] : ( aa_set_nat_int(groups705719431365010083at_int(semiri1314217659103216013at_int),aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = groups1705073143266064639nt_int(aTP_Lamm_db,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ) ).

% prod_int_eq
tff(fact_8111_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: code_integer,N: nat] : ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),one_one_Code_integer)) = aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(N),A) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_8112_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: code_natural,N: nat] : ( divide5121882707175180666atural(A,bit_se6611745700429515170atural(N,one_one_Code_natural)) = bit_se2751692689150723149atural(N,A) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_8113_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: int,N: nat] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),A) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_8114_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),A) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_8115_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: fun_nat_Code_integer,B2: code_integer,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I2)) )
       => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(F),S2) = B2 )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8116_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_Code_integer,B2: code_integer,I: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I2)) )
       => ( ( groups7873554091576472773nteger(F,S2) = B2 )
         => ( aa_set_int_o(member_int(I),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8117_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: fun_nat_rat,B2: rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(F),S2) = B2 )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8118_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_rat,B2: rat,I: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,S2) = B2 )
         => ( aa_set_int_o(member_int(I),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8119_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_nat,B2: nat,I: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
       => ( ( groups4541462559716669496nt_nat(F,S2) = B2 )
         => ( aa_set_int_o(member_int(I),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8120_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: fun_nat_int,B2: int,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I2)) )
       => ( ( aa_set_nat_int(groups3539618377306564664at_int(F),S2) = B2 )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8121_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: fun_nat_nat,B2: nat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,I2)) )
       => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),S2) = B2 )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8122_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_int,B2: int,I: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,I2)) )
       => ( ( groups4538972089207619220nt_int(F,S2) = B2 )
         => ( aa_set_int_o(member_int(I),S2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8123_sum__nonneg__leq__bound,axiom,
    ! [S2: set_set_nat,F: fun_se7586849273513198688nteger,B2: code_integer,I: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,S2)
     => ( ! [I2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(I2),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(F,I2)) )
       => ( ( groups9190459664516455967nteger(F,S2) = B2 )
         => ( aa_set_set_nat_o(member_set_nat(I),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8124_sum__nonneg__leq__bound,axiom,
    ! [S2: set_set_nat,F: fun_set_nat_rat,B2: rat,I: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,S2)
     => ( ! [I2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_set_nat_rat(F,I2)) )
       => ( ( groups7659867448343625626at_rat(F,S2) = B2 )
         => ( aa_set_set_nat_o(member_set_nat(I),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8125_sum__nonneg__0,axiom,
    ! [S2: set_nat,F: fun_nat_Code_integer,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I2)) )
       => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(F),S2) = zero_z3403309356797280102nteger )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S2)
           => ( aa_nat_Code_integer(F,I) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8126_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_Code_integer,I: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I2)) )
       => ( ( groups7873554091576472773nteger(F,S2) = zero_z3403309356797280102nteger )
         => ( aa_set_int_o(member_int(I),S2)
           => ( aa_int_Code_integer(F,I) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8127_sum__nonneg__0,axiom,
    ! [S2: set_nat,F: fun_nat_rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(F),S2) = zero_zero_rat )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S2)
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8128_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_rat,I: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,S2) = zero_zero_rat )
         => ( aa_set_int_o(member_int(I),S2)
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8129_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_nat,I: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
       => ( ( groups4541462559716669496nt_nat(F,S2) = zero_zero_nat )
         => ( aa_set_int_o(member_int(I),S2)
           => ( aa_int_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8130_sum__nonneg__0,axiom,
    ! [S2: set_nat,F: fun_nat_int,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I2)) )
       => ( ( aa_set_nat_int(groups3539618377306564664at_int(F),S2) = zero_zero_int )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S2)
           => ( aa_nat_int(F,I) = zero_zero_int ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8131_sum__nonneg__0,axiom,
    ! [S2: set_nat,F: fun_nat_nat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,I2)) )
       => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),S2) = zero_zero_nat )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S2)
           => ( aa_nat_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8132_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_int,I: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,I2)) )
       => ( ( groups4538972089207619220nt_int(F,S2) = zero_zero_int )
         => ( aa_set_int_o(member_int(I),S2)
           => ( aa_int_int(F,I) = zero_zero_int ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8133_sum__nonneg__0,axiom,
    ! [S2: set_set_nat,F: fun_se7586849273513198688nteger,I: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,S2)
     => ( ! [I2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(I2),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(F,I2)) )
       => ( ( groups9190459664516455967nteger(F,S2) = zero_z3403309356797280102nteger )
         => ( aa_set_set_nat_o(member_set_nat(I),S2)
           => ( aa_set871585252387937313nteger(F,I) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8134_sum__nonneg__0,axiom,
    ! [S2: set_set_nat,F: fun_set_nat_rat,I: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,S2)
     => ( ! [I2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_set_nat_rat(F,I2)) )
       => ( ( groups7659867448343625626at_rat(F,S2) = zero_zero_rat )
         => ( aa_set_set_nat_o(member_set_nat(I),S2)
           => ( aa_set_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8135_ivl__disj__un__one_I2_J,axiom,
    ! [L: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),U)
     => ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,set_ord_lessThan_rat(L)),set_or4029947393144176647an_rat(L,U)) = set_ord_lessThan_rat(U) ) ) ).

% ivl_disj_un_one(2)
tff(fact_8136_ivl__disj__un__one_I2_J,axiom,
    ! [L: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),U)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,L)),aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)) = aa_nat_set_nat(set_ord_lessThan_nat,U) ) ) ).

% ivl_disj_un_one(2)
tff(fact_8137_ivl__disj__un__one_I2_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),U)
     => ( sup_su848401254843788991nteger(set_or5754767410780653050nteger(L),set_or8404916559141939852nteger(L,U)) = set_or5754767410780653050nteger(U) ) ) ).

% ivl_disj_un_one(2)
tff(fact_8138_ivl__disj__un__one_I2_J,axiom,
    ! [L: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),U)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,set_ord_lessThan_int(L)),set_or4662586982721622107an_int(L,U)) = set_ord_lessThan_int(U) ) ) ).

% ivl_disj_un_one(2)
tff(fact_8139_ivl__disj__int__one_I4_J,axiom,
    ! [L: $o,U: $o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,set_ord_lessThan_o((L))),set_or8904488021354931149Most_o((L),(U))) = bot_bot_set_o ) ).

% ivl_disj_int_one(4)
tff(fact_8140_ivl__disj__int__one_I4_J,axiom,
    ! [L: nat,U: nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,L)),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = bot_bot_set_nat ) ).

% ivl_disj_int_one(4)
tff(fact_8141_ivl__disj__int__one_I4_J,axiom,
    ! [L: int,U: int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_ord_lessThan_int(L)),set_or1266510415728281911st_int(L,U)) = bot_bot_set_int ) ).

% ivl_disj_int_one(4)
tff(fact_8142_ivl__disj__int__one_I4_J,axiom,
    ! [L: code_integer,U: code_integer] : ( inf_in1364745209274528805nteger(set_or5754767410780653050nteger(L),set_or189985376899183464nteger(L,U)) = bot_bo3990330152332043303nteger ) ).

% ivl_disj_int_one(4)
tff(fact_8143_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : ( groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_sh(N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_zy(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ).

% prod.triangle_reindex
tff(fact_8144_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_int,N: nat] : ( aa_set4746584241299694929at_int(groups4075276357253098568at_int(produc6840382203811409530at_int(G)),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_sh(N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_aaa(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ).

% prod.triangle_reindex
tff(fact_8145_ivl__disj__int__one_I2_J,axiom,
    ! [L: $o,U: $o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,set_ord_lessThan_o((L))),set_or7139685690850216873Than_o((L),(U))) = bot_bot_set_o ) ).

% ivl_disj_int_one(2)
tff(fact_8146_ivl__disj__int__one_I2_J,axiom,
    ! [L: nat,U: nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,L)),aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)) = bot_bot_set_nat ) ).

% ivl_disj_int_one(2)
tff(fact_8147_ivl__disj__int__one_I2_J,axiom,
    ! [L: code_integer,U: code_integer] : ( inf_in1364745209274528805nteger(set_or5754767410780653050nteger(L),set_or8404916559141939852nteger(L,U)) = bot_bo3990330152332043303nteger ) ).

% ivl_disj_int_one(2)
tff(fact_8148_ivl__disj__int__one_I2_J,axiom,
    ! [L: int,U: int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_ord_lessThan_int(L)),set_or4662586982721622107an_int(L,U)) = bot_bot_set_int ) ).

% ivl_disj_int_one(2)
tff(fact_8149_sum_Ointer__restrict,axiom,
    ! [A2: set_int,G: fun_int_Code_integer,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7873554091576472773nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) = groups7873554091576472773nteger(aa_set5517701783735052070nteger(aTP_Lamm_aab(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8150_sum_Ointer__restrict,axiom,
    ! [A2: set_int,G: fun_int_rat,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) = groups3906332499630173760nt_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_aac(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8151_sum_Ointer__restrict,axiom,
    ! [A2: set_int,G: fun_int_nat,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) = groups4541462559716669496nt_nat(aa_set2264519855582034201nt_nat(aTP_Lamm_aad(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8152_sum_Ointer__restrict,axiom,
    ! [A2: set_nat,G: fun_nat_Code_integer,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_set753585224467312750nteger(aTP_Lamm_aae(G),B2)),A2) ) ) ).

% sum.inter_restrict
tff(fact_8153_sum_Ointer__restrict,axiom,
    ! [A2: set_nat,G: fun_nat_rat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_set3028864993726649449at_rat(aTP_Lamm_aaf(G),B2)),A2) ) ) ).

% sum.inter_restrict
tff(fact_8154_sum_Ointer__restrict,axiom,
    ! [A2: set_nat,G: fun_nat_int,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_int(groups3539618377306564664at_int(aa_set2359242532610849469at_int(aTP_Lamm_aag(G),B2)),A2) ) ) ).

% sum.inter_restrict
tff(fact_8155_sum_Ointer__restrict,axiom,
    ! [A2: set_nat,G: fun_nat_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_aah(G),B2)),A2) ) ) ).

% sum.inter_restrict
tff(fact_8156_sum_Ointer__restrict,axiom,
    ! [A2: set_int,G: fun_int_int,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_aai(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8157_sum_Ointer__restrict,axiom,
    ! [A2: set_set_nat,G: fun_se7586849273513198688nteger,B2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups9190459664516455967nteger(G,inf_inf_set_set_nat(A2,B2)) = groups9190459664516455967nteger(aa_set4557450419886607578nteger(aTP_Lamm_aaj(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8158_sum_Ointer__restrict,axiom,
    ! [A2: set_set_nat,G: fun_set_nat_rat,B2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups7659867448343625626at_rat(G,inf_inf_set_set_nat(A2,B2)) = groups7659867448343625626at_rat(aa_set8854944969235456213at_rat(aTP_Lamm_aak(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8159_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7873554091576472773nteger(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_aal(G)))) = groups7873554091576472773nteger(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8160_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_aam(G)))) = groups3906332499630173760nt_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8161_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_aan(G)))) = groups4541462559716669496nt_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8162_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),collect_nat(aTP_Lamm_aao(G)))) = aa_set871585252387937313nteger(groups7501900531339628137nteger(G),A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8163_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),collect_nat(aTP_Lamm_aap(G)))) = aa_set_nat_rat(groups2906978787729119204at_rat(G),A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8164_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),collect_nat(aTP_Lamm_aaq(G)))) = aa_set_nat_int(groups3539618377306564664at_int(G),A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8165_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),collect_nat(aTP_Lamm_aar(G)))) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8166_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4538972089207619220nt_int(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_aas(G)))) = groups4538972089207619220nt_int(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8167_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_set_nat,G: fun_se7586849273513198688nteger] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups9190459664516455967nteger(G,minus_2163939370556025621et_nat(A2,collect_set_nat(aTP_Lamm_aat(G)))) = groups9190459664516455967nteger(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8168_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_set_nat,G: fun_set_nat_rat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups7659867448343625626at_rat(G,minus_2163939370556025621et_nat(A2,collect_set_nat(aTP_Lamm_aau(G)))) = groups7659867448343625626at_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8169_prod_Ointer__restrict,axiom,
    ! [A2: set_int,G: fun_int_Code_integer,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) = groups3827104343326376752nteger(aa_set5517701783735052070nteger(aTP_Lamm_aav(G),B2),A2) ) ) ).

% prod.inter_restrict
tff(fact_8170_prod_Ointer__restrict,axiom,
    ! [A2: set_int,G: fun_int_assn,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) = groups7882442080178216443t_assn(aa_set5310648174735401585t_assn(aTP_Lamm_aaw(G),B2),A2) ) ) ).

% prod.inter_restrict
tff(fact_8171_prod_Ointer__restrict,axiom,
    ! [A2: set_int,G: fun_int_rat,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) = groups1072433553688619179nt_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_aax(G),B2),A2) ) ) ).

% prod.inter_restrict
tff(fact_8172_prod_Ointer__restrict,axiom,
    ! [A2: set_int,G: fun_int_nat,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) = groups1707563613775114915nt_nat(aa_set2264519855582034201nt_nat(aTP_Lamm_aay(G),B2),A2) ) ) ).

% prod.inter_restrict
tff(fact_8173_prod_Ointer__restrict,axiom,
    ! [A2: set_nat,G: fun_nat_Code_integer,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_set753585224467312750nteger(aTP_Lamm_aaz(G),B2)),A2) ) ) ).

% prod.inter_restrict
tff(fact_8174_prod_Ointer__restrict,axiom,
    ! [A2: set_nat,G: fun_nat_assn,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = groups6906906614972039071t_assn(aa_set7559964055741201337t_assn(aTP_Lamm_aba(G),B2),A2) ) ) ).

% prod.inter_restrict
tff(fact_8175_prod_Ointer__restrict,axiom,
    ! [A2: set_nat,G: fun_nat_rat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_rat(groups73079841787564623at_rat(aa_set3028864993726649449at_rat(aTP_Lamm_abb(G),B2)),A2) ) ) ).

% prod.inter_restrict
tff(fact_8176_prod_Ointer__restrict,axiom,
    ! [A2: set_nat,G: fun_nat_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_abc(G),B2)),A2) ) ) ).

% prod.inter_restrict
tff(fact_8177_prod_Ointer__restrict,axiom,
    ! [A2: set_nat,G: fun_nat_int,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_int(groups705719431365010083at_int(aa_set2359242532610849469at_int(aTP_Lamm_abd(G),B2)),A2) ) ) ).

% prod.inter_restrict
tff(fact_8178_prod_Ointer__restrict,axiom,
    ! [A2: set_int,G: fun_int_int,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1705073143266064639nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) = groups1705073143266064639nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_abe(G),B2),A2) ) ) ).

% prod.inter_restrict
tff(fact_8179_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3827104343326376752nteger(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_abf(G)))) = groups3827104343326376752nteger(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8180_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7882442080178216443t_assn(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_abg(G)))) = groups7882442080178216443t_assn(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8181_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1072433553688619179nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_abh(G)))) = groups1072433553688619179nt_rat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8182_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1707563613775114915nt_nat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_abi(G)))) = groups1707563613775114915nt_nat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8183_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),collect_nat(aTP_Lamm_abj(G)))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8184_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),collect_nat(aTP_Lamm_abk(G)))) = groups6906906614972039071t_assn(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8185_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),collect_nat(aTP_Lamm_abl(G)))) = aa_set_nat_rat(groups73079841787564623at_rat(G),A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8186_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),collect_nat(aTP_Lamm_abm(G)))) = aa_set_nat_nat(groups708209901874060359at_nat(G),A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8187_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),collect_nat(aTP_Lamm_abn(G)))) = aa_set_nat_int(groups705719431365010083at_int(G),A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8188_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1705073143266064639nt_int(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_abo(G)))) = groups1705073143266064639nt_int(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8189_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_if,M))) ) ).

% finite_divisors_nat
tff(fact_8190_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_8191_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,N: nat] :
      ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Q,X2)),aa_nat_nat(P,X2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lp(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ) ).

% sum_diff_distrib
tff(fact_8192_sum__pos,axiom,
    ! [I4: set_o,F: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,I4)
     => ( ( I4 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o((I2)),I4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(I2))) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups4406642042086082107nteger(F,I4)) ) ) ) ).

% sum_pos
tff(fact_8193_sum__pos,axiom,
    ! [I4: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I4)
     => ( ( I4 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(groups7501900531339628137nteger(F),I4)) ) ) ) ).

% sum_pos
tff(fact_8194_sum__pos,axiom,
    ! [I4: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,I4)
     => ( ( I4 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups7873554091576472773nteger(F,I4)) ) ) ) ).

% sum_pos
tff(fact_8195_sum__pos,axiom,
    ! [I4: set_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,I4)
     => ( ( I4 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o((I2)),I4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(F,(I2))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups7872700643590313910_o_rat(F,I4)) ) ) ) ).

% sum_pos
tff(fact_8196_sum__pos,axiom,
    ! [I4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I4)
     => ( ( I4 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I4)
             => 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),aa_set_nat_rat(groups2906978787729119204at_rat(F),I4)) ) ) ) ).

% sum_pos
tff(fact_8197_sum__pos,axiom,
    ! [I4: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I4)
     => ( ( I4 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I4)
             => 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,I4)) ) ) ) ).

% sum_pos
tff(fact_8198_sum__pos,axiom,
    ! [I4: set_o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,I4)
     => ( ( I4 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o((I2)),I4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(F,(I2))) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups8507830703676809646_o_nat(F,I4)) ) ) ) ).

% sum_pos
tff(fact_8199_sum__pos,axiom,
    ! [I4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I4)
     => ( ( I4 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I4)
             => 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,I4)) ) ) ) ).

% sum_pos
tff(fact_8200_sum__pos,axiom,
    ! [I4: set_o,F: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,I4)
     => ( ( I4 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o((I2)),I4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(F,(I2))) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups8505340233167759370_o_int(F,I4)) ) ) ) ).

% sum_pos
tff(fact_8201_sum__pos,axiom,
    ! [I4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,I4)
     => ( ( I4 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_set_nat_int(groups3539618377306564664at_int(F),I4)) ) ) ) ).

% sum_pos
tff(fact_8202_less__1__prod2,axiom,
    ! [I4: set_nat,I: nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I4)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I4)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I4)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(F,I2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),I4)) ) ) ) ) ).

% less_1_prod2
tff(fact_8203_less__1__prod2,axiom,
    ! [I4: set_int,I: int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,I4)
     => ( aa_set_int_o(member_int(I),I4)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I4)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(F,I2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3827104343326376752nteger(F,I4)) ) ) ) ) ).

% less_1_prod2
tff(fact_8204_less__1__prod2,axiom,
    ! [I4: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I4)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I4)
       => ( 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(aa_nat_fun_set_nat_o(member_nat,I2),I4)
               => 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),aa_set_nat_rat(groups73079841787564623at_rat(F),I4)) ) ) ) ) ).

% less_1_prod2
tff(fact_8205_less__1__prod2,axiom,
    ! [I4: set_int,I: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I4)
     => ( aa_set_int_o(member_int(I),I4)
       => ( 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),I4)
               => 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,I4)) ) ) ) ) ).

% less_1_prod2
tff(fact_8206_less__1__prod2,axiom,
    ! [I4: set_nat,I: nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,I4)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_set_nat_int(groups705719431365010083at_int(F),I4)) ) ) ) ) ).

% less_1_prod2
tff(fact_8207_less__1__prod2,axiom,
    ! [I4: set_int,I: int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,I4)
     => ( aa_set_int_o(member_int(I),I4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups1705073143266064639nt_int(F,I4)) ) ) ) ) ).

% less_1_prod2
tff(fact_8208_less__1__prod2,axiom,
    ! [I4: set_set_nat,I: set_nat,F: fun_se7586849273513198688nteger] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,I4)
     => ( aa_set_set_nat_o(member_set_nat(I),I4)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_set871585252387937313nteger(F,I))
         => ( ! [I2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(I2),I4)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_set871585252387937313nteger(F,I2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups2761809935038513290nteger(F,I4)) ) ) ) ) ).

% less_1_prod2
tff(fact_8209_less__1__prod2,axiom,
    ! [I4: set_set_nat,I: set_nat,F: fun_set_nat_rat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,I4)
     => ( aa_set_set_nat_o(member_set_nat(I),I4)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_set_nat_rat(F,I))
         => ( ! [I2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(I2),I4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_set_nat_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups3613417700093529605at_rat(F,I4)) ) ) ) ) ).

% less_1_prod2
tff(fact_8210_less__1__prod2,axiom,
    ! [I4: set_set_nat,I: set_nat,F: fun_set_nat_int] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,I4)
     => ( aa_set_set_nat_o(member_set_nat(I),I4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_set_nat_int(F,I))
         => ( ! [I2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(I2),I4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_set_nat_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups4246057289670975065at_int(F,I4)) ) ) ) ) ).

% less_1_prod2
tff(fact_8211_less__1__prod2,axiom,
    ! [I4: set_Pr1261947904930325089at_nat,I: product_prod_nat_nat,F: fun_Pr6026251430685297nteger] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,I4)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I),I4)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Pro4764841431850000930nteger(F,I))
         => ( ! [I2: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I2),I4)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Pro4764841431850000930nteger(F,I2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups1230400874837758585nteger(F,I4)) ) ) ) ) ).

% less_1_prod2
tff(fact_8212_less__1__prod,axiom,
    ! [I4: set_o,F: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,I4)
     => ( ( I4 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o((I2)),I4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_o_Code_integer(F,(I2))) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups7694694392188491536nteger(F,I4)) ) ) ) ).

% less_1_prod
tff(fact_8213_less__1__prod,axiom,
    ! [I4: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I4)
     => ( ( I4 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(F,I2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),I4)) ) ) ) ).

% less_1_prod
tff(fact_8214_less__1__prod,axiom,
    ! [I4: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,I4)
     => ( ( I4 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(F,I2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3827104343326376752nteger(F,I4)) ) ) ) ).

% less_1_prod
tff(fact_8215_less__1__prod,axiom,
    ! [I4: set_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,I4)
     => ( ( I4 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o((I2)),I4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_o_rat(F,(I2))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups2869687844427037835_o_rat(F,I4)) ) ) ) ).

% less_1_prod
tff(fact_8216_less__1__prod,axiom,
    ! [I4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I4)
     => ( ( I4 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I4)
             => 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),aa_set_nat_rat(groups73079841787564623at_rat(F),I4)) ) ) ) ).

% less_1_prod
tff(fact_8217_less__1__prod,axiom,
    ! [I4: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I4)
     => ( ( I4 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I4)
             => 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,I4)) ) ) ) ).

% less_1_prod
tff(fact_8218_less__1__prod,axiom,
    ! [I4: set_o,F: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,I4)
     => ( ( I4 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o((I2)),I4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_o_int(F,(I2))) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups3502327434004483295_o_int(F,I4)) ) ) ) ).

% less_1_prod
tff(fact_8219_less__1__prod,axiom,
    ! [I4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,I4)
     => ( ( I4 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(F,I2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_set_nat_int(groups705719431365010083at_int(F),I4)) ) ) ) ).

% less_1_prod
tff(fact_8220_less__1__prod,axiom,
    ! [I4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,I4)
     => ( ( I4 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(F,I2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups1705073143266064639nt_int(F,I4)) ) ) ) ).

% less_1_prod
tff(fact_8221_less__1__prod,axiom,
    ! [I4: set_set_nat,F: fun_se7586849273513198688nteger] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,I4)
     => ( ( I4 != bot_bot_set_set_nat )
       => ( ! [I2: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(I2),I4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_set871585252387937313nteger(F,I2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups2761809935038513290nteger(F,I4)) ) ) ) ).

% less_1_prod
tff(fact_8222_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ma(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% sum.lessThan_Suc_shift
tff(fact_8223_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_mb(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% sum.lessThan_Suc_shift
tff(fact_8224_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lw(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% sum.lessThan_Suc_shift
tff(fact_8225_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ob(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,M)) ) ).

% sum_lessThan_telescope'
tff(fact_8226_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_oc(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,M)) ) ).

% sum_lessThan_telescope'
tff(fact_8227_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ly(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,M)),aa_nat_rat(F,zero_zero_nat)) ) ).

% sum_lessThan_telescope
tff(fact_8228_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lz(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,zero_zero_nat)) ) ).

% sum_lessThan_telescope
tff(fact_8229_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_1_neg
tff(fact_8230_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% sgn_1_neg
tff(fact_8231_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% sgn_1_neg
tff(fact_8232_sgn__if,axiom,
    ! [X: int] :
      ( aa_int_int(sgn_sgn_int,X) = $ite(
          X = zero_zero_int,
          zero_zero_int,
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ) ).

% sgn_if
tff(fact_8233_sgn__if,axiom,
    ! [X: code_integer] :
      ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X) = $ite(
          X = zero_z3403309356797280102nteger,
          zero_z3403309356797280102nteger,
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ) ) ).

% sgn_if
tff(fact_8234_sgn__if,axiom,
    ! [X: rat] :
      ( aa_rat_rat(sgn_sgn_rat,X) = $ite(
          X = zero_zero_rat,
          zero_zero_rat,
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ) ).

% sgn_if
tff(fact_8235_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,zero_zero_nat)),aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_qn(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_8236_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_assn,N: nat] : ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,zero_zero_nat)),groups6906906614972039071t_assn(aTP_Lamm_qo(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_8237_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_ma(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_8238_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = 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_lw(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_8239_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_mb(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_8240_sum_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H2: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),minus_minus_set_int(T,S))
             => ( aa_int_Code_integer(H2,I2) = zero_z3403309356797280102nteger ) )
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),minus_minus_set_int(S,T))
               => ( aa_int_Code_integer(G,I2) = zero_z3403309356797280102nteger ) )
           => ( ! [X2: int] :
                  ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_Code_integer(G,X2) = aa_int_Code_integer(H2,X2) ) )
             => ( groups7873554091576472773nteger(G,S) = groups7873554091576472773nteger(H2,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8241_sum_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H2: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),minus_minus_set_int(T,S))
             => ( aa_int_rat(H2,I2) = zero_zero_rat ) )
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),minus_minus_set_int(S,T))
               => ( aa_int_rat(G,I2) = zero_zero_rat ) )
           => ( ! [X2: int] :
                  ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_rat(G,X2) = aa_int_rat(H2,X2) ) )
             => ( groups3906332499630173760nt_rat(G,S) = groups3906332499630173760nt_rat(H2,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8242_sum_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H2: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),minus_minus_set_int(T,S))
             => ( aa_int_nat(H2,I2) = zero_zero_nat ) )
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),minus_minus_set_int(S,T))
               => ( aa_int_nat(G,I2) = zero_zero_nat ) )
           => ( ! [X2: int] :
                  ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_nat(G,X2) = aa_int_nat(H2,X2) ) )
             => ( groups4541462559716669496nt_nat(G,S) = groups4541462559716669496nt_nat(H2,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8243_sum_Omono__neutral__cong,axiom,
    ! [T: set_nat,S: set_nat,H2: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(finite_finite_nat,S)
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_Code_integer(H2,I2) = zero_z3403309356797280102nteger ) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T))
               => ( aa_nat_Code_integer(G,I2) = zero_z3403309356797280102nteger ) )
           => ( ! [X2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T))
                 => ( aa_nat_Code_integer(G,X2) = aa_nat_Code_integer(H2,X2) ) )
             => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),S) = aa_set871585252387937313nteger(groups7501900531339628137nteger(H2),T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8244_sum_Omono__neutral__cong,axiom,
    ! [T: set_nat,S: set_nat,H2: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(finite_finite_nat,S)
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_rat(H2,I2) = zero_zero_rat ) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T))
               => ( aa_nat_rat(G,I2) = zero_zero_rat ) )
           => ( ! [X2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T))
                 => ( aa_nat_rat(G,X2) = aa_nat_rat(H2,X2) ) )
             => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),S) = aa_set_nat_rat(groups2906978787729119204at_rat(H2),T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8245_sum_Omono__neutral__cong,axiom,
    ! [T: set_nat,S: set_nat,H2: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(finite_finite_nat,S)
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_int(H2,I2) = zero_zero_int ) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T))
               => ( aa_nat_int(G,I2) = zero_zero_int ) )
           => ( ! [X2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T))
                 => ( aa_nat_int(G,X2) = aa_nat_int(H2,X2) ) )
             => ( aa_set_nat_int(groups3539618377306564664at_int(G),S) = aa_set_nat_int(groups3539618377306564664at_int(H2),T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8246_sum_Omono__neutral__cong,axiom,
    ! [T: set_nat,S: set_nat,H2: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(finite_finite_nat,S)
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_nat(H2,I2) = zero_zero_nat ) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T))
               => ( aa_nat_nat(G,I2) = zero_zero_nat ) )
           => ( ! [X2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T))
                 => ( aa_nat_nat(G,X2) = aa_nat_nat(H2,X2) ) )
             => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),S) = aa_set_nat_nat(groups3542108847815614940at_nat(H2),T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8247_sum_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H2: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),minus_minus_set_int(T,S))
             => ( aa_int_int(H2,I2) = zero_zero_int ) )
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),minus_minus_set_int(S,T))
               => ( aa_int_int(G,I2) = zero_zero_int ) )
           => ( ! [X2: int] :
                  ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_int(G,X2) = aa_int_int(H2,X2) ) )
             => ( groups4538972089207619220nt_int(G,S) = groups4538972089207619220nt_int(H2,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8248_sum_Omono__neutral__cong,axiom,
    ! [T: set_set_nat,S: set_set_nat,H2: fun_se7586849273513198688nteger,G: fun_se7586849273513198688nteger] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,T)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,S)
       => ( ! [I2: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(I2),minus_2163939370556025621et_nat(T,S))
             => ( aa_set871585252387937313nteger(H2,I2) = zero_z3403309356797280102nteger ) )
         => ( ! [I2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(I2),minus_2163939370556025621et_nat(S,T))
               => ( aa_set871585252387937313nteger(G,I2) = zero_z3403309356797280102nteger ) )
           => ( ! [X2: set_nat] :
                  ( aa_set_set_nat_o(member_set_nat(X2),inf_inf_set_set_nat(S,T))
                 => ( aa_set871585252387937313nteger(G,X2) = aa_set871585252387937313nteger(H2,X2) ) )
             => ( groups9190459664516455967nteger(G,S) = groups9190459664516455967nteger(H2,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8249_sum_Omono__neutral__cong,axiom,
    ! [T: set_set_nat,S: set_set_nat,H2: fun_set_nat_rat,G: fun_set_nat_rat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,T)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,S)
       => ( ! [I2: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(I2),minus_2163939370556025621et_nat(T,S))
             => ( aa_set_nat_rat(H2,I2) = zero_zero_rat ) )
         => ( ! [I2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(I2),minus_2163939370556025621et_nat(S,T))
               => ( aa_set_nat_rat(G,I2) = zero_zero_rat ) )
           => ( ! [X2: set_nat] :
                  ( aa_set_set_nat_o(member_set_nat(X2),inf_inf_set_set_nat(S,T))
                 => ( aa_set_nat_rat(G,X2) = aa_set_nat_rat(H2,X2) ) )
             => ( groups7659867448343625626at_rat(G,S) = groups7659867448343625626at_rat(H2,T) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8250_prod_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups3827104343326376752nteger(G,A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G,minus_minus_set_int(A2,B2))),groups3827104343326376752nteger(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8251_prod_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_assn] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups7882442080178216443t_assn(G,A2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(G,minus_minus_set_int(A2,B2))),groups7882442080178216443t_assn(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8252_prod_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups1072433553688619179nt_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G,minus_minus_set_int(A2,B2))),groups1072433553688619179nt_rat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8253_prod_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups1707563613775114915nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,minus_minus_set_int(A2,B2))),groups1707563613775114915nt_nat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8254_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8255_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( groups6906906614972039071t_assn(G,A2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),groups6906906614972039071t_assn(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8256_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( aa_set_nat_rat(groups73079841787564623at_rat(G),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),aa_set_nat_rat(groups73079841787564623at_rat(G),B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8257_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),aa_set_nat_nat(groups708209901874060359at_nat(G),B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8258_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( aa_set_nat_int(groups705719431365010083at_int(G),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),aa_set_nat_int(groups705719431365010083at_int(G),B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8259_prod_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups1705073143266064639nt_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,minus_minus_set_int(A2,B2))),groups1705073143266064639nt_int(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8260_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_int,S: set_int,G: fun_int_Code_integer,H2: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_Code_integer(G,X2) = one_one_Code_integer ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ( aa_int_Code_integer(G,X2) = aa_int_Code_integer(H2,X2) ) )
           => ( groups3827104343326376752nteger(G,T) = groups3827104343326376752nteger(H2,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8261_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_int,S: set_int,G: fun_int_assn,H2: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_assn(G,X2) = one_one_assn ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ( aa_int_assn(G,X2) = aa_int_assn(H2,X2) ) )
           => ( groups7882442080178216443t_assn(G,T) = groups7882442080178216443t_assn(H2,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8262_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_int,S: set_int,G: fun_int_rat,H2: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_rat(G,X2) = one_one_rat ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ( aa_int_rat(G,X2) = aa_int_rat(H2,X2) ) )
           => ( groups1072433553688619179nt_rat(G,T) = groups1072433553688619179nt_rat(H2,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8263_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_int,S: set_int,G: fun_int_nat,H2: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_nat(G,X2) = one_one_nat ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ( aa_int_nat(G,X2) = aa_int_nat(H2,X2) ) )
           => ( groups1707563613775114915nt_nat(G,T) = groups1707563613775114915nt_nat(H2,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8264_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_nat,S: set_nat,G: fun_nat_Code_integer,H2: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_Code_integer(G,X2) = one_one_Code_integer ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => ( aa_nat_Code_integer(G,X2) = aa_nat_Code_integer(H2,X2) ) )
           => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),T) = aa_set871585252387937313nteger(groups3455450783089532116nteger(H2),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8265_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_nat,S: set_nat,G: fun_nat_assn,H2: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_assn(G,X2) = one_one_assn ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => ( aa_nat_assn(G,X2) = aa_nat_assn(H2,X2) ) )
           => ( groups6906906614972039071t_assn(G,T) = groups6906906614972039071t_assn(H2,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8266_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_nat,S: set_nat,G: fun_nat_rat,H2: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_rat(G,X2) = one_one_rat ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => ( aa_nat_rat(G,X2) = aa_nat_rat(H2,X2) ) )
           => ( aa_set_nat_rat(groups73079841787564623at_rat(G),T) = aa_set_nat_rat(groups73079841787564623at_rat(H2),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8267_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_nat,S: set_nat,G: fun_nat_nat,H2: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_nat(G,X2) = one_one_nat ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => ( aa_nat_nat(G,X2) = aa_nat_nat(H2,X2) ) )
           => ( aa_set_nat_nat(groups708209901874060359at_nat(G),T) = aa_set_nat_nat(groups708209901874060359at_nat(H2),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8268_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_nat,S: set_nat,G: fun_nat_int,H2: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_int(G,X2) = one_one_int ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => ( aa_nat_int(G,X2) = aa_nat_int(H2,X2) ) )
           => ( aa_set_nat_int(groups705719431365010083at_int(G),T) = aa_set_nat_int(groups705719431365010083at_int(H2),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8269_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_int,S: set_int,G: fun_int_int,H2: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_int(G,X2) = one_one_int ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ( aa_int_int(G,X2) = aa_int_int(H2,X2) ) )
           => ( groups1705073143266064639nt_int(G,T) = groups1705073143266064639nt_int(H2,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8270_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_int,S: set_int,H2: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_Code_integer(H2,X2) = one_one_Code_integer ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ( aa_int_Code_integer(G,X2) = aa_int_Code_integer(H2,X2) ) )
           => ( groups3827104343326376752nteger(G,S) = groups3827104343326376752nteger(H2,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8271_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_int,S: set_int,H2: fun_int_assn,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_assn(H2,X2) = one_one_assn ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ( aa_int_assn(G,X2) = aa_int_assn(H2,X2) ) )
           => ( groups7882442080178216443t_assn(G,S) = groups7882442080178216443t_assn(H2,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8272_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_int,S: set_int,H2: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_rat(H2,X2) = one_one_rat ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ( aa_int_rat(G,X2) = aa_int_rat(H2,X2) ) )
           => ( groups1072433553688619179nt_rat(G,S) = groups1072433553688619179nt_rat(H2,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8273_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_int,S: set_int,H2: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_nat(H2,X2) = one_one_nat ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ( aa_int_nat(G,X2) = aa_int_nat(H2,X2) ) )
           => ( groups1707563613775114915nt_nat(G,S) = groups1707563613775114915nt_nat(H2,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8274_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_nat,S: set_nat,H2: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_Code_integer(H2,X2) = one_one_Code_integer ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => ( aa_nat_Code_integer(G,X2) = aa_nat_Code_integer(H2,X2) ) )
           => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),S) = aa_set871585252387937313nteger(groups3455450783089532116nteger(H2),T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8275_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_nat,S: set_nat,H2: fun_nat_assn,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_assn(H2,X2) = one_one_assn ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => ( aa_nat_assn(G,X2) = aa_nat_assn(H2,X2) ) )
           => ( groups6906906614972039071t_assn(G,S) = groups6906906614972039071t_assn(H2,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8276_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_nat,S: set_nat,H2: fun_nat_rat,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,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_rat(H2,X2) = one_one_rat ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => ( aa_nat_rat(G,X2) = aa_nat_rat(H2,X2) ) )
           => ( aa_set_nat_rat(groups73079841787564623at_rat(G),S) = aa_set_nat_rat(groups73079841787564623at_rat(H2),T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8277_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_nat,S: set_nat,H2: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_nat(H2,X2) = one_one_nat ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => ( aa_nat_nat(G,X2) = aa_nat_nat(H2,X2) ) )
           => ( aa_set_nat_nat(groups708209901874060359at_nat(G),S) = aa_set_nat_nat(groups708209901874060359at_nat(H2),T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8278_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_nat,S: set_nat,H2: fun_nat_int,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,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_int(H2,X2) = one_one_int ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
               => ( aa_nat_int(G,X2) = aa_nat_int(H2,X2) ) )
           => ( aa_set_nat_int(groups705719431365010083at_int(G),S) = aa_set_nat_int(groups705719431365010083at_int(H2),T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8279_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_int,S: set_int,H2: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_int(H2,X2) = one_one_int ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ( aa_int_int(G,X2) = aa_int_int(H2,X2) ) )
           => ( groups1705073143266064639nt_int(G,S) = groups1705073143266064639nt_int(H2,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8280_prod_Omono__neutral__right,axiom,
    ! [T: set_int,S: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_Code_integer(G,X2) = one_one_Code_integer ) )
         => ( groups3827104343326376752nteger(G,T) = groups3827104343326376752nteger(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8281_prod_Omono__neutral__right,axiom,
    ! [T: set_int,S: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_assn(G,X2) = one_one_assn ) )
         => ( groups7882442080178216443t_assn(G,T) = groups7882442080178216443t_assn(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8282_prod_Omono__neutral__right,axiom,
    ! [T: set_int,S: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_rat(G,X2) = one_one_rat ) )
         => ( groups1072433553688619179nt_rat(G,T) = groups1072433553688619179nt_rat(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8283_prod_Omono__neutral__right,axiom,
    ! [T: set_int,S: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_nat(G,X2) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(G,T) = groups1707563613775114915nt_nat(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8284_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_Code_integer(G,X2) = one_one_Code_integer ) )
         => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),T) = aa_set871585252387937313nteger(groups3455450783089532116nteger(G),S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8285_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_assn(G,X2) = one_one_assn ) )
         => ( groups6906906614972039071t_assn(G,T) = groups6906906614972039071t_assn(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8286_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S: 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,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_rat(G,X2) = one_one_rat ) )
         => ( aa_set_nat_rat(groups73079841787564623at_rat(G),T) = aa_set_nat_rat(groups73079841787564623at_rat(G),S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8287_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_nat(G,X2) = one_one_nat ) )
         => ( aa_set_nat_nat(groups708209901874060359at_nat(G),T) = aa_set_nat_nat(groups708209901874060359at_nat(G),S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8288_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S: 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,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_int(G,X2) = one_one_int ) )
         => ( aa_set_nat_int(groups705719431365010083at_int(G),T) = aa_set_nat_int(groups705719431365010083at_int(G),S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8289_prod_Omono__neutral__right,axiom,
    ! [T: set_int,S: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_int(G,X2) = one_one_int ) )
         => ( groups1705073143266064639nt_int(G,T) = groups1705073143266064639nt_int(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8290_prod_Omono__neutral__left,axiom,
    ! [T: set_int,S: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_Code_integer(G,X2) = one_one_Code_integer ) )
         => ( groups3827104343326376752nteger(G,S) = groups3827104343326376752nteger(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8291_prod_Omono__neutral__left,axiom,
    ! [T: set_int,S: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_assn(G,X2) = one_one_assn ) )
         => ( groups7882442080178216443t_assn(G,S) = groups7882442080178216443t_assn(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8292_prod_Omono__neutral__left,axiom,
    ! [T: set_int,S: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_rat(G,X2) = one_one_rat ) )
         => ( groups1072433553688619179nt_rat(G,S) = groups1072433553688619179nt_rat(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8293_prod_Omono__neutral__left,axiom,
    ! [T: set_int,S: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_nat(G,X2) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(G,S) = groups1707563613775114915nt_nat(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8294_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_Code_integer(G,X2) = one_one_Code_integer ) )
         => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),S) = aa_set871585252387937313nteger(groups3455450783089532116nteger(G),T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8295_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_assn(G,X2) = one_one_assn ) )
         => ( groups6906906614972039071t_assn(G,S) = groups6906906614972039071t_assn(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8296_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S: 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,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_rat(G,X2) = one_one_rat ) )
         => ( aa_set_nat_rat(groups73079841787564623at_rat(G),S) = aa_set_nat_rat(groups73079841787564623at_rat(G),T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8297_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_nat(G,X2) = one_one_nat ) )
         => ( aa_set_nat_nat(groups708209901874060359at_nat(G),S) = aa_set_nat_nat(groups708209901874060359at_nat(G),T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8298_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S: 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,S),T)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_int(G,X2) = one_one_int ) )
         => ( aa_set_nat_int(groups705719431365010083at_int(G),S) = aa_set_nat_int(groups705719431365010083at_int(G),T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8299_prod_Omono__neutral__left,axiom,
    ! [T: set_int,S: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T,S))
             => ( aa_int_int(G,X2) = one_one_int ) )
         => ( groups1705073143266064639nt_int(G,S) = groups1705073143266064639nt_int(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8300_prod_Osame__carrierI,axiom,
    ! [C2: set_int,A2: set_int,B2: set_int,G: fun_int_Code_integer,H2: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),C2)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),C2)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C2,A2))
               => ( aa_int_Code_integer(G,A4) = one_one_Code_integer ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C2,B2))
                 => ( aa_int_Code_integer(H2,B4) = one_one_Code_integer ) )
             => ( ( groups3827104343326376752nteger(G,C2) = groups3827104343326376752nteger(H2,C2) )
               => ( groups3827104343326376752nteger(G,A2) = groups3827104343326376752nteger(H2,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8301_prod_Osame__carrierI,axiom,
    ! [C2: set_int,A2: set_int,B2: set_int,G: fun_int_assn,H2: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),C2)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),C2)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C2,A2))
               => ( aa_int_assn(G,A4) = one_one_assn ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C2,B2))
                 => ( aa_int_assn(H2,B4) = one_one_assn ) )
             => ( ( groups7882442080178216443t_assn(G,C2) = groups7882442080178216443t_assn(H2,C2) )
               => ( groups7882442080178216443t_assn(G,A2) = groups7882442080178216443t_assn(H2,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8302_prod_Osame__carrierI,axiom,
    ! [C2: set_int,A2: set_int,B2: set_int,G: fun_int_rat,H2: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),C2)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),C2)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C2,A2))
               => ( aa_int_rat(G,A4) = one_one_rat ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C2,B2))
                 => ( aa_int_rat(H2,B4) = one_one_rat ) )
             => ( ( groups1072433553688619179nt_rat(G,C2) = groups1072433553688619179nt_rat(H2,C2) )
               => ( groups1072433553688619179nt_rat(G,A2) = groups1072433553688619179nt_rat(H2,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8303_prod_Osame__carrierI,axiom,
    ! [C2: set_int,A2: set_int,B2: set_int,G: fun_int_nat,H2: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),C2)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),C2)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C2,A2))
               => ( aa_int_nat(G,A4) = one_one_nat ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C2,B2))
                 => ( aa_int_nat(H2,B4) = one_one_nat ) )
             => ( ( groups1707563613775114915nt_nat(G,C2) = groups1707563613775114915nt_nat(H2,C2) )
               => ( groups1707563613775114915nt_nat(G,A2) = groups1707563613775114915nt_nat(H2,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8304_prod_Osame__carrierI,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_Code_integer,H2: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C2)
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A2))
               => ( aa_nat_Code_integer(G,A4) = one_one_Code_integer ) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),B2))
                 => ( aa_nat_Code_integer(H2,B4) = one_one_Code_integer ) )
             => ( ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),C2) = aa_set871585252387937313nteger(groups3455450783089532116nteger(H2),C2) )
               => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2) = aa_set871585252387937313nteger(groups3455450783089532116nteger(H2),B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8305_prod_Osame__carrierI,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_assn,H2: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C2)
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A2))
               => ( aa_nat_assn(G,A4) = one_one_assn ) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),B2))
                 => ( aa_nat_assn(H2,B4) = one_one_assn ) )
             => ( ( groups6906906614972039071t_assn(G,C2) = groups6906906614972039071t_assn(H2,C2) )
               => ( groups6906906614972039071t_assn(G,A2) = groups6906906614972039071t_assn(H2,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8306_prod_Osame__carrierI,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_rat,H2: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C2)
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A2))
               => ( aa_nat_rat(G,A4) = one_one_rat ) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),B2))
                 => ( aa_nat_rat(H2,B4) = one_one_rat ) )
             => ( ( aa_set_nat_rat(groups73079841787564623at_rat(G),C2) = aa_set_nat_rat(groups73079841787564623at_rat(H2),C2) )
               => ( aa_set_nat_rat(groups73079841787564623at_rat(G),A2) = aa_set_nat_rat(groups73079841787564623at_rat(H2),B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8307_prod_Osame__carrierI,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_nat,H2: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C2)
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A2))
               => ( aa_nat_nat(G,A4) = one_one_nat ) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),B2))
                 => ( aa_nat_nat(H2,B4) = one_one_nat ) )
             => ( ( aa_set_nat_nat(groups708209901874060359at_nat(G),C2) = aa_set_nat_nat(groups708209901874060359at_nat(H2),C2) )
               => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A2) = aa_set_nat_nat(groups708209901874060359at_nat(H2),B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8308_prod_Osame__carrierI,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_int,H2: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C2)
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A2))
               => ( aa_nat_int(G,A4) = one_one_int ) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),B2))
                 => ( aa_nat_int(H2,B4) = one_one_int ) )
             => ( ( aa_set_nat_int(groups705719431365010083at_int(G),C2) = aa_set_nat_int(groups705719431365010083at_int(H2),C2) )
               => ( aa_set_nat_int(groups705719431365010083at_int(G),A2) = aa_set_nat_int(groups705719431365010083at_int(H2),B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8309_prod_Osame__carrierI,axiom,
    ! [C2: set_int,A2: set_int,B2: set_int,G: fun_int_int,H2: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),C2)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),C2)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C2,A2))
               => ( aa_int_int(G,A4) = one_one_int ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C2,B2))
                 => ( aa_int_int(H2,B4) = one_one_int ) )
             => ( ( groups1705073143266064639nt_int(G,C2) = groups1705073143266064639nt_int(H2,C2) )
               => ( groups1705073143266064639nt_int(G,A2) = groups1705073143266064639nt_int(H2,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8310_prod_Osame__carrier,axiom,
    ! [C2: set_int,A2: set_int,B2: set_int,G: fun_int_Code_integer,H2: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),C2)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),C2)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C2,A2))
               => ( aa_int_Code_integer(G,A4) = one_one_Code_integer ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C2,B2))
                 => ( aa_int_Code_integer(H2,B4) = one_one_Code_integer ) )
             => ( ( groups3827104343326376752nteger(G,A2) = groups3827104343326376752nteger(H2,B2) )
              <=> ( groups3827104343326376752nteger(G,C2) = groups3827104343326376752nteger(H2,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8311_prod_Osame__carrier,axiom,
    ! [C2: set_int,A2: set_int,B2: set_int,G: fun_int_assn,H2: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),C2)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),C2)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C2,A2))
               => ( aa_int_assn(G,A4) = one_one_assn ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C2,B2))
                 => ( aa_int_assn(H2,B4) = one_one_assn ) )
             => ( ( groups7882442080178216443t_assn(G,A2) = groups7882442080178216443t_assn(H2,B2) )
              <=> ( groups7882442080178216443t_assn(G,C2) = groups7882442080178216443t_assn(H2,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8312_prod_Osame__carrier,axiom,
    ! [C2: set_int,A2: set_int,B2: set_int,G: fun_int_rat,H2: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),C2)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),C2)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C2,A2))
               => ( aa_int_rat(G,A4) = one_one_rat ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C2,B2))
                 => ( aa_int_rat(H2,B4) = one_one_rat ) )
             => ( ( groups1072433553688619179nt_rat(G,A2) = groups1072433553688619179nt_rat(H2,B2) )
              <=> ( groups1072433553688619179nt_rat(G,C2) = groups1072433553688619179nt_rat(H2,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8313_prod_Osame__carrier,axiom,
    ! [C2: set_int,A2: set_int,B2: set_int,G: fun_int_nat,H2: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),C2)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),C2)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C2,A2))
               => ( aa_int_nat(G,A4) = one_one_nat ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C2,B2))
                 => ( aa_int_nat(H2,B4) = one_one_nat ) )
             => ( ( groups1707563613775114915nt_nat(G,A2) = groups1707563613775114915nt_nat(H2,B2) )
              <=> ( groups1707563613775114915nt_nat(G,C2) = groups1707563613775114915nt_nat(H2,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8314_prod_Osame__carrier,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_Code_integer,H2: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C2)
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A2))
               => ( aa_nat_Code_integer(G,A4) = one_one_Code_integer ) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),B2))
                 => ( aa_nat_Code_integer(H2,B4) = one_one_Code_integer ) )
             => ( ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2) = aa_set871585252387937313nteger(groups3455450783089532116nteger(H2),B2) )
              <=> ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),C2) = aa_set871585252387937313nteger(groups3455450783089532116nteger(H2),C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8315_prod_Osame__carrier,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_assn,H2: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C2)
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A2))
               => ( aa_nat_assn(G,A4) = one_one_assn ) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),B2))
                 => ( aa_nat_assn(H2,B4) = one_one_assn ) )
             => ( ( groups6906906614972039071t_assn(G,A2) = groups6906906614972039071t_assn(H2,B2) )
              <=> ( groups6906906614972039071t_assn(G,C2) = groups6906906614972039071t_assn(H2,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8316_prod_Osame__carrier,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_rat,H2: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C2)
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A2))
               => ( aa_nat_rat(G,A4) = one_one_rat ) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),B2))
                 => ( aa_nat_rat(H2,B4) = one_one_rat ) )
             => ( ( aa_set_nat_rat(groups73079841787564623at_rat(G),A2) = aa_set_nat_rat(groups73079841787564623at_rat(H2),B2) )
              <=> ( aa_set_nat_rat(groups73079841787564623at_rat(G),C2) = aa_set_nat_rat(groups73079841787564623at_rat(H2),C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8317_prod_Osame__carrier,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_nat,H2: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C2)
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A2))
               => ( aa_nat_nat(G,A4) = one_one_nat ) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),B2))
                 => ( aa_nat_nat(H2,B4) = one_one_nat ) )
             => ( ( aa_set_nat_nat(groups708209901874060359at_nat(G),A2) = aa_set_nat_nat(groups708209901874060359at_nat(H2),B2) )
              <=> ( aa_set_nat_nat(groups708209901874060359at_nat(G),C2) = aa_set_nat_nat(groups708209901874060359at_nat(H2),C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8318_prod_Osame__carrier,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_int,H2: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C2)
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A2))
               => ( aa_nat_int(G,A4) = one_one_int ) )
           => ( ! [B4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),B2))
                 => ( aa_nat_int(H2,B4) = one_one_int ) )
             => ( ( aa_set_nat_int(groups705719431365010083at_int(G),A2) = aa_set_nat_int(groups705719431365010083at_int(H2),B2) )
              <=> ( aa_set_nat_int(groups705719431365010083at_int(G),C2) = aa_set_nat_int(groups705719431365010083at_int(H2),C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8319_prod_Osame__carrier,axiom,
    ! [C2: set_int,A2: set_int,B2: set_int,G: fun_int_int,H2: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),C2)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B2),C2)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C2,A2))
               => ( aa_int_int(G,A4) = one_one_int ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C2,B2))
                 => ( aa_int_int(H2,B4) = one_one_int ) )
             => ( ( groups1705073143266064639nt_int(G,A2) = groups1705073143266064639nt_int(H2,B2) )
              <=> ( groups1705073143266064639nt_int(G,C2) = groups1705073143266064639nt_int(H2,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8320_sum_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2))),groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G,A2)),groups3906332499630173760nt_rat(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8321_sum_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2))),groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,A2)),groups4541462559716669496nt_nat(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8322_sum_Ounion__inter,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),A2)),aa_set_nat_rat(groups2906978787729119204at_rat(G),B2)) ) ) ) ).

% sum.union_inter
tff(fact_8323_sum_Ounion__inter,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),A2)),aa_set_nat_int(groups3539618377306564664at_int(G),B2)) ) ) ) ).

% sum.union_inter
tff(fact_8324_sum_Ounion__inter,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B2)) ) ) ) ).

% sum.union_inter
tff(fact_8325_sum_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2))),groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A2)),groups4538972089207619220nt_int(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8326_sum_Ounion__inter,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2))),groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(G,A2)),groups342789780944988191at_rat(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8327_sum_Ounion__inter,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups977919841031483927at_nat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2))),groups977919841031483927at_nat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups977919841031483927at_nat(G,A2)),groups977919841031483927at_nat(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8328_sum_Ounion__inter,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G: fun_Pr7647638413817135296at_int] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set4746584241299694929at_int(groups975429370522433651at_int(G),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2))),aa_set4746584241299694929at_int(groups975429370522433651at_int(G),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set4746584241299694929at_int(groups975429370522433651at_int(G),A2)),aa_set4746584241299694929at_int(groups975429370522433651at_int(G),B2)) ) ) ) ).

% sum.union_inter
tff(fact_8329_sum_Ounion__inter,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,G: fun_Pr4065951289917319378at_rat] :
      ( finite1918287321285529104at_nat(A2)
     => ( finite1918287321285529104at_nat(B2)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6424767709179651461at_rat(G,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2))),groups6424767709179651461at_rat(G,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),B2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6424767709179651461at_rat(G,A2)),groups6424767709179651461at_rat(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8330_sum_OInt__Diff,axiom,
    ! [A2: set_int,G: fun_int_rat,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))),groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8331_sum_OInt__Diff,axiom,
    ! [A2: set_int,G: fun_int_nat,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))),groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8332_sum_OInt__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,G: fun_Pr8317260874932935276at_rat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( groups342789780944988191at_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))),groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2))) ) ) ).

% sum.Int_Diff
tff(fact_8333_sum_OInt__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,G: fun_Pr2602117396471556196at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( groups977919841031483927at_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups977919841031483927at_nat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))),groups977919841031483927at_nat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2))) ) ) ).

% sum.Int_Diff
tff(fact_8334_sum_OInt__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,G: fun_Pr7647638413817135296at_int,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set4746584241299694929at_int(groups975429370522433651at_int(G),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set4746584241299694929at_int(groups975429370522433651at_int(G),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))),aa_set4746584241299694929at_int(groups975429370522433651at_int(G),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2))) ) ) ).

% sum.Int_Diff
tff(fact_8335_sum_OInt__Diff,axiom,
    ! [A2: set_nat,G: fun_nat_rat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))) ) ) ).

% sum.Int_Diff
tff(fact_8336_sum_OInt__Diff,axiom,
    ! [A2: set_nat,G: fun_nat_int,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))) ) ) ).

% sum.Int_Diff
tff(fact_8337_sum_OInt__Diff,axiom,
    ! [A2: set_nat,G: fun_nat_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))) ) ) ).

% sum.Int_Diff
tff(fact_8338_sum_OInt__Diff,axiom,
    ! [A2: set_int,G: fun_int_int,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4538972089207619220nt_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))),groups4538972089207619220nt_int(G,minus_minus_set_int(A2,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8339_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lw(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_8340_prod_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2))),groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G,A2)),groups3827104343326376752nteger(G,B2)) ) ) ) ).

% prod.union_inter
tff(fact_8341_prod_Ounion__inter,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),B2)) ) ) ) ).

% prod.union_inter
tff(fact_8342_prod_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2))),groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(G,A2)),groups7882442080178216443t_assn(G,B2)) ) ) ) ).

% prod.union_inter
tff(fact_8343_prod_Ounion__inter,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))),groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,A2)),groups6906906614972039071t_assn(G,B2)) ) ) ) ).

% prod.union_inter
tff(fact_8344_prod_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2))),groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G,A2)),groups1072433553688619179nt_rat(G,B2)) ) ) ) ).

% prod.union_inter
tff(fact_8345_prod_Ounion__inter,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),A2)),aa_set_nat_rat(groups73079841787564623at_rat(G),B2)) ) ) ) ).

% prod.union_inter
tff(fact_8346_prod_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2))),groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,A2)),groups1707563613775114915nt_nat(G,B2)) ) ) ) ).

% prod.union_inter
tff(fact_8347_prod_Ounion__inter,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),A2)),aa_set_nat_nat(groups708209901874060359at_nat(G),B2)) ) ) ) ).

% prod.union_inter
tff(fact_8348_prod_Ounion__inter,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))),aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),A2)),aa_set_nat_int(groups705719431365010083at_int(G),B2)) ) ) ) ).

% prod.union_inter
tff(fact_8349_prod_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2))),groups1705073143266064639nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,A2)),groups1705073143266064639nt_int(G,B2)) ) ) ) ).

% prod.union_inter
tff(fact_8350_prod_OInt__Diff,axiom,
    ! [A2: set_int,G: fun_int_Code_integer,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3827104343326376752nteger(G,A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))),groups3827104343326376752nteger(G,minus_minus_set_int(A2,B2))) ) ) ).

% prod.Int_Diff
tff(fact_8351_prod_OInt__Diff,axiom,
    ! [A2: set_nat,G: fun_nat_Code_integer,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))) ) ) ).

% prod.Int_Diff
tff(fact_8352_prod_OInt__Diff,axiom,
    ! [A2: set_int,G: fun_int_assn,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7882442080178216443t_assn(G,A2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))),groups7882442080178216443t_assn(G,minus_minus_set_int(A2,B2))) ) ) ).

% prod.Int_Diff
tff(fact_8353_prod_OInt__Diff,axiom,
    ! [A2: set_nat,G: fun_nat_assn,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups6906906614972039071t_assn(G,A2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))),groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))) ) ) ).

% prod.Int_Diff
tff(fact_8354_prod_OInt__Diff,axiom,
    ! [A2: set_int,G: fun_int_rat,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1072433553688619179nt_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))),groups1072433553688619179nt_rat(G,minus_minus_set_int(A2,B2))) ) ) ).

% prod.Int_Diff
tff(fact_8355_prod_OInt__Diff,axiom,
    ! [A2: set_nat,G: fun_nat_rat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))) ) ) ).

% prod.Int_Diff
tff(fact_8356_prod_OInt__Diff,axiom,
    ! [A2: set_int,G: fun_int_nat,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1707563613775114915nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))),groups1707563613775114915nt_nat(G,minus_minus_set_int(A2,B2))) ) ) ).

% prod.Int_Diff
tff(fact_8357_prod_OInt__Diff,axiom,
    ! [A2: set_nat,G: fun_nat_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))) ) ) ).

% prod.Int_Diff
tff(fact_8358_prod_OInt__Diff,axiom,
    ! [A2: set_nat,G: fun_nat_int,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))),aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))) ) ) ).

% prod.Int_Diff
tff(fact_8359_prod_OInt__Diff,axiom,
    ! [A2: set_int,G: fun_int_int,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1705073143266064639nt_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))),groups1705073143266064639nt_int(G,minus_minus_set_int(A2,B2))) ) ) ).

% prod.Int_Diff
tff(fact_8360_prod_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H2: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),minus_minus_set_int(T,S))
             => ( aa_int_Code_integer(H2,I2) = one_one_Code_integer ) )
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),minus_minus_set_int(S,T))
               => ( aa_int_Code_integer(G,I2) = one_one_Code_integer ) )
           => ( ! [X2: int] :
                  ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_Code_integer(G,X2) = aa_int_Code_integer(H2,X2) ) )
             => ( groups3827104343326376752nteger(G,S) = groups3827104343326376752nteger(H2,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_8361_prod_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H2: fun_int_assn,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),minus_minus_set_int(T,S))
             => ( aa_int_assn(H2,I2) = one_one_assn ) )
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),minus_minus_set_int(S,T))
               => ( aa_int_assn(G,I2) = one_one_assn ) )
           => ( ! [X2: int] :
                  ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_assn(G,X2) = aa_int_assn(H2,X2) ) )
             => ( groups7882442080178216443t_assn(G,S) = groups7882442080178216443t_assn(H2,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_8362_prod_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H2: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),minus_minus_set_int(T,S))
             => ( aa_int_rat(H2,I2) = one_one_rat ) )
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),minus_minus_set_int(S,T))
               => ( aa_int_rat(G,I2) = one_one_rat ) )
           => ( ! [X2: int] :
                  ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_rat(G,X2) = aa_int_rat(H2,X2) ) )
             => ( groups1072433553688619179nt_rat(G,S) = groups1072433553688619179nt_rat(H2,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_8363_prod_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H2: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),minus_minus_set_int(T,S))
             => ( aa_int_nat(H2,I2) = one_one_nat ) )
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),minus_minus_set_int(S,T))
               => ( aa_int_nat(G,I2) = one_one_nat ) )
           => ( ! [X2: int] :
                  ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_nat(G,X2) = aa_int_nat(H2,X2) ) )
             => ( groups1707563613775114915nt_nat(G,S) = groups1707563613775114915nt_nat(H2,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_8364_prod_Omono__neutral__cong,axiom,
    ! [T: set_nat,S: set_nat,H2: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(finite_finite_nat,S)
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_Code_integer(H2,I2) = one_one_Code_integer ) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T))
               => ( aa_nat_Code_integer(G,I2) = one_one_Code_integer ) )
           => ( ! [X2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T))
                 => ( aa_nat_Code_integer(G,X2) = aa_nat_Code_integer(H2,X2) ) )
             => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),S) = aa_set871585252387937313nteger(groups3455450783089532116nteger(H2),T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_8365_prod_Omono__neutral__cong,axiom,
    ! [T: set_nat,S: set_nat,H2: fun_nat_assn,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(finite_finite_nat,S)
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_assn(H2,I2) = one_one_assn ) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T))
               => ( aa_nat_assn(G,I2) = one_one_assn ) )
           => ( ! [X2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T))
                 => ( aa_nat_assn(G,X2) = aa_nat_assn(H2,X2) ) )
             => ( groups6906906614972039071t_assn(G,S) = groups6906906614972039071t_assn(H2,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_8366_prod_Omono__neutral__cong,axiom,
    ! [T: set_nat,S: set_nat,H2: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(finite_finite_nat,S)
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_rat(H2,I2) = one_one_rat ) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T))
               => ( aa_nat_rat(G,I2) = one_one_rat ) )
           => ( ! [X2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T))
                 => ( aa_nat_rat(G,X2) = aa_nat_rat(H2,X2) ) )
             => ( aa_set_nat_rat(groups73079841787564623at_rat(G),S) = aa_set_nat_rat(groups73079841787564623at_rat(H2),T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_8367_prod_Omono__neutral__cong,axiom,
    ! [T: set_nat,S: set_nat,H2: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(finite_finite_nat,S)
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_nat(H2,I2) = one_one_nat ) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T))
               => ( aa_nat_nat(G,I2) = one_one_nat ) )
           => ( ! [X2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T))
                 => ( aa_nat_nat(G,X2) = aa_nat_nat(H2,X2) ) )
             => ( aa_set_nat_nat(groups708209901874060359at_nat(G),S) = aa_set_nat_nat(groups708209901874060359at_nat(H2),T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_8368_prod_Omono__neutral__cong,axiom,
    ! [T: set_nat,S: set_nat,H2: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(finite_finite_nat,S)
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T),S))
             => ( aa_nat_int(H2,I2) = one_one_int ) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T))
               => ( aa_nat_int(G,I2) = one_one_int ) )
           => ( ! [X2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T))
                 => ( aa_nat_int(G,X2) = aa_nat_int(H2,X2) ) )
             => ( aa_set_nat_int(groups705719431365010083at_int(G),S) = aa_set_nat_int(groups705719431365010083at_int(H2),T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_8369_prod_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H2: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),minus_minus_set_int(T,S))
             => ( aa_int_int(H2,I2) = one_one_int ) )
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),minus_minus_set_int(S,T))
               => ( aa_int_int(G,I2) = one_one_int ) )
           => ( ! [X2: int] :
                  ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T))
                 => ( aa_int_int(G,X2) = aa_int_int(H2,X2) ) )
             => ( groups1705073143266064639nt_int(G,S) = groups1705073143266064639nt_int(H2,T) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_8370_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_lw(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_8371_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_mb(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_8372_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lw(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_8373_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : ( aa_set_nat_int(groups705719431365010083at_int(semiri1314217659103216013at_int),aa_nat_set_nat(set_or1269000886237332187st_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_db,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_8374_sum_Onat__group,axiom,
    ! [G: fun_nat_nat,K: nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_abp(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K))) ) ).

% sum.nat_group
tff(fact_8375_prod_Onat__group,axiom,
    ! [G: fun_nat_nat,K: nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_abq(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K))) ) ).

% prod.nat_group
tff(fact_8376_prod_Onat__group,axiom,
    ! [G: fun_nat_int,K: nat,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_abr(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K))) ) ).

% prod.nat_group
tff(fact_8377_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_abs(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lu(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ).

% sum.nested_swap'
tff(fact_8378_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_abt(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qr(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ).

% prod.nested_swap'
tff(fact_8379_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_abu(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qt(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ).

% prod.nested_swap'
tff(fact_8380_Iio__Int__singleton,axiom,
    ! [K: $o,X: $o] :
      ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,set_ord_lessThan_o((K))),insert_o((X),bot_bot_set_o)) = $ite(aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),(K)),insert_o((X),bot_bot_set_o),bot_bot_set_o) ) ).

% Iio_Int_singleton
tff(fact_8381_Iio__Int__singleton,axiom,
    ! [K: literal,X: literal] :
      ( inf_inf_set_literal(set_or2436086697161274615iteral(K),insert_literal(X,bot_bot_set_literal)) = $ite(aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),K),insert_literal(X,bot_bot_set_literal),bot_bot_set_literal) ) ).

% Iio_Int_singleton
tff(fact_8382_Iio__Int__singleton,axiom,
    ! [K: rat,X: rat] :
      ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(inf_inf_set_rat,set_ord_lessThan_rat(K)),insert_rat(X,bot_bot_set_rat)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),K),insert_rat(X,bot_bot_set_rat),bot_bot_set_rat) ) ).

% Iio_Int_singleton
tff(fact_8383_Iio__Int__singleton,axiom,
    ! [K: int,X: int] :
      ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_ord_lessThan_int(K)),insert_int(X,bot_bot_set_int)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X),K),insert_int(X,bot_bot_set_int),bot_bot_set_int) ) ).

% Iio_Int_singleton
tff(fact_8384_Iio__Int__singleton,axiom,
    ! [K: nat,X: nat] :
      ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,K)),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),K),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat),bot_bot_set_nat) ) ).

% Iio_Int_singleton
tff(fact_8385_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: $o] : ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,set_ord_lessThan_o((U))),insert_o((U),bot_bot_set_o)) = set_ord_atMost_o((U)) ) ).

% ivl_disj_un_singleton(2)
tff(fact_8386_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: int] : ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,set_ord_lessThan_int(U)),insert_int(U,bot_bot_set_int)) = set_ord_atMost_int(U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_8387_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: nat] : ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,U)),aa_set_nat_set_nat(insert_nat(U),bot_bot_set_nat)) = aa_nat_set_nat(set_ord_atMost_nat,U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_8388_ivl__disj__un__one_I4_J,axiom,
    ! [L: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),U)
     => ( aa_set_rat_set_rat(aa_set3814230239291249893et_rat(sup_sup_set_rat,set_ord_lessThan_rat(L)),set_or633870826150836451st_rat(L,U)) = set_ord_atMost_rat(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_8389_ivl__disj__un__one_I4_J,axiom,
    ! [L: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),U)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,L)),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_set_nat(set_ord_atMost_nat,U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_8390_ivl__disj__un__one_I4_J,axiom,
    ! [L: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),U)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,set_ord_lessThan_int(L)),set_or1266510415728281911st_int(L,U)) = set_ord_atMost_int(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_8391_ivl__disj__un__one_I4_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),U)
     => ( sup_su848401254843788991nteger(set_or5754767410780653050nteger(L),set_or189985376899183464nteger(L,U)) = set_or9101266186257409494nteger(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_8392_sum_OIf__cases,axiom,
    ! [A2: set_int,P: fun_int_o,H2: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_abv(P),H2),G),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(H2,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P)))),groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),aa_set_int_set_int(uminus1532241313380277803et_int,collect_int(P))))) ) ) ).

% sum.If_cases
tff(fact_8393_sum_OIf__cases,axiom,
    ! [A2: set_int,P: fun_int_o,H2: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_abw(P),H2),G),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(H2,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P)))),groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),aa_set_int_set_int(uminus1532241313380277803et_int,collect_int(P))))) ) ) ).

% sum.If_cases
tff(fact_8394_sum_OIf__cases,axiom,
    ! [A2: set_nat,P: fun_nat_o,H2: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_abx(P),H2),G)),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(H2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P)))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,collect_nat(P))))) ) ) ).

% sum.If_cases
tff(fact_8395_sum_OIf__cases,axiom,
    ! [A2: set_nat,P: fun_nat_o,H2: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_aby(P),H2),G)),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(H2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P)))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,collect_nat(P))))) ) ) ).

% sum.If_cases
tff(fact_8396_sum_OIf__cases,axiom,
    ! [A2: set_nat,P: fun_nat_o,H2: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_abz(P),H2),G)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(H2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P)))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,collect_nat(P))))) ) ) ).

% sum.If_cases
tff(fact_8397_sum_OIf__cases,axiom,
    ! [A2: set_int,P: fun_int_o,H2: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_aca(P),H2),G),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(H2,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P)))),groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),aa_set_int_set_int(uminus1532241313380277803et_int,collect_int(P))))) ) ) ).

% sum.If_cases
tff(fact_8398_sum_OIf__cases,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_o,H2: fun_set_nat_rat,G: fun_set_nat_rat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups7659867448343625626at_rat(aa_fun8478617152544252148at_rat(aa_fun2377924995230810828at_rat(aTP_Lamm_acb(P),H2),G),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups7659867448343625626at_rat(H2,inf_inf_set_set_nat(A2,collect_set_nat(P)))),groups7659867448343625626at_rat(G,inf_inf_set_set_nat(A2,uminus613421341184616069et_nat(collect_set_nat(P))))) ) ) ).

% sum.If_cases
tff(fact_8399_sum_OIf__cases,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_o,H2: fun_set_nat_nat,G: fun_set_nat_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aa_fun8118849421014274228at_nat(aTP_Lamm_acc(P),H2),G),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8294997508430121362at_nat(H2,inf_inf_set_set_nat(A2,collect_set_nat(P)))),groups8294997508430121362at_nat(G,inf_inf_set_set_nat(A2,uminus613421341184616069et_nat(collect_set_nat(P))))) ) ) ).

% sum.If_cases
tff(fact_8400_sum_OIf__cases,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_o,H2: fun_set_nat_int,G: fun_set_nat_int] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups8292507037921071086at_int(aa_fun3600621996379952540at_int(aa_fun4894436714634582984at_int(aTP_Lamm_acd(P),H2),G),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups8292507037921071086at_int(H2,inf_inf_set_set_nat(A2,collect_set_nat(P)))),groups8292507037921071086at_int(G,inf_inf_set_set_nat(A2,uminus613421341184616069et_nat(collect_set_nat(P))))) ) ) ).

% sum.If_cases
tff(fact_8401_sum_OIf__cases,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: fun_Pr4796943543331071514_nat_o,H2: fun_Pr8317260874932935276at_rat,G: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( groups342789780944988191at_rat(aa_fun1488288443938188604at_rat(aa_fun126448503906362355at_rat(aTP_Lamm_ace(P),H2),G),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(H2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),collec3392354462482085612at_nat(P)))),groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,collec3392354462482085612at_nat(P))))) ) ) ).

% sum.If_cases
tff(fact_8402_prod_OIf__cases,axiom,
    ! [A2: set_int,P: fun_int_o,H2: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3827104343326376752nteger(aa_fun5304498221708291146nteger(aa_fun3938006231294128461nteger(aTP_Lamm_acf(P),H2),G),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(H2,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P)))),groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),aa_set_int_set_int(uminus1532241313380277803et_int,collect_int(P))))) ) ) ).

% prod.If_cases
tff(fact_8403_prod_OIf__cases,axiom,
    ! [A2: set_nat,P: fun_nat_o,H2: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_acg(P),H2),G)),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(H2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P)))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,collect_nat(P))))) ) ) ).

% prod.If_cases
tff(fact_8404_prod_OIf__cases,axiom,
    ! [A2: set_int,P: fun_int_o,H2: fun_int_assn,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aa_fun7795160494388424622t_assn(aTP_Lamm_ach(P),H2),G),A2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(H2,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P)))),groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),aa_set_int_set_int(uminus1532241313380277803et_int,collect_int(P))))) ) ) ).

% prod.If_cases
tff(fact_8405_prod_OIf__cases,axiom,
    ! [A2: set_nat,P: fun_nat_o,H2: fun_nat_assn,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_aci(P),H2),G),A2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(H2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P)))),groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,collect_nat(P))))) ) ) ).

% prod.If_cases
tff(fact_8406_prod_OIf__cases,axiom,
    ! [A2: set_int,P: fun_int_o,H2: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_abv(P),H2),G),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(H2,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P)))),groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),aa_set_int_set_int(uminus1532241313380277803et_int,collect_int(P))))) ) ) ).

% prod.If_cases
tff(fact_8407_prod_OIf__cases,axiom,
    ! [A2: set_nat,P: fun_nat_o,H2: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_abx(P),H2),G)),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(H2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P)))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,collect_nat(P))))) ) ) ).

% prod.If_cases
tff(fact_8408_prod_OIf__cases,axiom,
    ! [A2: set_int,P: fun_int_o,H2: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_abw(P),H2),G),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(H2,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P)))),groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),aa_set_int_set_int(uminus1532241313380277803et_int,collect_int(P))))) ) ) ).

% prod.If_cases
tff(fact_8409_prod_OIf__cases,axiom,
    ! [A2: set_nat,P: fun_nat_o,H2: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_abz(P),H2),G)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(H2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P)))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,collect_nat(P))))) ) ) ).

% prod.If_cases
tff(fact_8410_prod_OIf__cases,axiom,
    ! [A2: set_nat,P: fun_nat_o,H2: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_aby(P),H2),G)),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(H2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P)))),aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,collect_nat(P))))) ) ) ).

% prod.If_cases
tff(fact_8411_prod_OIf__cases,axiom,
    ! [A2: set_int,P: fun_int_o,H2: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_aca(P),H2),G),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(H2,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P)))),groups1705073143266064639nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),aa_set_int_set_int(uminus1532241313380277803et_int,collect_int(P))))) ) ) ).

% prod.If_cases
tff(fact_8412_one__diff__power__eq,axiom,
    ! [X: code_integer,N: nat] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),X)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% one_diff_power_eq
tff(fact_8413_one__diff__power__eq,axiom,
    ! [X: rat,N: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% one_diff_power_eq
tff(fact_8414_one__diff__power__eq,axiom,
    ! [X: int,N: nat] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% one_diff_power_eq
tff(fact_8415_power__diff__1__eq,axiom,
    ! [X: code_integer,N: nat] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),one_one_Code_integer)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% power_diff_1_eq
tff(fact_8416_power__diff__1__eq,axiom,
    ! [X: rat,N: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),N)),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% power_diff_1_eq
tff(fact_8417_power__diff__1__eq,axiom,
    ! [X: int,N: nat] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% power_diff_1_eq
tff(fact_8418_geometric__sum,axiom,
    ! [X: rat,N: nat] :
      ( ( X != one_one_rat )
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),N)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_8419_prod__mono__strict,axiom,
    ! [A2: set_o,F: fun_o_Code_integer,G: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o((I2)),A2)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(I2)))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_o_Code_integer(F,(I2))),aa_o_Code_integer(G,(I2))) ) )
       => ( ( A2 != bot_bot_set_o )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups7694694392188491536nteger(F,A2)),groups7694694392188491536nteger(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8420_prod__mono__strict,axiom,
    ! [A2: set_nat,F: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),A2)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I2))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,I2)),aa_nat_Code_integer(G,I2)) ) )
       => ( ( A2 != bot_bot_set_nat )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A2)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8421_prod__mono__strict,axiom,
    ! [A2: set_int,F: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),A2)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I2))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,I2)),aa_int_Code_integer(G,I2)) ) )
       => ( ( A2 != bot_bot_set_int )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups3827104343326376752nteger(F,A2)),groups3827104343326376752nteger(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8422_prod__mono__strict,axiom,
    ! [A2: set_o,F: fun_o_rat,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o((I2)),A2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(I2)))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(F,(I2))),aa_o_rat(G,(I2))) ) )
       => ( ( A2 != bot_bot_set_o )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2869687844427037835_o_rat(F,A2)),groups2869687844427037835_o_rat(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8423_prod__mono__strict,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),A2)
           => ( 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)) ) )
       => ( ( A2 != bot_bot_set_nat )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_set_nat_rat(groups73079841787564623at_rat(F),A2)),aa_set_nat_rat(groups73079841787564623at_rat(G),A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8424_prod__mono__strict,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),A2)
           => ( 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)) ) )
       => ( ( A2 != bot_bot_set_int )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1072433553688619179nt_rat(F,A2)),groups1072433553688619179nt_rat(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8425_prod__mono__strict,axiom,
    ! [A2: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o((I2)),A2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(I2)))
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(F,(I2))),aa_o_nat(G,(I2))) ) )
       => ( ( A2 != bot_bot_set_o )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups3504817904513533571_o_nat(F,A2)),groups3504817904513533571_o_nat(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8426_prod__mono__strict,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),A2)
           => ( 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,aa_int_nat(F,I2)),aa_int_nat(G,I2)) ) )
       => ( ( A2 != bot_bot_set_int )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8427_prod__mono__strict,axiom,
    ! [A2: set_o,F: fun_o_int,G: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o((I2)),A2)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(F,(I2)))
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(F,(I2))),aa_o_int(G,(I2))) ) )
       => ( ( A2 != bot_bot_set_o )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3502327434004483295_o_int(F,A2)),groups3502327434004483295_o_int(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8428_prod__mono__strict,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),A2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,I2))
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(G,I2)) ) )
       => ( ( A2 != bot_bot_set_nat )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),aa_set_nat_nat(groups708209901874060359at_nat(G),A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8429_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ma(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% sum.atMost_shift
tff(fact_8430_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,N: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_mb(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% sum.atMost_shift
tff(fact_8431_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = 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_lw(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% sum.atMost_shift
tff(fact_8432_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
             => ( aa_int_Code_integer(G,X2) = zero_z3403309356797280102nteger ) )
         => ( groups7873554091576472773nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,groups7873554091576472773nteger(G,A2)),groups7873554091576472773nteger(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8433_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
             => ( aa_int_rat(G,X2) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G,A2)),groups3906332499630173760nt_rat(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8434_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
             => ( aa_int_nat(G,X2) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,A2)),groups4541462559716669496nt_nat(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8435_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
             => ( aa_nat_Code_integer(G,X2) = zero_z3403309356797280102nteger ) )
         => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(G),A2)),aa_set871585252387937313nteger(groups7501900531339628137nteger(G),B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8436_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
             => ( aa_nat_rat(G,X2) = zero_zero_rat ) )
         => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),A2)),aa_set_nat_rat(groups2906978787729119204at_rat(G),B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8437_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
             => ( aa_nat_int(G,X2) = zero_zero_int ) )
         => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),A2)),aa_set_nat_int(groups3539618377306564664at_int(G),B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8438_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
             => ( aa_nat_nat(G,X2) = zero_zero_nat ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8439_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
             => ( aa_int_int(G,X2) = zero_zero_int ) )
         => ( groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A2)),groups4538972089207619220nt_int(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8440_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G: fun_Pr6026251430685297nteger] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( ! [X2: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
             => ( aa_Pro4764841431850000930nteger(G,X2) = zero_z3403309356797280102nteger ) )
         => ( groups196832835297161892nteger(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,groups196832835297161892nteger(G,A2)),groups196832835297161892nteger(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8441_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( ! [X2: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
             => ( aa_Pro3608043709957902237at_rat(G,X2) = zero_zero_rat ) )
         => ( groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(G,A2)),groups342789780944988191at_rat(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8442_sum_Oremove,axiom,
    ! [A2: set_o,X: $o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( groups7872700643590313910_o_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(G,(X))),groups7872700643590313910_o_rat(G,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ) ).

% sum.remove
tff(fact_8443_sum_Oremove,axiom,
    ! [A2: set_o,X: $o,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( groups8507830703676809646_o_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(G,(X))),groups8507830703676809646_o_nat(G,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ) ).

% sum.remove
tff(fact_8444_sum_Oremove,axiom,
    ! [A2: set_o,X: $o,G: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( groups8505340233167759370_o_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(G,(X))),groups8505340233167759370_o_int(G,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ) ).

% sum.remove
tff(fact_8445_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(X),A2)
       => ( groups3906332499630173760nt_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G,X)),groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_8446_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(X),A2)
       => ( groups4541462559716669496nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,X)),groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_8447_sum_Oremove,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,X)),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) ) ) ) ).

% sum.remove
tff(fact_8448_sum_Oremove,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2)
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,X)),aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) ) ) ) ).

% sum.remove
tff(fact_8449_sum_Oremove,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,X)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) ) ) ) ).

% sum.remove
tff(fact_8450_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(X),A2)
       => ( groups4538972089207619220nt_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(G,X)),groups4538972089207619220nt_int(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_8451_sum_Oremove,axiom,
    ! [A2: set_set_nat,X: set_nat,G: fun_set_nat_rat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(X),A2)
       => ( groups7659867448343625626at_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(G,X)),groups7659867448343625626at_rat(G,minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat)))) ) ) ) ).

% sum.remove
tff(fact_8452_sum_Oinsert__remove,axiom,
    ! [A2: set_o,G: fun_o_rat,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups7872700643590313910_o_rat(G,insert_o((X),A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(G,(X))),groups7872700643590313910_o_rat(G,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ).

% sum.insert_remove
tff(fact_8453_sum_Oinsert__remove,axiom,
    ! [A2: set_o,G: fun_o_nat,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups8507830703676809646_o_nat(G,insert_o((X),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(G,(X))),groups8507830703676809646_o_nat(G,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ).

% sum.insert_remove
tff(fact_8454_sum_Oinsert__remove,axiom,
    ! [A2: set_o,G: fun_o_int,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups8505340233167759370_o_int(G,insert_o((X),A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(G,(X))),groups8505340233167759370_o_int(G,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ).

% sum.insert_remove
tff(fact_8455_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_rat,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G,X)),groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_8456_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_nat,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,insert_int(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,X)),groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_8457_sum_Oinsert__remove,axiom,
    ! [A2: set_nat,G: fun_nat_rat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(insert_nat(X),A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,X)),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) ) ) ).

% sum.insert_remove
tff(fact_8458_sum_Oinsert__remove,axiom,
    ! [A2: set_nat,G: fun_nat_int,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(insert_nat(X),A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,X)),aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) ) ) ).

% sum.insert_remove
tff(fact_8459_sum_Oinsert__remove,axiom,
    ! [A2: set_nat,G: fun_nat_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(insert_nat(X),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,X)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) ) ) ).

% sum.insert_remove
tff(fact_8460_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4538972089207619220nt_int(G,insert_int(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(G,X)),groups4538972089207619220nt_int(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_8461_sum_Oinsert__remove,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,G: fun_Pr8317260874932935276at_rat,X: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( groups342789780944988191at_rat(G,insert8211810215607154385at_nat(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Pro3608043709957902237at_rat(G,X)),groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))) ) ) ).

% sum.insert_remove
tff(fact_8462_prod_OatMost__shift,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,zero_zero_nat)),aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_qn(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% prod.atMost_shift
tff(fact_8463_prod_OatMost__shift,axiom,
    ! [G: fun_nat_assn,N: nat] : ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,zero_zero_nat)),groups6906906614972039071t_assn(aTP_Lamm_qo(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% prod.atMost_shift
tff(fact_8464_prod_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_ma(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% prod.atMost_shift
tff(fact_8465_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = 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_lw(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% prod.atMost_shift
tff(fact_8466_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,N: nat] : ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_mb(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% prod.atMost_shift
tff(fact_8467_sum__diff1,axiom,
    ! [A2: set_o,F: fun_o_rat,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups7872700643590313910_o_rat(F,minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups7872700643590313910_o_rat(F,A2)),aa_o_rat(F,(A))),groups7872700643590313910_o_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_8468_sum__diff1,axiom,
    ! [A2: set_int,F: fun_int_rat,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups3906332499630173760nt_rat(F,A2)),aa_int_rat(F,A)),groups3906332499630173760nt_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_8469_sum__diff1,axiom,
    ! [A2: set_o,F: fun_o_int,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups8505340233167759370_o_int(F,minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups8505340233167759370_o_int(F,A2)),aa_o_int(F,(A))),groups8505340233167759370_o_int(F,A2)) ) ) ).

% sum_diff1
tff(fact_8470_sum__diff1,axiom,
    ! [A2: set_nat,F: fun_nat_rat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),A2)),aa_nat_rat(F,A)),aa_set_nat_rat(groups2906978787729119204at_rat(F),A2)) ) ) ).

% sum_diff1
tff(fact_8471_sum__diff1,axiom,
    ! [A2: set_nat,F: fun_nat_int,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(F),A2)),aa_nat_int(F,A)),aa_set_nat_int(groups3539618377306564664at_int(F),A2)) ) ) ).

% sum_diff1
tff(fact_8472_sum__diff1,axiom,
    ! [A2: set_int,F: fun_int_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),aa_int_int(F,A)),groups4538972089207619220nt_int(F,A2)) ) ) ).

% sum_diff1
tff(fact_8473_sum__diff1,axiom,
    ! [A2: set_set_nat,F: fun_set_nat_rat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups7659867448343625626at_rat(F,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = $ite(aa_set_set_nat_o(member_set_nat(A),A2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups7659867448343625626at_rat(F,A2)),aa_set_nat_rat(F,A)),groups7659867448343625626at_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_8474_sum__diff1,axiom,
    ! [A2: set_set_nat,F: fun_set_nat_int,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( groups8292507037921071086at_int(F,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = $ite(aa_set_set_nat_o(member_set_nat(A),A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups8292507037921071086at_int(F,A2)),aa_set_nat_int(F,A)),groups8292507037921071086at_int(F,A2)) ) ) ).

% sum_diff1
tff(fact_8475_sum__diff1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: fun_Pr8317260874932935276at_rat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups342789780944988191at_rat(F,A2)),aa_Pro3608043709957902237at_rat(F,A)),groups342789780944988191at_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_8476_sum__diff1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: fun_Pr7647638413817135296at_int,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set4746584241299694929at_int(groups975429370522433651at_int(F),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set4746584241299694929at_int(groups975429370522433651at_int(F),A2)),aa_Pro4240683299535347697at_int(F,A)),aa_set4746584241299694929at_int(groups975429370522433651at_int(F),A2)) ) ) ).

% sum_diff1
tff(fact_8477_sum__Un,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2))),groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ) ).

% sum_Un
tff(fact_8478_sum__Un,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(F,A2)),groups342789780944988191at_rat(F,B2))),groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) ) ) ) ).

% sum_Un
tff(fact_8479_sum__Un,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,F: fun_Pr4065951289917319378at_rat] :
      ( finite1918287321285529104at_nat(A2)
     => ( finite1918287321285529104at_nat(B2)
       => ( groups6424767709179651461at_rat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6424767709179651461at_rat(F,A2)),groups6424767709179651461at_rat(F,B2))),groups6424767709179651461at_rat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),B2))) ) ) ) ).

% sum_Un
tff(fact_8480_sum__Un,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,F: fun_Pr6968369006614801746at_rat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A2)
     => ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B2)
       => ( groups3225780264831618053at_rat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3225780264831618053at_rat(F,A2)),groups3225780264831618053at_rat(F,B2))),groups3225780264831618053at_rat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2))) ) ) ) ).

% sum_Un
tff(fact_8481_sum__Un,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),A2)),aa_set_nat_rat(groups2906978787729119204at_rat(F),B2))),aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ) ).

% sum_Un
tff(fact_8482_sum__Un,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F: fun_Pr7647638413817135296at_int] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( aa_set4746584241299694929at_int(groups975429370522433651at_int(F),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set4746584241299694929at_int(groups975429370522433651at_int(F),A2)),aa_set4746584241299694929at_int(groups975429370522433651at_int(F),B2))),aa_set4746584241299694929at_int(groups975429370522433651at_int(F),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) ) ) ) ).

% sum_Un
tff(fact_8483_sum__Un,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,F: fun_Pr3396328828801519398at_int] :
      ( finite1918287321285529104at_nat(A2)
     => ( finite1918287321285529104at_nat(B2)
       => ( groups7057407298757096921at_int(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7057407298757096921at_int(F,A2)),groups7057407298757096921at_int(F,B2))),groups7057407298757096921at_int(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),B2))) ) ) ) ).

% sum_Un
tff(fact_8484_sum__Un,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,F: fun_Pr6298746545499001766at_int] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A2)
     => ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B2)
       => ( groups3858419854409063513at_int(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3858419854409063513at_int(F,A2)),groups3858419854409063513at_int(F,B2))),groups3858419854409063513at_int(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2))) ) ) ) ).

% sum_Un
tff(fact_8485_sum__Un,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(F),A2)),aa_set_nat_int(groups3539618377306564664at_int(F),B2))),aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ) ).

% sum_Un
tff(fact_8486_sum__Un,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups4538972089207619220nt_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(F,B2))),groups4538972089207619220nt_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ) ).

% sum_Un
tff(fact_8487_sum_Ounion__disjoint,axiom,
    ! [A2: set_o,B2: set_o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,B2)
       => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o )
         => ( groups7872700643590313910_o_rat(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups7872700643590313910_o_rat(G,A2)),groups7872700643590313910_o_rat(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8488_sum_Ounion__disjoint,axiom,
    ! [A2: set_o,B2: set_o,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,B2)
       => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o )
         => ( groups8507830703676809646_o_nat(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8507830703676809646_o_nat(G,A2)),groups8507830703676809646_o_nat(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8489_sum_Ounion__disjoint,axiom,
    ! [A2: set_o,B2: set_o,G: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,B2)
       => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o )
         => ( groups8505340233167759370_o_int(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups8505340233167759370_o_int(G,A2)),groups8505340233167759370_o_int(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8490_sum_Ounion__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
         => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),A2)),aa_set_nat_rat(groups2906978787729119204at_rat(G),B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8491_sum_Ounion__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
         => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),A2)),aa_set_nat_int(groups3539618377306564664at_int(G),B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8492_sum_Ounion__disjoint,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
         => ( groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G,A2)),groups3906332499630173760nt_rat(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8493_sum_Ounion__disjoint,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
         => ( groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,A2)),groups4541462559716669496nt_nat(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8494_sum_Ounion__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8495_sum_Ounion__disjoint,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
         => ( groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A2)),groups4538972089207619220nt_int(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8496_sum_Ounion__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
         => ( groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(G,A2)),groups342789780944988191at_rat(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8497_prod_Oremove,axiom,
    ! [A2: set_o,X: $o,G: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( groups7694694392188491536nteger(G,A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(G,(X))),groups7694694392188491536nteger(G,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ) ).

% prod.remove
tff(fact_8498_prod_Oremove,axiom,
    ! [A2: set_int,X: int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(X),A2)
       => ( groups3827104343326376752nteger(G,A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(G,X)),groups3827104343326376752nteger(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_8499_prod_Oremove,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2)
       => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,X)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_8500_prod_Oremove,axiom,
    ! [A2: set_o,X: $o,G: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( groups5301882518646026715o_assn(G,A2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(G,(X))),groups5301882518646026715o_assn(G,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ) ).

% prod.remove
tff(fact_8501_prod_Oremove,axiom,
    ! [A2: set_int,X: int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(X),A2)
       => ( groups7882442080178216443t_assn(G,A2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(G,X)),groups7882442080178216443t_assn(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_8502_prod_Oremove,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2)
       => ( groups6906906614972039071t_assn(G,A2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,X)),groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_8503_prod_Oremove,axiom,
    ! [A2: set_o,X: $o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( groups2869687844427037835_o_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(G,(X))),groups2869687844427037835_o_rat(G,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ) ).

% prod.remove
tff(fact_8504_prod_Oremove,axiom,
    ! [A2: set_int,X: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(X),A2)
       => ( groups1072433553688619179nt_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G,X)),groups1072433553688619179nt_rat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_8505_prod_Oremove,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A2)
       => ( aa_set_nat_rat(groups73079841787564623at_rat(G),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,X)),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_8506_prod_Oremove,axiom,
    ! [A2: set_o,X: $o,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( groups3504817904513533571_o_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(G,(X))),groups3504817904513533571_o_nat(G,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ) ).

% prod.remove
tff(fact_8507_prod_Oinsert__remove,axiom,
    ! [A2: set_o,G: fun_o_Code_integer,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups7694694392188491536nteger(G,insert_o((X),A2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(G,(X))),groups7694694392188491536nteger(G,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ).

% prod.insert_remove
tff(fact_8508_prod_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_Code_integer,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3827104343326376752nteger(G,insert_int(X,A2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(G,X)),groups3827104343326376752nteger(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_8509_prod_Oinsert__remove,axiom,
    ! [A2: set_nat,G: fun_nat_Code_integer,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(insert_nat(X),A2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,X)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_8510_prod_Oinsert__remove,axiom,
    ! [A2: set_o,G: fun_o_assn,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups5301882518646026715o_assn(G,insert_o((X),A2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(G,(X))),groups5301882518646026715o_assn(G,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ).

% prod.insert_remove
tff(fact_8511_prod_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_assn,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7882442080178216443t_assn(G,insert_int(X,A2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(G,X)),groups7882442080178216443t_assn(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_8512_prod_Oinsert__remove,axiom,
    ! [A2: set_nat,G: fun_nat_assn,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups6906906614972039071t_assn(G,aa_set_nat_set_nat(insert_nat(X),A2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,X)),groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_8513_prod_Oinsert__remove,axiom,
    ! [A2: set_o,G: fun_o_rat,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups2869687844427037835_o_rat(G,insert_o((X),A2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(G,(X))),groups2869687844427037835_o_rat(G,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ).

% prod.insert_remove
tff(fact_8514_prod_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_rat,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1072433553688619179nt_rat(G,insert_int(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G,X)),groups1072433553688619179nt_rat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_8515_prod_Oinsert__remove,axiom,
    ! [A2: set_nat,G: fun_nat_rat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(insert_nat(X),A2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,X)),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_8516_prod_Oinsert__remove,axiom,
    ! [A2: set_o,G: fun_o_nat,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups3504817904513533571_o_nat(G,insert_o((X),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(G,(X))),groups3504817904513533571_o_nat(G,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ).

% prod.insert_remove
tff(fact_8517_prod_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
             => ( aa_int_Code_integer(G,X2) = one_one_Code_integer ) )
         => ( groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G,A2)),groups3827104343326376752nteger(G,B2)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_8518_prod_Ounion__inter__neutral,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
             => ( aa_nat_Code_integer(G,X2) = one_one_Code_integer ) )
         => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),B2)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_8519_prod_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
             => ( aa_int_assn(G,X2) = one_one_assn ) )
         => ( groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(G,A2)),groups7882442080178216443t_assn(G,B2)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_8520_prod_Ounion__inter__neutral,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
             => ( aa_nat_assn(G,X2) = one_one_assn ) )
         => ( groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,A2)),groups6906906614972039071t_assn(G,B2)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_8521_prod_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
             => ( aa_int_rat(G,X2) = one_one_rat ) )
         => ( groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G,A2)),groups1072433553688619179nt_rat(G,B2)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_8522_prod_Ounion__inter__neutral,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
             => ( aa_nat_rat(G,X2) = one_one_rat ) )
         => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),A2)),aa_set_nat_rat(groups73079841787564623at_rat(G),B2)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_8523_prod_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
             => ( aa_int_nat(G,X2) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,A2)),groups1707563613775114915nt_nat(G,B2)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_8524_prod_Ounion__inter__neutral,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
             => ( aa_nat_nat(G,X2) = one_one_nat ) )
         => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),A2)),aa_set_nat_nat(groups708209901874060359at_nat(G),B2)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_8525_prod_Ounion__inter__neutral,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
             => ( aa_nat_int(G,X2) = one_one_int ) )
         => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),A2)),aa_set_nat_int(groups705719431365010083at_int(G),B2)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_8526_prod_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
             => ( aa_int_int(G,X2) = one_one_int ) )
         => ( groups1705073143266064639nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,A2)),groups1705073143266064639nt_int(G,B2)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_8527_prod_Ounion__disjoint,axiom,
    ! [A2: set_o,B2: set_o,G: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,B2)
       => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o )
         => ( groups7694694392188491536nteger(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups7694694392188491536nteger(G,A2)),groups7694694392188491536nteger(G,B2)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_8528_prod_Ounion__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
         => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A2)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),B2)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_8529_prod_Ounion__disjoint,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
         => ( groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G,A2)),groups3827104343326376752nteger(G,B2)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_8530_prod_Ounion__disjoint,axiom,
    ! [A2: set_o,B2: set_o,G: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,B2)
       => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o )
         => ( groups5301882518646026715o_assn(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups5301882518646026715o_assn(G,A2)),groups5301882518646026715o_assn(G,B2)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_8531_prod_Ounion__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
         => ( groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,A2)),groups6906906614972039071t_assn(G,B2)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_8532_prod_Ounion__disjoint,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
         => ( groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(G,A2)),groups7882442080178216443t_assn(G,B2)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_8533_prod_Ounion__disjoint,axiom,
    ! [A2: set_o,B2: set_o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,B2)
       => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o )
         => ( groups2869687844427037835_o_rat(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2869687844427037835_o_rat(G,A2)),groups2869687844427037835_o_rat(G,B2)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_8534_prod_Ounion__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
         => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),A2)),aa_set_nat_rat(groups73079841787564623at_rat(G),B2)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_8535_prod_Ounion__disjoint,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
         => ( groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G,A2)),groups1072433553688619179nt_rat(G,B2)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_8536_prod_Ounion__disjoint,axiom,
    ! [A2: set_o,B2: set_o,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,B2)
       => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o )
         => ( groups3504817904513533571_o_nat(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3504817904513533571_o_nat(G,A2)),groups3504817904513533571_o_nat(G,B2)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_8537_sum_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,B2))),groups3906332499630173760nt_rat(G,minus_minus_set_int(B2,A2)))),groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8538_sum_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,B2))),groups4541462559716669496nt_nat(G,minus_minus_set_int(B2,A2)))),groups4541462559716669496nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8539_sum_Ounion__diff2,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8540_sum_Ounion__diff2,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8541_sum_Ounion__diff2,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8542_sum_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,minus_minus_set_int(A2,B2))),groups4538972089207619220nt_int(G,minus_minus_set_int(B2,A2)))),groups4538972089207619220nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8543_sum_Ounion__diff2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2))),groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),A2)))),groups342789780944988191at_rat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8544_sum_Ounion__diff2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( groups977919841031483927at_nat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups977919841031483927at_nat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2))),groups977919841031483927at_nat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),A2)))),groups977919841031483927at_nat(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8545_sum_Ounion__diff2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G: fun_Pr7647638413817135296at_int] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( aa_set4746584241299694929at_int(groups975429370522433651at_int(G),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set4746584241299694929at_int(groups975429370522433651at_int(G),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2))),aa_set4746584241299694929at_int(groups975429370522433651at_int(G),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),A2)))),aa_set4746584241299694929at_int(groups975429370522433651at_int(G),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8546_sum_Ounion__diff2,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,G: fun_Pr4065951289917319378at_rat] :
      ( finite1918287321285529104at_nat(A2)
     => ( finite1918287321285529104at_nat(B2)
       => ( groups6424767709179651461at_rat(G,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6424767709179651461at_rat(G,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),B2))),groups6424767709179651461at_rat(G,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,B2),A2)))),groups6424767709179651461at_rat(G,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8547_sum__Un2,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2))
     => ( groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,B2))),groups3906332499630173760nt_rat(F,minus_minus_set_int(B2,A2)))),groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ).

% sum_Un2
tff(fact_8548_sum__Un2,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2))
     => ( groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,B2))),groups4541462559716669496nt_nat(F,minus_minus_set_int(B2,A2)))),groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ).

% sum_Un2
tff(fact_8549_sum__Un2,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)))),aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ).

% sum_Un2
tff(fact_8550_sum__Un2,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
     => ( aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)))),aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ).

% sum_Un2
tff(fact_8551_sum__Un2,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ).

% sum_Un2
tff(fact_8552_sum__Un2,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2))
     => ( groups4538972089207619220nt_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(F,minus_minus_set_int(A2,B2))),groups4538972089207619220nt_int(F,minus_minus_set_int(B2,A2)))),groups4538972089207619220nt_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ).

% sum_Un2
tff(fact_8553_sum__Un2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2))
     => ( groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2))),groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),A2)))),groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) ) ) ).

% sum_Un2
tff(fact_8554_sum__Un2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2))
     => ( groups977919841031483927at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups977919841031483927at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2))),groups977919841031483927at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),A2)))),groups977919841031483927at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) ) ) ).

% sum_Un2
tff(fact_8555_sum__Un2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F: fun_Pr7647638413817135296at_int] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2))
     => ( aa_set4746584241299694929at_int(groups975429370522433651at_int(F),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set4746584241299694929at_int(groups975429370522433651at_int(F),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2))),aa_set4746584241299694929at_int(groups975429370522433651at_int(F),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),A2)))),aa_set4746584241299694929at_int(groups975429370522433651at_int(F),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) ) ) ).

% sum_Un2
tff(fact_8556_sum__Un2,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,F: fun_Pr4065951289917319378at_rat] :
      ( finite1918287321285529104at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2))
     => ( groups6424767709179651461at_rat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6424767709179651461at_rat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A2),B2))),groups6424767709179651461at_rat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,B2),A2)))),groups6424767709179651461at_rat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),B2))) ) ) ).

% sum_Un2
tff(fact_8557_prod_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G,minus_minus_set_int(A2,B2))),groups3827104343326376752nteger(G,minus_minus_set_int(B2,A2)))),groups3827104343326376752nteger(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ) ).

% prod.union_diff2
tff(fact_8558_prod_Ounion__diff2,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)))),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ) ).

% prod.union_diff2
tff(fact_8559_prod_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(G,minus_minus_set_int(A2,B2))),groups7882442080178216443t_assn(G,minus_minus_set_int(B2,A2)))),groups7882442080178216443t_assn(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ) ).

% prod.union_diff2
tff(fact_8560_prod_Ounion__diff2,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)))),groups6906906614972039071t_assn(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ) ).

% prod.union_diff2
tff(fact_8561_prod_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G,minus_minus_set_int(A2,B2))),groups1072433553688619179nt_rat(G,minus_minus_set_int(B2,A2)))),groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ) ).

% prod.union_diff2
tff(fact_8562_prod_Ounion__diff2,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)))),aa_set_nat_rat(groups73079841787564623at_rat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ) ).

% prod.union_diff2
tff(fact_8563_prod_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,minus_minus_set_int(A2,B2))),groups1707563613775114915nt_nat(G,minus_minus_set_int(B2,A2)))),groups1707563613775114915nt_nat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ) ).

% prod.union_diff2
tff(fact_8564_prod_Ounion__diff2,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ) ).

% prod.union_diff2
tff(fact_8565_prod_Ounion__diff2,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)))),aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ) ).

% prod.union_diff2
tff(fact_8566_prod_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups1705073143266064639nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,minus_minus_set_int(A2,B2))),groups1705073143266064639nt_int(G,minus_minus_set_int(B2,A2)))),groups1705073143266064639nt_int(G,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ) ).

% prod.union_diff2
tff(fact_8567_sum__Un__nat,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B2))),groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8568_sum__Un__nat,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( groups977919841031483927at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups977919841031483927at_nat(F,A2)),groups977919841031483927at_nat(F,B2))),groups977919841031483927at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8569_sum__Un__nat,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,F: fun_Pr7574179848310716106at_nat] :
      ( finite1918287321285529104at_nat(A2)
     => ( finite1918287321285529104at_nat(B2)
       => ( groups7059897769266147197at_nat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7059897769266147197at_nat(F,A2)),groups7059897769266147197at_nat(F,B2))),groups7059897769266147197at_nat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8570_sum__Un__nat,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,F: fun_Pr1253225528153422666at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A2)
     => ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B2)
       => ( groups3860910324918113789at_nat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups3860910324918113789at_nat(F,A2)),groups3860910324918113789at_nat(F,B2))),groups3860910324918113789at_nat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8571_sum__Un__nat,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B2))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8572_sum_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: fun_o_rat,C: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups7872700643590313910_o_rat(aa_fun4336163974892599420_o_rat(aa_fun2919298112881465802_o_rat(aTP_Lamm_acj((A)),B),C),S) = $ite(aa_set_o_o(member_o((A)),S),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(B,(A))),groups7872700643590313910_o_rat(C,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))),groups7872700643590313910_o_rat(C,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))) ) ) ).

% sum.delta_remove
tff(fact_8573_sum_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: fun_o_nat,C: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups8507830703676809646_o_nat(aa_fun6176012476734143100_o_nat(aa_fun6251207968994965954_o_nat(aTP_Lamm_ack((A)),B),C),S) = $ite(aa_set_o_o(member_o((A)),S),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(B,(A))),groups8507830703676809646_o_nat(C,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))),groups8507830703676809646_o_nat(C,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))) ) ) ).

% sum.delta_remove
tff(fact_8574_sum_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: fun_o_int,C: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups8505340233167759370_o_int(aa_fun4727766026823762812_o_int(aa_fun5648456818326528798_o_int(aTP_Lamm_acl((A)),B),C),S) = $ite(aa_set_o_o(member_o((A)),S),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(B,(A))),groups8505340233167759370_o_int(C,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))),groups8505340233167759370_o_int(C,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))) ) ) ).

% sum.delta_remove
tff(fact_8575_sum_Odelta__remove,axiom,
    ! [S: set_int,A: int,B: fun_int_rat,C: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_acm(A),B),C),S) = $ite(aa_set_int_o(member_int(A),S),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(B,A)),groups3906332499630173760nt_rat(C,minus_minus_set_int(S,insert_int(A,bot_bot_set_int)))),groups3906332499630173760nt_rat(C,minus_minus_set_int(S,insert_int(A,bot_bot_set_int)))) ) ) ).

% sum.delta_remove
tff(fact_8576_sum_Odelta__remove,axiom,
    ! [S: set_int,A: int,B: fun_int_nat,C: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_acn(A),B),C),S) = $ite(aa_set_int_o(member_int(A),S),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(B,A)),groups4541462559716669496nt_nat(C,minus_minus_set_int(S,insert_int(A,bot_bot_set_int)))),groups4541462559716669496nt_nat(C,minus_minus_set_int(S,insert_int(A,bot_bot_set_int)))) ) ) ).

% sum.delta_remove
tff(fact_8577_sum_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_rat,C: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_aco(A),B),C)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(B,A)),aa_set_nat_rat(groups2906978787729119204at_rat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))),aa_set_nat_rat(groups2906978787729119204at_rat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))) ) ) ).

% sum.delta_remove
tff(fact_8578_sum_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_int,C: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_acp(A),B),C)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(B,A)),aa_set_nat_int(groups3539618377306564664at_int(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))),aa_set_nat_int(groups3539618377306564664at_int(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))) ) ) ).

% sum.delta_remove
tff(fact_8579_sum_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_nat,C: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_acq(A),B),C)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(B,A)),aa_set_nat_nat(groups3542108847815614940at_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))),aa_set_nat_nat(groups3542108847815614940at_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))) ) ) ).

% sum.delta_remove
tff(fact_8580_sum_Odelta__remove,axiom,
    ! [S: set_int,A: int,B: fun_int_int,C: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_acr(A),B),C),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(B,A)),groups4538972089207619220nt_int(C,minus_minus_set_int(S,insert_int(A,bot_bot_set_int)))),groups4538972089207619220nt_int(C,minus_minus_set_int(S,insert_int(A,bot_bot_set_int)))) ) ) ).

% sum.delta_remove
tff(fact_8581_sum_Odelta__remove,axiom,
    ! [S: set_set_nat,A: set_nat,B: fun_set_nat_rat,C: fun_set_nat_rat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,S)
     => ( groups7659867448343625626at_rat(aa_fun8478617152544252148at_rat(aa_fun2377924995230810828at_rat(aTP_Lamm_acs(A),B),C),S) = $ite(aa_set_set_nat_o(member_set_nat(A),S),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(B,A)),groups7659867448343625626at_rat(C,minus_2163939370556025621et_nat(S,insert_set_nat(A,bot_bot_set_set_nat)))),groups7659867448343625626at_rat(C,minus_2163939370556025621et_nat(S,insert_set_nat(A,bot_bot_set_set_nat)))) ) ) ).

% sum.delta_remove
tff(fact_8582_sum__div__partition,axiom,
    ! [A2: set_int,F: fun_int_nat,B: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups4541462559716669496nt_nat(F,A2)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_act(F),B),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(aa_nat_fun_int_o(aTP_Lamm_acu(F),B))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(aa_nat_fun_int_o(aTP_Lamm_acv(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8583_sum__div__partition,axiom,
    ! [A2: set_nat,F: fun_nat_int,B: int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_set_nat_int(groups3539618377306564664at_int(F),A2)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_acw(F),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(aa_int_fun_nat_o(aTP_Lamm_acx(F),B))))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(aa_int_fun_nat_o(aTP_Lamm_acy(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8584_sum__div__partition,axiom,
    ! [A2: set_int,F: fun_int_Code_integer,B: code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(groups7873554091576472773nteger(F,A2)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,groups7873554091576472773nteger(aa_Cod1949805079358391713nteger(aTP_Lamm_acz(F),B),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(aa_Cod8477617969808203498_int_o(aTP_Lamm_ada(F),B))))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(groups7873554091576472773nteger(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(aa_Cod8477617969808203498_int_o(aTP_Lamm_adb(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8585_sum__div__partition,axiom,
    ! [A2: set_nat,F: fun_nat_Code_integer,B: code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A2)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_adc(F),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(aa_Cod1075288278386115654_nat_o(aTP_Lamm_add(F),B))))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_set871585252387937313nteger(groups7501900531339628137nteger(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(aa_Cod1075288278386115654_nat_o(aTP_Lamm_ade(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8586_sum__div__partition,axiom,
    ! [A2: set_int,F: fun_int_Code_natural,B: code_natural] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( divide5121882707175180666atural(groups6697149243333190288atural(F,A2),B) = plus_p4538020629002901425atural(groups6697149243333190288atural(aa_Cod5692187177200718263atural(aTP_Lamm_adf(F),B),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(aa_Cod7299285730463426143_int_o(aTP_Lamm_adg(F),B)))),divide5121882707175180666atural(groups6697149243333190288atural(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(aa_Cod7299285730463426143_int_o(aTP_Lamm_adh(F),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_8587_sum__div__partition,axiom,
    ! [A2: set_nat,F: fun_nat_Code_natural,B: code_natural] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( divide5121882707175180666atural(groups6325495683096345652atural(F,A2),B) = plus_p4538020629002901425atural(groups6325495683096345652atural(aa_Cod5320533616963873627atural(aTP_Lamm_adi(F),B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(aa_Cod9120328075896114107_nat_o(aTP_Lamm_adj(F),B)))),divide5121882707175180666atural(groups6325495683096345652atural(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(aa_Cod9120328075896114107_nat_o(aTP_Lamm_adk(F),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_8588_sum__div__partition,axiom,
    ! [A2: set_nat,F: fun_nat_nat,B: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_adl(F),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_adm(F),B))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_adn(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8589_sum__div__partition,axiom,
    ! [A2: set_int,F: fun_int_int,B: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,groups4538972089207619220nt_int(F,A2)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_ado(F),B),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(aa_int_fun_int_o(aTP_Lamm_adp(F),B))))),aa_int_int(aa_int_fun_int_int(divide_divide_int,groups4538972089207619220nt_int(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(aa_int_fun_int_o(aTP_Lamm_adq(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8590_sum__div__partition,axiom,
    ! [A2: set_set_nat,F: fun_set_nat_nat,B: nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups8294997508430121362at_nat(F,A2)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8294997508430121362at_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_adr(F),B),inf_inf_set_set_nat(A2,collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_ads(F),B))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups8294997508430121362at_nat(F,inf_inf_set_set_nat(A2,collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_adt(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8591_sum__div__partition,axiom,
    ! [A2: set_set_nat,F: fun_set_nat_int,B: int] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,groups8292507037921071086at_int(F,A2)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups8292507037921071086at_int(aa_int2448287375063835929at_int(aTP_Lamm_adu(F),B),inf_inf_set_set_nat(A2,collect_set_nat(aa_int_fun_set_nat_o(aTP_Lamm_adv(F),B))))),aa_int_int(aa_int_fun_int_int(divide_divide_int,groups8292507037921071086at_int(F,inf_inf_set_set_nat(A2,collect_set_nat(aa_int_fun_set_nat_o(aTP_Lamm_adw(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8592_prod_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: fun_o_Code_integer,C: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups7694694392188491536nteger(aa_fun8520278569782659900nteger(aa_fun5804704703283808911nteger(aTP_Lamm_adx((A)),B),C),S) = $ite(aa_set_o_o(member_o((A)),S),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(B,(A))),groups7694694392188491536nteger(C,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))),groups7694694392188491536nteger(C,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))) ) ) ).

% prod.delta_remove
tff(fact_8593_prod_Odelta__remove,axiom,
    ! [S: set_int,A: int,B: fun_int_Code_integer,C: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups3827104343326376752nteger(aa_fun5304498221708291146nteger(aa_fun3938006231294128461nteger(aTP_Lamm_ady(A),B),C),S) = $ite(aa_set_int_o(member_int(A),S),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(B,A)),groups3827104343326376752nteger(C,minus_minus_set_int(S,insert_int(A,bot_bot_set_int)))),groups3827104343326376752nteger(C,minus_minus_set_int(S,insert_int(A,bot_bot_set_int)))) ) ) ).

% prod.delta_remove
tff(fact_8594_prod_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_Code_integer,C: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_adz(A),B),C)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(B,A)),aa_set871585252387937313nteger(groups3455450783089532116nteger(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))),aa_set871585252387937313nteger(groups3455450783089532116nteger(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))) ) ) ).

% prod.delta_remove
tff(fact_8595_prod_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: fun_o_assn,C: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups5301882518646026715o_assn(aa_fun4536517337817438332o_assn(aa_fun3110273112593859354o_assn(aTP_Lamm_aea((A)),B),C),S) = $ite(aa_set_o_o(member_o((A)),S),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(B,(A))),groups5301882518646026715o_assn(C,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))),groups5301882518646026715o_assn(C,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))) ) ) ).

% prod.delta_remove
tff(fact_8596_prod_Odelta__remove,axiom,
    ! [S: set_int,A: int,B: fun_int_assn,C: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aa_fun7795160494388424622t_assn(aTP_Lamm_aeb(A),B),C),S) = $ite(aa_set_int_o(member_int(A),S),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(B,A)),groups7882442080178216443t_assn(C,minus_minus_set_int(S,insert_int(A,bot_bot_set_int)))),groups7882442080178216443t_assn(C,minus_minus_set_int(S,insert_int(A,bot_bot_set_int)))) ) ) ).

% prod.delta_remove
tff(fact_8597_prod_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_assn,C: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_aec(A),B),C),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(B,A)),groups6906906614972039071t_assn(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))),groups6906906614972039071t_assn(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))) ) ) ).

% prod.delta_remove
tff(fact_8598_prod_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: fun_o_rat,C: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups2869687844427037835_o_rat(aa_fun4336163974892599420_o_rat(aa_fun2919298112881465802_o_rat(aTP_Lamm_acj((A)),B),C),S) = $ite(aa_set_o_o(member_o((A)),S),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(B,(A))),groups2869687844427037835_o_rat(C,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))),groups2869687844427037835_o_rat(C,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))) ) ) ).

% prod.delta_remove
tff(fact_8599_prod_Odelta__remove,axiom,
    ! [S: set_int,A: int,B: fun_int_rat,C: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_acm(A),B),C),S) = $ite(aa_set_int_o(member_int(A),S),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(B,A)),groups1072433553688619179nt_rat(C,minus_minus_set_int(S,insert_int(A,bot_bot_set_int)))),groups1072433553688619179nt_rat(C,minus_minus_set_int(S,insert_int(A,bot_bot_set_int)))) ) ) ).

% prod.delta_remove
tff(fact_8600_prod_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_rat,C: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_aco(A),B),C)),S) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(B,A)),aa_set_nat_rat(groups73079841787564623at_rat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))),aa_set_nat_rat(groups73079841787564623at_rat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))) ) ) ).

% prod.delta_remove
tff(fact_8601_prod_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: fun_o_nat,C: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups3504817904513533571_o_nat(aa_fun6176012476734143100_o_nat(aa_fun6251207968994965954_o_nat(aTP_Lamm_ack((A)),B),C),S) = $ite(aa_set_o_o(member_o((A)),S),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(B,(A))),groups3504817904513533571_o_nat(C,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))),groups3504817904513533571_o_nat(C,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))) ) ) ).

% prod.delta_remove
tff(fact_8602_sum__gp__strict,axiom,
    ! [X: rat,N: nat] :
      ( aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = $ite(X = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,N),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),N))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X))) ) ).

% sum_gp_strict
tff(fact_8603_power__diff__sumr2,axiom,
    ! [X: code_integer,N: nat,Y: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),Y)),aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_aed(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% power_diff_sumr2
tff(fact_8604_power__diff__sumr2,axiom,
    ! [X: rat,N: nat,Y: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(Y),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)),aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_aee(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% power_diff_sumr2
tff(fact_8605_power__diff__sumr2,axiom,
    ! [X: int,N: nat,Y: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_aef(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% power_diff_sumr2
tff(fact_8606_diff__power__eq__sum,axiom,
    ! [X: code_integer,N: nat,Y: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_nat_nat(suc,N))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),Y)),aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_aeg(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ) ).

% diff_power_eq_sum
tff(fact_8607_diff__power__eq__sum,axiom,
    ! [X: rat,N: nat,Y: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)),aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_aeh(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ) ).

% diff_power_eq_sum
tff(fact_8608_diff__power__eq__sum,axiom,
    ! [X: int,N: nat,Y: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_nat_nat(suc,N))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_aei(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ) ).

% diff_power_eq_sum
tff(fact_8609_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q3: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L))
       => ( ( K = aa_int_int(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(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_8610_member__le__sum,axiom,
    ! [I: $o,A2: set_o,F: fun_o_Code_integer] :
      ( aa_set_o_o(member_o((I)),A2)
     => ( ! [X2: $o] :
            ( aa_set_o_o(member_o((X2)),minus_minus_set_o(A2,insert_o((I),bot_bot_set_o)))
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(X2))) )
       => ( aa_set_o_o(finite_finite_o,A2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(F,(I))),groups4406642042086082107nteger(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8611_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(member_int(I),A2)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),minus_minus_set_int(A2,insert_int(I,bot_bot_set_int)))
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X2)) )
       => ( aa_set_int_o(finite_finite_int,A2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,I)),groups7873554091576472773nteger(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8612_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),A2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)))
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X2)) )
       => ( aa_set_nat_o(finite_finite_nat,A2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,I)),aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A2)) ) ) ) ).

% member_le_sum
tff(fact_8613_member__le__sum,axiom,
    ! [I: $o,A2: set_o,F: fun_o_rat] :
      ( aa_set_o_o(member_o((I)),A2)
     => ( ! [X2: $o] :
            ( aa_set_o_o(member_o((X2)),minus_minus_set_o(A2,insert_o((I),bot_bot_set_o)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(X2))) )
       => ( aa_set_o_o(finite_finite_o,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(I))),groups7872700643590313910_o_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8614_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(member_int(I),A2)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),minus_minus_set_int(A2,insert_int(I,bot_bot_set_int)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2)) )
       => ( aa_set_int_o(finite_finite_int,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),groups3906332499630173760nt_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8615_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),A2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2)) )
       => ( aa_set_nat_o(finite_finite_nat,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),aa_set_nat_rat(groups2906978787729119204at_rat(F),A2)) ) ) ) ).

% member_le_sum
tff(fact_8616_member__le__sum,axiom,
    ! [I: $o,A2: set_o,F: fun_o_nat] :
      ( aa_set_o_o(member_o((I)),A2)
     => ( ! [X2: $o] :
            ( aa_set_o_o(member_o((X2)),minus_minus_set_o(A2,insert_o((I),bot_bot_set_o)))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(X2))) )
       => ( aa_set_o_o(finite_finite_o,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(I))),groups8507830703676809646_o_nat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8617_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(member_int(I),A2)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),minus_minus_set_int(A2,insert_int(I,bot_bot_set_int)))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X2)) )
       => ( aa_set_int_o(finite_finite_int,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I)),groups4541462559716669496nt_nat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8618_member__le__sum,axiom,
    ! [I: $o,A2: set_o,F: fun_o_int] :
      ( aa_set_o_o(member_o((I)),A2)
     => ( ! [X2: $o] :
            ( aa_set_o_o(member_o((X2)),minus_minus_set_o(A2,insert_o((I),bot_bot_set_o)))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(F,(X2))) )
       => ( aa_set_o_o(finite_finite_o,A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(I))),groups8505340233167759370_o_int(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8619_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),A2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X2)) )
       => ( aa_set_nat_o(finite_finite_nat,A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I)),aa_set_nat_int(groups3539618377306564664at_int(F),A2)) ) ) ) ).

% member_le_sum
tff(fact_8620_prod__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),B2)
       => ( ! [B4: int] :
              ( aa_set_int_o(member_int(B4),minus_minus_set_int(B2,A2))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(F,B4)) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),A2)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,A4)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3827104343326376752nteger(F,A2)),groups3827104343326376752nteger(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_8621_prod__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),B2)
       => ( ! [B4: int] :
              ( aa_set_int_o(member_int(B4),minus_minus_set_int(B2,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,B4)) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),A2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,A4)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A2)),groups1072433553688619179nt_rat(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_8622_prod__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
       => ( ! [B4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(F,B4)) )
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),A2)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,A4)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A2)),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),B2)) ) ) ) ) ).

% prod_mono2
tff(fact_8623_prod__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
       => ( ! [B4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,B4)) )
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),A2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,A4)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups73079841787564623at_rat(F),A2)),aa_set_nat_rat(groups73079841787564623at_rat(F),B2)) ) ) ) ) ).

% prod_mono2
tff(fact_8624_prod__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
       => ( ! [B4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(F,B4)) )
         => ( ! [A4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),A2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,A4)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups705719431365010083at_int(F),A2)),aa_set_nat_int(groups705719431365010083at_int(F),B2)) ) ) ) ) ).

% prod_mono2
tff(fact_8625_prod__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),B2)
       => ( ! [B4: int] :
              ( aa_set_int_o(member_int(B4),minus_minus_set_int(B2,A2))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(F,B4)) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),A2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,A4)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1705073143266064639nt_int(F,A2)),groups1705073143266064639nt_int(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_8626_prod__mono2,axiom,
    ! [B2: set_set_nat,A2: set_set_nat,F: fun_se7586849273513198688nteger] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
       => ( ! [B4: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(B4),minus_2163939370556025621et_nat(B2,A2))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_set871585252387937313nteger(F,B4)) )
         => ( ! [A4: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(A4),A2)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(F,A4)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups2761809935038513290nteger(F,A2)),groups2761809935038513290nteger(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_8627_prod__mono2,axiom,
    ! [B2: set_set_nat,A2: set_set_nat,F: fun_set_nat_rat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
       => ( ! [B4: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(B4),minus_2163939370556025621et_nat(B2,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_set_nat_rat(F,B4)) )
         => ( ! [A4: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(A4),A2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_set_nat_rat(F,A4)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3613417700093529605at_rat(F,A2)),groups3613417700093529605at_rat(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_8628_prod__mono2,axiom,
    ! [B2: set_set_nat,A2: set_set_nat,F: fun_set_nat_int] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
       => ( ! [B4: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(B4),minus_2163939370556025621et_nat(B2,A2))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_set_nat_int(F,B4)) )
         => ( ! [A4: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(A4),A2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_set_nat_int(F,A4)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4246057289670975065at_int(F,A2)),groups4246057289670975065at_int(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_8629_prod__mono2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,F: fun_Pr6026251430685297nteger] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),B2)
       => ( ! [B4: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),A2))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Pro4764841431850000930nteger(F,B4)) )
         => ( ! [A4: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A4),A2)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Pro4764841431850000930nteger(F,A4)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups1230400874837758585nteger(F,A2)),groups1230400874837758585nteger(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_8630_prod__diff1,axiom,
    ! [A2: set_o,F: fun_o_nat,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( aa_o_nat(F,(A)) != zero_zero_nat )
       => ( groups3504817904513533571_o_nat(F,minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups3504817904513533571_o_nat(F,A2)),aa_o_nat(F,(A))),groups3504817904513533571_o_nat(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8631_prod__diff1,axiom,
    ! [A2: set_int,F: fun_int_nat,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( aa_int_nat(F,A) != zero_zero_nat )
       => ( groups1707563613775114915nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups1707563613775114915nt_nat(F,A2)),aa_int_nat(F,A)),groups1707563613775114915nt_nat(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8632_prod__diff1,axiom,
    ! [A2: set_o,F: fun_o_int,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( aa_o_int(F,(A)) != zero_zero_int )
       => ( groups3502327434004483295_o_int(F,minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,groups3502327434004483295_o_int(F,A2)),aa_o_int(F,(A))),groups3502327434004483295_o_int(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8633_prod__diff1,axiom,
    ! [A2: set_o,F: fun_o_Code_integer,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( aa_o_Code_integer(F,(A)) != zero_z3403309356797280102nteger )
       => ( groups7694694392188491536nteger(F,minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(groups7694694392188491536nteger(F,A2)),aa_o_Code_integer(F,(A))),groups7694694392188491536nteger(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8634_prod__diff1,axiom,
    ! [A2: set_int,F: fun_int_Code_integer,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( aa_int_Code_integer(F,A) != zero_z3403309356797280102nteger )
       => ( groups3827104343326376752nteger(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(groups3827104343326376752nteger(F,A2)),aa_int_Code_integer(F,A)),groups3827104343326376752nteger(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8635_prod__diff1,axiom,
    ! [A2: set_nat,F: fun_nat_Code_integer,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_nat_Code_integer(F,A) != zero_z3403309356797280102nteger )
       => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A2)),aa_nat_Code_integer(F,A)),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A2)) ) ) ) ).

% prod_diff1
tff(fact_8636_prod__diff1,axiom,
    ! [A2: set_o,F: fun_o_rat,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( aa_o_rat(F,(A)) != zero_zero_rat )
       => ( groups2869687844427037835_o_rat(F,minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,groups2869687844427037835_o_rat(F,A2)),aa_o_rat(F,(A))),groups2869687844427037835_o_rat(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8637_prod__diff1,axiom,
    ! [A2: set_int,F: fun_int_rat,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( aa_int_rat(F,A) != zero_zero_rat )
       => ( groups1072433553688619179nt_rat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,groups1072433553688619179nt_rat(F,A2)),aa_int_rat(F,A)),groups1072433553688619179nt_rat(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8638_prod__diff1,axiom,
    ! [A2: set_nat,F: fun_nat_rat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_nat_rat(F,A) != zero_zero_rat )
       => ( aa_set_nat_rat(groups73079841787564623at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_set_nat_rat(groups73079841787564623at_rat(F),A2)),aa_nat_rat(F,A)),aa_set_nat_rat(groups73079841787564623at_rat(F),A2)) ) ) ) ).

% prod_diff1
tff(fact_8639_prod__diff1,axiom,
    ! [A2: set_o,F: fun_o_Code_natural,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( aa_o_Code_natural(F,(A)) != zero_z2226904508553997617atural )
       => ( groups6518289543945209051atural(F,minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A2),divide5121882707175180666atural(groups6518289543945209051atural(F,A2),aa_o_Code_natural(F,(A))),groups6518289543945209051atural(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8640_prod__Un,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
             => ( aa_int_rat(F,X2) != zero_zero_rat ) )
         => ( groups1072433553688619179nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(F,A2)),groups1072433553688619179nt_rat(F,B2))),groups1072433553688619179nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))) ) ) ) ) ).

% prod_Un
tff(fact_8641_prod__Un,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( ! [X2: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
             => ( aa_Pro3608043709957902237at_rat(F,X2) != zero_zero_rat ) )
         => ( groups3442636767675653108at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups3442636767675653108at_rat(F,A2)),groups3442636767675653108at_rat(F,B2))),groups3442636767675653108at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) ) ) ) ) ).

% prod_Un
tff(fact_8642_prod__Un,axiom,
    ! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,F: fun_Pr4065951289917319378at_rat] :
      ( finite1918287321285529104at_nat(A2)
     => ( finite1918287321285529104at_nat(B2)
       => ( ! [X2: produc4166570645942440679at_nat] :
              ( member6689249552917799696at_nat(X2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),B2))
             => ( aa_Pro6772482835110228483at_rat(F,X2) != zero_zero_rat ) )
         => ( groups5938585286922990810at_rat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups5938585286922990810at_rat(F,A2)),groups5938585286922990810at_rat(F,B2))),groups5938585286922990810at_rat(F,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A2),B2))) ) ) ) ) ).

% prod_Un
tff(fact_8643_prod__Un,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,F: fun_Pr6968369006614801746at_rat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A2)
     => ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B2)
       => ( ! [X2: produc3843707927480180839at_nat] :
              ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2))
             => ( aa_Pro5677984708781969027at_rat(F,X2) != zero_zero_rat ) )
         => ( groups8874911130973611098at_rat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups8874911130973611098at_rat(F,A2)),groups8874911130973611098at_rat(F,B2))),groups8874911130973611098at_rat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2))) ) ) ) ) ).

% prod_Un
tff(fact_8644_prod__Un,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
             => ( aa_nat_rat(F,X2) != zero_zero_rat ) )
         => ( aa_set_nat_rat(groups73079841787564623at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(F),A2)),aa_set_nat_rat(groups73079841787564623at_rat(F),B2))),aa_set_nat_rat(groups73079841787564623at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ) ) ).

% prod_Un
tff(fact_8645_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : ( aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,N)),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat)) ) ).

% atLeast1_lessThan_eq_remove0
tff(fact_8646_one__diff__power__eq_H,axiom,
    ! [X: code_integer,N: nat] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),X)),aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aTP_Lamm_aej(X),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% one_diff_power_eq'
tff(fact_8647_one__diff__power__eq_H,axiom,
    ! [X: rat,N: nat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_aek(X),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% one_diff_power_eq'
tff(fact_8648_one__diff__power__eq_H,axiom,
    ! [X: int,N: nat] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ael(X),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% one_diff_power_eq'
tff(fact_8649_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) ) )
       => ( ! [Q5: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q5),A22) ) ) )
         => ~ ! [R4: int,Q5: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R4) )
               => ( ( aa_int_int(sgn_sgn_int,R4) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R4)),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,Q5),A22)),R4) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_8650_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K4: int] :
            ( ( A1 = K4 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K4) ) )
        | ? [L4: int,K4: int,Q6: int] :
            ( ( A1 = K4 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q6),zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K4 = aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L4) ) )
        | ? [R5: int,L4: int,K4: int,Q6: int] :
            ( ( A1 = K4 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q6),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))
            & ( K4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_8651_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_8652_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : ( groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_od(N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_zy(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% prod.triangle_reindex_eq
tff(fact_8653_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_int,N: nat] : ( aa_set4746584241299694929at_int(groups4075276357253098568at_int(produc6840382203811409530at_int(G)),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_od(N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_aaa(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ).

% prod.triangle_reindex_eq
tff(fact_8654_drop__bit__rec,axiom,
    ! [N: nat,A: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),A) = $ite(N = zero_zero_nat,A,aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ).

% drop_bit_rec
tff(fact_8655_drop__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      ( aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(N),A) = $ite(N = zero_zero_nat,A,aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% drop_bit_rec
tff(fact_8656_drop__bit__rec,axiom,
    ! [N: nat,A: code_natural] :
      ( bit_se2751692689150723149atural(N,A) = $ite(N = zero_zero_nat,A,bit_se2751692689150723149atural(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))))) ) ).

% drop_bit_rec
tff(fact_8657_drop__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),A) = $ite(N = zero_zero_nat,A,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) ).

% drop_bit_rec
tff(fact_8658_finite__compl,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,A2))
      <=> aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,top_to4669805908274784177at_nat) ) ) ).

% finite_compl
tff(fact_8659_finite__compl,axiom,
    ! [A2: set_list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( finite8100373058378681591st_nat(aa_set1363491528093000170st_nat(uminus3195874150345416415st_nat,A2))
      <=> finite8100373058378681591st_nat(top_top_set_list_nat) ) ) ).

% finite_compl
tff(fact_8660_finite__compl,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,aa_set_o_set_o(uminus_uminus_set_o,A2))
      <=> aa_set_o_o(finite_finite_o,top_top_set_o) ) ) ).

% finite_compl
tff(fact_8661_finite__compl,axiom,
    ! [A2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(finite_finite_rat,aa_set_rat_set_rat(uminus2201863774496077783et_rat,A2))
      <=> aa_set_rat_o(finite_finite_rat,top_top_set_rat) ) ) ).

% finite_compl
tff(fact_8662_finite__compl,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2))
      <=> aa_set_nat_o(finite_finite_nat,top_top_set_nat) ) ) ).

% finite_compl
tff(fact_8663_finite__compl,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(uminus1532241313380277803et_int,A2))
      <=> aa_set_int_o(finite_finite_int,top_top_set_int) ) ) ).

% finite_compl
tff(fact_8664_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gf,K))) ).

% finite_Collect_le_nat
tff(fact_8665_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gg,K))) ).

% finite_Collect_less_nat
tff(fact_8666_finite__Collect__not,axiom,
    ! [P: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(P))
     => ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aTP_Lamm_az(P)))
      <=> aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,top_to6034159466715884489nt_int) ) ) ).

% finite_Collect_not
tff(fact_8667_finite__Collect__not,axiom,
    ! [P: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P))
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_ba(P)))
      <=> aa_set_set_nat_o(finite1152437895449049373et_nat,top_top_set_set_nat) ) ) ).

% finite_Collect_not
tff(fact_8668_finite__Collect__not,axiom,
    ! [P: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(P))
     => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(aTP_Lamm_bd(P)))
      <=> aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,top_to9044565752620676511_nat_o) ) ) ).

% finite_Collect_not
tff(fact_8669_finite__Collect__not,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P))
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_aem,P)))
      <=> aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,top_to4669805908274784177at_nat) ) ) ).

% finite_Collect_not
tff(fact_8670_finite__Collect__not,axiom,
    ! [P: fun_list_nat_o] :
      ( finite8100373058378681591st_nat(collect_list_nat(P))
     => ( finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_aen(P)))
      <=> finite8100373058378681591st_nat(top_top_set_list_nat) ) ) ).

% finite_Collect_not
tff(fact_8671_finite__Collect__not,axiom,
    ! [P: fun_o_o] :
      ( aa_set_o_o(finite_finite_o,collect_o(P))
     => ( aa_set_o_o(finite_finite_o,collect_o(aTP_Lamm_aeo(P)))
      <=> aa_set_o_o(finite_finite_o,top_top_set_o) ) ) ).

% finite_Collect_not
tff(fact_8672_finite__Collect__not,axiom,
    ! [P: fun_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,collect_rat(P))
     => ( aa_set_rat_o(finite_finite_rat,collect_rat(aTP_Lamm_aep(P)))
      <=> aa_set_rat_o(finite_finite_rat,top_top_set_rat) ) ) ).

% finite_Collect_not
tff(fact_8673_finite__Collect__not,axiom,
    ! [P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_bb(P)))
      <=> aa_set_nat_o(finite_finite_nat,top_top_set_nat) ) ) ).

% finite_Collect_not
tff(fact_8674_finite__Collect__not,axiom,
    ! [P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(P))
     => ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_bc(P)))
      <=> aa_set_int_o(finite_finite_int,top_top_set_int) ) ) ).

% finite_Collect_not
tff(fact_8675_finite__Collect__subsets,axiom,
    ! [A2: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(finite2998713641127702882nt_int,A2)
     => aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aa_set4993530631398190431_int_o(aTP_Lamm_nr,A2))) ) ).

% finite_Collect_subsets
tff(fact_8676_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite6197958912794628473et_int(collect_set_int(aa_set5425576183772390127_int_o(aTP_Lamm_fz,A2))) ) ).

% finite_Collect_subsets
tff(fact_8677_finite__Collect__subsets,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => finite9047747110432174090at_nat(collec5514110066124741708at_nat(aa_set7675215768510388063_nat_o(aTP_Lamm_fv,A2))) ) ).

% finite_Collect_subsets
tff(fact_8678_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_gd,A2))) ) ).

% finite_Collect_subsets
tff(fact_8679_finite__finite__vimage__IntI,axiom,
    ! [F3: set_o,H2: fun_int_o,A2: set_int] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( ! [Y2: $o] :
            ( aa_set_o_o(member_o((Y2)),F3)
           => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,vimage_int_o(H2,insert_o((Y2),bot_bot_set_o))),A2)) )
       => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,vimage_int_o(H2,F3)),A2)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_8680_finite__finite__vimage__IntI,axiom,
    ! [F3: set_o,H2: fun_nat_o,A2: set_nat] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( ! [Y2: $o] :
            ( aa_set_o_o(member_o((Y2)),F3)
           => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage_nat_o(H2,insert_o((Y2),bot_bot_set_o))),A2)) )
       => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage_nat_o(H2,F3)),A2)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_8681_finite__finite__vimage__IntI,axiom,
    ! [F3: set_nat,H2: fun_int_nat,A2: set_int] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( ! [Y2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y2),F3)
           => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,vimage_int_nat(H2,aa_set_nat_set_nat(insert_nat(Y2),bot_bot_set_nat))),A2)) )
       => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,vimage_int_nat(H2,F3)),A2)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_8682_finite__finite__vimage__IntI,axiom,
    ! [F3: set_nat,H2: fun_nat_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( ! [Y2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y2),F3)
           => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(vimage_nat_nat(H2),aa_set_nat_set_nat(insert_nat(Y2),bot_bot_set_nat))),A2)) )
       => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(vimage_nat_nat(H2),F3)),A2)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_8683_finite__finite__vimage__IntI,axiom,
    ! [F3: set_int,H2: fun_int_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( ! [Y2: int] :
            ( aa_set_int_o(member_int(Y2),F3)
           => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,vimage_int_int(H2,insert_int(Y2,bot_bot_set_int))),A2)) )
       => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,vimage_int_int(H2,F3)),A2)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_8684_finite__finite__vimage__IntI,axiom,
    ! [F3: set_int,H2: fun_nat_int,A2: set_nat] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( ! [Y2: int] :
            ( aa_set_int_o(member_int(Y2),F3)
           => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage_nat_int(H2,insert_int(Y2,bot_bot_set_int))),A2)) )
       => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage_nat_int(H2,F3)),A2)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_8685_finite__finite__vimage__IntI,axiom,
    ! [F3: set_set_nat,H2: fun_int_set_nat,A2: set_int] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( ! [Y2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(Y2),F3)
           => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,vimage_int_set_nat(H2,insert_set_nat(Y2,bot_bot_set_set_nat))),A2)) )
       => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,vimage_int_set_nat(H2,F3)),A2)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_8686_finite__finite__vimage__IntI,axiom,
    ! [F3: set_set_nat,H2: fun_nat_set_nat,A2: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( ! [Y2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(Y2),F3)
           => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage_nat_set_nat(H2,insert_set_nat(Y2,bot_bot_set_set_nat))),A2)) )
       => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage_nat_set_nat(H2,F3)),A2)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_8687_finite__finite__vimage__IntI,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,H2: fun_in4514472496968388910at_nat,A2: set_int] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => ( ! [Y2: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y2),F3)
           => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,vimage1827798767263827160at_nat(H2,insert8211810215607154385at_nat(Y2,bot_bo2099793752762293965at_nat))),A2)) )
       => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,vimage1827798767263827160at_nat(H2,F3)),A2)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_8688_finite__finite__vimage__IntI,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,H2: fun_na1476630412504255114at_nat,A2: set_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => ( ! [Y2: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y2),F3)
           => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage8013328719654469172at_nat(H2,insert8211810215607154385at_nat(Y2,bot_bo2099793752762293965at_nat))),A2)) )
       => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage8013328719654469172at_nat(H2,F3)),A2)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_8689_finite__Collect__conjI,axiom,
    ! [P: fun_se7249947736802449410_int_o,Q: fun_se7249947736802449410_int_o] :
      ( ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(P))
        | aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(Q)) )
     => aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_bu(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_8690_finite__Collect__conjI,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P))
        | aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Q)) )
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bv(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_8691_finite__Collect__conjI,axiom,
    ! [P: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] :
      ( ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(P))
        | aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(Q)) )
     => aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_bx(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_8692_finite__Collect__conjI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
        | 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_bz(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_8693_finite__Collect__conjI,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ( aa_set_int_o(finite_finite_int,collect_int(P))
        | 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_bw(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_8694_finite__Collect__conjI,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P))
        | aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Q)) )
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_by(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_8695_finite__Collect__disjI,axiom,
    ! [P: fun_se7249947736802449410_int_o,Q: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_cq(P),Q)))
    <=> ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(P))
        & aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_8696_finite__Collect__disjI,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_cr(P),Q)))
    <=> ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P))
        & aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_8697_finite__Collect__disjI,axiom,
    ! [P: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_ct(P),Q)))
    <=> ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(P))
        & aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_8698_finite__Collect__disjI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cw(P),Q)))
    <=> ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
        & aa_set_nat_o(finite_finite_nat,collect_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_8699_finite__Collect__disjI,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cs(P),Q)))
    <=> ( aa_set_int_o(finite_finite_int,collect_int(P))
        & aa_set_int_o(finite_finite_int,collect_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_8700_finite__Collect__disjI,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_aeq(P),Q)))
    <=> ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P))
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_8701_finite__interval__int1,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_aer(A),B))) ).

% finite_interval_int1
tff(fact_8702_finite__interval__int4,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_aes(A),B))) ).

% finite_interval_int4
tff(fact_8703_finite__Plus__UNIV__iff,axiom,
    ( finite6699802884135759036um_o_o(top_to1686961084667892491um_o_o)
  <=> ( aa_set_o_o(finite_finite_o,top_top_set_o)
      & aa_set_o_o(finite_finite_o,top_top_set_o) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_8704_finite__Plus__UNIV__iff,axiom,
    ( finite2301497163391418442_o_rat(top_to8915897456388626257_o_rat)
  <=> ( aa_set_o_o(finite_finite_o,top_top_set_o)
      & aa_set_rat_o(finite_finite_rat,top_top_set_rat) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_8705_finite__Plus__UNIV__iff,axiom,
    ( finite5809725721784815170_o_nat(top_to6072511757011528009_o_nat)
  <=> ( aa_set_o_o(finite_finite_o,top_top_set_o)
      & aa_set_nat_o(finite_finite_nat,top_top_set_nat) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_8706_finite__Plus__UNIV__iff,axiom,
    ( finite1631874702275618462_o_int(top_to3582428612079801253_o_int)
  <=> ( aa_set_o_o(finite_finite_o,top_top_set_o)
      & aa_set_int_o(finite_finite_int,top_top_set_int) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_8707_finite__Plus__UNIV__iff,axiom,
    ( finite5580927318523740128_rat_o(top_to6370680384054995943_rat_o)
  <=> ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
      & aa_set_o_o(finite_finite_o,top_top_set_o) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_8708_finite__Plus__UNIV__iff,axiom,
    ( finite2362923481445498406at_rat(top_to5104522503381003637at_rat)
  <=> ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
      & aa_set_rat_o(finite_finite_rat,top_top_set_rat) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_8709_finite__Plus__UNIV__iff,axiom,
    ( finite5871152039838895134at_nat(top_to2261136804003905389at_nat)
  <=> ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
      & aa_set_nat_o(finite_finite_nat,top_top_set_nat) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_8710_finite__Plus__UNIV__iff,axiom,
    ( finite1693301020329698426at_int(top_to8994425695926954441at_int)
  <=> ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
      & aa_set_int_o(finite_finite_int,top_top_set_int) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_8711_finite__Plus__UNIV__iff,axiom,
    ( finite94888208985532392_nat_o(top_to7120114879189831663_nat_o)
  <=> ( aa_set_nat_o(finite_finite_nat,top_top_set_nat)
      & aa_set_o_o(finite_finite_o,top_top_set_o) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_8712_finite__Plus__UNIV__iff,axiom,
    ( finite2679478125380364318at_rat(top_to281834657035230061at_rat)
  <=> ( aa_set_nat_o(finite_finite_nat,top_top_set_nat)
      & aa_set_rat_o(finite_finite_rat,top_top_set_rat) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_8713_finite__Int,axiom,
    ! [F3: set_int,G2: set_int] :
      ( ( aa_set_int_o(finite_finite_int,F3)
        | aa_set_int_o(finite_finite_int,G2) )
     => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,F3),G2)) ) ).

% finite_Int
tff(fact_8714_finite__Int,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
        | aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,G2) )
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,F3),G2)) ) ).

% finite_Int
tff(fact_8715_finite__Int,axiom,
    ! [F3: set_nat,G2: set_nat] :
      ( ( aa_set_nat_o(finite_finite_nat,F3)
        | aa_set_nat_o(finite_finite_nat,G2) )
     => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,F3),G2)) ) ).

% finite_Int
tff(fact_8716_finite__Un,axiom,
    ! [F3: set_int,G2: set_int] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,F3),G2))
    <=> ( aa_set_int_o(finite_finite_int,F3)
        & aa_set_int_o(finite_finite_int,G2) ) ) ).

% finite_Un
tff(fact_8717_finite__Un,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,F3),G2))
    <=> ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,G2) ) ) ).

% finite_Un
tff(fact_8718_finite__Un,axiom,
    ! [F3: set_Pr8551490117392284871at_nat,G2: set_Pr8551490117392284871at_nat] :
      ( finite1918287321285529104at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,F3),G2))
    <=> ( finite1918287321285529104at_nat(F3)
        & finite1918287321285529104at_nat(G2) ) ) ).

% finite_Un
tff(fact_8719_finite__Un,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,F3),G2))
    <=> ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
        & aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,G2) ) ) ).

% finite_Un
tff(fact_8720_finite__Un,axiom,
    ! [F3: set_nat,G2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,F3),G2))
    <=> ( aa_set_nat_o(finite_finite_nat,F3)
        & aa_set_nat_o(finite_finite_nat,G2) ) ) ).

% finite_Un
tff(fact_8721_finite__interval__int2,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_aet(A),B))) ).

% finite_interval_int2
tff(fact_8722_finite__interval__int3,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_aeu(A),B))) ).

% finite_interval_int3
tff(fact_8723_infinite__UNIV__int,axiom,
    ~ aa_set_int_o(finite_finite_int,top_top_set_int) ).

% infinite_UNIV_int
tff(fact_8724_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_id,I))) ) ).

% finite_divisors_int
tff(fact_8725_infinite__UNIV__nat,axiom,
    ~ aa_set_nat_o(finite_finite_nat,top_top_set_nat) ).

% infinite_UNIV_nat
tff(fact_8726_not__finite__existsD,axiom,
    ! [P: fun_se7249947736802449410_int_o] :
      ( ~ aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(P))
     => ? [X_1: set_Pr958786334691620121nt_int] : aa_set6170482117830564987_int_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_8727_not__finite__existsD,axiom,
    ! [P: fun_set_nat_o] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P))
     => ? [X_1: set_nat] : aa_set_nat_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_8728_not__finite__existsD,axiom,
    ! [P: fun_fu5135969500543783852at_o_o] :
      ( ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(P))
     => ? [X_1: fun_Pr6460944699146269935_nat_o] : aa_fun1675921005893230629at_o_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_8729_not__finite__existsD,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ? [X_1: nat] : aa_nat_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_8730_not__finite__existsD,axiom,
    ! [P: fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,collect_int(P))
     => ? [X_1: int] : aa_int_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_8731_not__finite__existsD,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P))
     => ? [X_1: product_prod_nat_nat] : aa_Pro6150637604445964947_nat_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_8732_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_nat,R: fun_nat_fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
             => ? [Xa3: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),B2)
                  & aa_nat_o(aa_nat_fun_nat_o(R,X2),Xa3) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),B2)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_vm(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8733_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_int,R: fun_nat_fun_int_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
             => ? [Xa3: int] :
                  ( aa_set_int_o(member_int(Xa3),B2)
                  & aa_int_o(aa_nat_fun_int_o(R,X2),Xa3) ) )
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),B2)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_vf(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8734_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_nat,R: fun_int_fun_nat_o] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A2)
             => ? [Xa3: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),B2)
                  & aa_nat_o(aa_int_fun_nat_o(R,X2),Xa3) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),B2)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_vb(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8735_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_int,R: fun_int_fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A2)
             => ? [Xa3: int] :
                  ( aa_set_int_o(member_int(Xa3),B2)
                  & aa_int_o(aa_int_fun_int_o(R,X2),Xa3) ) )
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),B2)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_vs(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8736_pigeonhole__infinite__rel,axiom,
    ! [A2: set_set_nat,B2: set_nat,R: fun_se382663106265794036_nat_o] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(X2),A2)
             => ? [Xa3: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),B2)
                  & aa_nat_o(aa_set_nat_fun_nat_o(R,X2),Xa3) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),B2)
              & ~ aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_nat_fun_set_nat_o(aa_fun9105140013458882876_nat_o(aTP_Lamm_vx(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8737_pigeonhole__infinite__rel,axiom,
    ! [A2: set_set_nat,B2: set_int,R: fun_se7360890175450123928_int_o] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(X2),A2)
             => ? [Xa3: int] :
                  ( aa_set_int_o(member_int(Xa3),B2)
                  & aa_int_o(aa_set_nat_fun_int_o(R,X2),Xa3) ) )
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),B2)
              & ~ aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_int_fun_set_nat_o(aa_fun9086678093774146108_nat_o(aTP_Lamm_wc(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8738_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_Pr1261947904930325089at_nat,R: fun_na3970208099681795815_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
             => ? [Xa3: product_prod_nat_nat] :
                  ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa3),B2)
                  & aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(R,X2),Xa3) ) )
         => ? [X2: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),B2)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_Pro4733806262894120092_nat_o(aa_fun5389978208584994248_nat_o(aTP_Lamm_aev(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8739_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_Pr1261947904930325089at_nat,R: fun_in2375804543815556931_nat_o] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A2)
             => ? [Xa3: product_prod_nat_nat] :
                  ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa3),B2)
                  & aa_Pro6150637604445964947_nat_o(aa_int6101145382336329604_nat_o(R,X2),Xa3) ) )
         => ? [X2: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),B2)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_Pro2912763917461432128_int_o(aa_fun8623471109766342856_int_o(aTP_Lamm_aew(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8740_pigeonhole__infinite__rel,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_nat,R: fun_Pr6984107991879650211_nat_o] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),A2)
             => ? [Xa3: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),B2)
                  & aa_nat_o(aa_Pro4733806262894120092_nat_o(R,X2),Xa3) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),B2)
              & ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_nat7695548938202568488_nat_o(aa_fun2489310642977492936_nat_o(aTP_Lamm_aex(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8741_pigeonhole__infinite__rel,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_int,R: fun_Pr4738963024209204295_int_o] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),A2)
             => ? [Xa3: int] :
                  ( aa_set_int_o(member_int(Xa3),B2)
                  & aa_int_o(aa_Pro2912763917461432128_int_o(R,X2),Xa3) ) )
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),B2)
              & ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_int6101145382336329604_nat_o(aa_fun1144309370076579528_nat_o(aTP_Lamm_aey(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8742_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = $ite(
          K = zero_z3403309356797280102nteger,
          zero_z3403309356797280102nteger,
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) ) ) ).

% sgn_integer_code
tff(fact_8743_finite_OemptyI,axiom,
    aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,bot_bo2099793752762293965at_nat) ).

% finite.emptyI
tff(fact_8744_finite_OemptyI,axiom,
    aa_set_o_o(finite_finite_o,bot_bot_set_o) ).

% finite.emptyI
tff(fact_8745_finite_OemptyI,axiom,
    aa_set_nat_o(finite_finite_nat,bot_bot_set_nat) ).

% finite.emptyI
tff(fact_8746_finite_OemptyI,axiom,
    aa_set_int_o(finite_finite_int,bot_bot_set_int) ).

% finite.emptyI
tff(fact_8747_infinite__imp__nonempty,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
     => ( S != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
tff(fact_8748_infinite__imp__nonempty,axiom,
    ! [S: set_o] :
      ( ~ aa_set_o_o(finite_finite_o,S)
     => ( S != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
tff(fact_8749_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_8750_infinite__imp__nonempty,axiom,
    ! [S: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
     => ( S != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_8751_finite__UNIV,axiom,
    aa_set_o_o(finite_finite_o,top_top_set_o) ).

% finite_UNIV
tff(fact_8752_ex__new__if__finite,axiom,
    ! [A2: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,top_to6034159466715884489nt_int)
     => ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,A2)
       => ? [A4: set_Pr958786334691620121nt_int] : ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A4),A2) ) ) ).

% ex_new_if_finite
tff(fact_8753_ex__new__if__finite,axiom,
    ! [A2: set_set_nat] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,top_top_set_set_nat)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
       => ? [A4: set_nat] : ~ aa_set_set_nat_o(member_set_nat(A4),A2) ) ) ).

% ex_new_if_finite
tff(fact_8754_ex__new__if__finite,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,top_to9044565752620676511_nat_o)
     => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
       => ? [A4: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A4),A2) ) ) ).

% ex_new_if_finite
tff(fact_8755_ex__new__if__finite,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,top_to4669805908274784177at_nat)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
       => ? [A4: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A4),A2) ) ) ).

% ex_new_if_finite
tff(fact_8756_ex__new__if__finite,axiom,
    ! [A2: set_list_nat] :
      ( ~ finite8100373058378681591st_nat(top_top_set_list_nat)
     => ( finite8100373058378681591st_nat(A2)
       => ? [A4: list_nat] : ~ member_list_nat(A4,A2) ) ) ).

% ex_new_if_finite
tff(fact_8757_ex__new__if__finite,axiom,
    ! [A2: set_o] :
      ( ~ aa_set_o_o(finite_finite_o,top_top_set_o)
     => ( aa_set_o_o(finite_finite_o,A2)
       => ? [A4: $o] : ~ aa_set_o_o(member_o((A4)),A2) ) ) ).

% ex_new_if_finite
tff(fact_8758_ex__new__if__finite,axiom,
    ! [A2: set_rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,top_top_set_rat)
     => ( aa_set_rat_o(finite_finite_rat,A2)
       => ? [A4: rat] : ~ aa_set_rat_o(member_rat(A4),A2) ) ) ).

% ex_new_if_finite
tff(fact_8759_ex__new__if__finite,axiom,
    ! [A2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,top_top_set_nat)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ? [A4: nat] : ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),A2) ) ) ).

% ex_new_if_finite
tff(fact_8760_ex__new__if__finite,axiom,
    ! [A2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,top_top_set_int)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ? [A4: int] : ~ aa_set_int_o(member_int(A4),A2) ) ) ).

% ex_new_if_finite
tff(fact_8761_infinite__UNIV__char__0,axiom,
    ~ aa_set_rat_o(finite_finite_rat,top_top_set_rat) ).

% infinite_UNIV_char_0
tff(fact_8762_infinite__UNIV__char__0,axiom,
    ~ aa_set_nat_o(finite_finite_nat,top_top_set_nat) ).

% infinite_UNIV_char_0
tff(fact_8763_infinite__UNIV__char__0,axiom,
    ~ aa_set_int_o(finite_finite_int,top_top_set_int) ).

% infinite_UNIV_char_0
tff(fact_8764_finite__UnI,axiom,
    ! [F3: set_int,G2: set_int] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(finite_finite_int,G2)
       => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,F3),G2)) ) ) ).

% finite_UnI
tff(fact_8765_finite__UnI,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,G2)
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,F3),G2)) ) ) ).

% finite_UnI
tff(fact_8766_finite__UnI,axiom,
    ! [F3: set_Pr8551490117392284871at_nat,G2: set_Pr8551490117392284871at_nat] :
      ( finite1918287321285529104at_nat(F3)
     => ( finite1918287321285529104at_nat(G2)
       => finite1918287321285529104at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,F3),G2)) ) ) ).

% finite_UnI
tff(fact_8767_finite__UnI,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
     => ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,G2)
       => aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,F3),G2)) ) ) ).

% finite_UnI
tff(fact_8768_finite__UnI,axiom,
    ! [F3: set_nat,G2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(finite_finite_nat,G2)
       => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,F3),G2)) ) ) ).

% finite_UnI
tff(fact_8769_Un__infinite,axiom,
    ! [S: set_int,T: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
     => ~ aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,S),T)) ) ).

% Un_infinite
tff(fact_8770_Un__infinite,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
     => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T)) ) ).

% Un_infinite
tff(fact_8771_Un__infinite,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
      ( ~ finite1918287321285529104at_nat(S)
     => ~ finite1918287321285529104at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,S),T)) ) ).

% Un_infinite
tff(fact_8772_Un__infinite,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,S)
     => ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T)) ) ).

% Un_infinite
tff(fact_8773_Un__infinite,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T)) ) ).

% Un_infinite
tff(fact_8774_infinite__Un,axiom,
    ! [S: set_int,T: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,S),T))
    <=> ( ~ aa_set_int_o(finite_finite_int,S)
        | ~ aa_set_int_o(finite_finite_int,T) ) ) ).

% infinite_Un
tff(fact_8775_infinite__Un,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T))
    <=> ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
        | ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,T) ) ) ).

% infinite_Un
tff(fact_8776_infinite__Un,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
      ( ~ finite1918287321285529104at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,S),T))
    <=> ( ~ finite1918287321285529104at_nat(S)
        | ~ finite1918287321285529104at_nat(T) ) ) ).

% infinite_Un
tff(fact_8777_infinite__Un,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T))
    <=> ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,S)
        | ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,T) ) ) ).

% infinite_Un
tff(fact_8778_infinite__Un,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T))
    <=> ( ~ aa_set_nat_o(finite_finite_nat,S)
        | ~ aa_set_nat_o(finite_finite_nat,T) ) ) ).

% infinite_Un
tff(fact_8779_finite__has__minimal,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ? [X2: $o] :
            ( aa_set_o_o(member_o((X2)),A2)
            & ! [Xa3: $o] :
                ( aa_set_o_o(member_o((Xa3)),A2)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa3)),(X2))
                 => ( (X2)
                  <=> (Xa3) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8780_finite__has__minimal,axiom,
    ! [A2: set_filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( ( A2 != bot_bo498966703094740906er_nat )
       => ? [X2: filter_nat] :
            ( member_filter_nat(X2,A2)
            & ! [Xa3: filter_nat] :
                ( member_filter_nat(Xa3,A2)
               => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Xa3),X2)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8781_finite__has__minimal,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X2),A2)
            & ! [Xa3: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa3),A2)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa3),X2)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8782_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A2)
            & ! [Xa3: rat] :
                ( aa_set_rat_o(member_rat(Xa3),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa3),X2)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8783_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
            & ! [Xa3: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa3),X2)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8784_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa3),X2)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8785_finite__has__maximal,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ? [X2: $o] :
            ( aa_set_o_o(member_o((X2)),A2)
            & ! [Xa3: $o] :
                ( aa_set_o_o(member_o((Xa3)),A2)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X2)),(Xa3))
                 => ( (X2)
                  <=> (Xa3) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8786_finite__has__maximal,axiom,
    ! [A2: set_filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( ( A2 != bot_bo498966703094740906er_nat )
       => ? [X2: filter_nat] :
            ( member_filter_nat(X2,A2)
            & ! [Xa3: filter_nat] :
                ( member_filter_nat(Xa3,A2)
               => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X2),Xa3)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8787_finite__has__maximal,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X2),A2)
            & ! [Xa3: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa3),A2)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),Xa3)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8788_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A2)
            & ! [Xa3: rat] :
                ( aa_set_rat_o(member_rat(Xa3),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Xa3)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8789_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),A2)
            & ! [Xa3: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa3)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8790_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Xa3)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8791_finite_Ocases,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A)
     => ( ( A != bot_bo228742789529271731at_nat )
       => ~ ! [A6: set_Pr4329608150637261639at_nat] :
              ( ? [A4: produc3843707927480180839at_nat] : ( A = insert9069300056098147895at_nat(A4,A6) )
             => ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A6) ) ) ) ).

% finite.cases
tff(fact_8792_finite_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A)
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ~ ! [A6: set_Pr1261947904930325089at_nat] :
              ( ? [A4: product_prod_nat_nat] : ( A = insert8211810215607154385at_nat(A4,A6) )
             => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A6) ) ) ) ).

% finite.cases
tff(fact_8793_finite_Ocases,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ~ ! [A6: set_o] :
              ( ? [A4: $o] : ( A = insert_o((A4),A6) )
             => ~ aa_set_o_o(finite_finite_o,A6) ) ) ) ).

% finite.cases
tff(fact_8794_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A6: set_nat] :
              ( ? [A4: nat] : ( A = aa_set_nat_set_nat(insert_nat(A4),A6) )
             => ~ aa_set_nat_o(finite_finite_nat,A6) ) ) ) ).

% finite.cases
tff(fact_8795_finite_Ocases,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ~ ! [A6: set_int] :
              ( ? [A4: int] : ( A = insert_int(A4,A6) )
             => ~ aa_set_int_o(finite_finite_int,A6) ) ) ) ).

% finite.cases
tff(fact_8796_finite_Osimps,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A)
    <=> ( ( A = bot_bo228742789529271731at_nat )
        | ? [A7: set_Pr4329608150637261639at_nat,A8: produc3843707927480180839at_nat] :
            ( ( A = insert9069300056098147895at_nat(A8,A7) )
            & aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A7) ) ) ) ).

% finite.simps
tff(fact_8797_finite_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A)
    <=> ( ( A = bot_bo2099793752762293965at_nat )
        | ? [A7: set_Pr1261947904930325089at_nat,A8: product_prod_nat_nat] :
            ( ( A = insert8211810215607154385at_nat(A8,A7) )
            & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A7) ) ) ) ).

% finite.simps
tff(fact_8798_finite_Osimps,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
    <=> ( ( A = bot_bot_set_o )
        | ? [A7: set_o,A8: $o] :
            ( ( A = insert_o((A8),A7) )
            & aa_set_o_o(finite_finite_o,A7) ) ) ) ).

% finite.simps
tff(fact_8799_finite_Osimps,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
    <=> ( ( A = bot_bot_set_nat )
        | ? [A7: set_nat,A8: nat] :
            ( ( A = aa_set_nat_set_nat(insert_nat(A8),A7) )
            & aa_set_nat_o(finite_finite_nat,A7) ) ) ) ).

% finite.simps
tff(fact_8800_finite_Osimps,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
    <=> ( ( A = bot_bot_set_int )
        | ? [A7: set_int,A8: int] :
            ( ( A = insert_int(A8,A7) )
            & aa_set_int_o(finite_finite_int,A7) ) ) ) ).

% finite.simps
tff(fact_8801_finite__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
     => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
       => ( ! [X2: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
             => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X2),F4)
               => ( aa_set5844364881195198285_nat_o(P,F4)
                 => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X2,F4)) ) ) )
         => aa_set5844364881195198285_nat_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_8802_finite__induct,axiom,
    ! [F3: set_se6260736226359567993nt_int,P: fun_se9106460625052465826_int_o] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,F3)
     => ( aa_set6820988702716878619_int_o(P,bot_bo1488462491386950373nt_int)
       => ( ! [X2: set_Pr958786334691620121nt_int,F4: set_se6260736226359567993nt_int] :
              ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,F4)
             => ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),F4)
               => ( aa_set6820988702716878619_int_o(P,F4)
                 => aa_set6820988702716878619_int_o(P,insert8897473484851387113nt_int(X2,F4)) ) ) )
         => aa_set6820988702716878619_int_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_8803_finite__induct,axiom,
    ! [F3: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [X2: set_nat,F4: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
             => ( ~ aa_set_set_nat_o(member_set_nat(X2),F4)
               => ( aa_set_set_nat_o(P,F4)
                 => aa_set_set_nat_o(P,insert_set_nat(X2,F4)) ) ) )
         => aa_set_set_nat_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_8804_finite__induct,axiom,
    ! [F3: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
     => ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
       => ( ! [X2: fun_Pr6460944699146269935_nat_o,F4: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F4)
             => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),F4)
               => ( aa_set5091322549519198405at_o_o(P,F4)
                 => aa_set5091322549519198405at_o_o(P,insert4923026906511862463_nat_o(X2,F4)) ) ) )
         => aa_set5091322549519198405at_o_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_8805_finite__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
       => ( ! [X2: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
             => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),F4)
               => ( aa_set3702741264012733747_nat_o(P,F4)
                 => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X2,F4)) ) ) )
         => aa_set3702741264012733747_nat_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_8806_finite__induct,axiom,
    ! [F3: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X2: $o,F4: set_o] :
              ( aa_set_o_o(finite_finite_o,F4)
             => ( ~ aa_set_o_o(member_o((X2)),F4)
               => ( aa_set_o_o(P,F4)
                 => aa_set_o_o(P,insert_o((X2),F4)) ) ) )
         => aa_set_o_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_8807_finite__induct,axiom,
    ! [F3: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X2: nat,F4: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F4)
             => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),F4)
               => ( aa_set_nat_o(P,F4)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X2),F4)) ) ) )
         => aa_set_nat_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_8808_finite__induct,axiom,
    ! [F3: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X2: int,F4: set_int] :
              ( aa_set_int_o(finite_finite_int,F4)
             => ( ~ aa_set_int_o(member_int(X2),F4)
               => ( aa_set_int_o(P,F4)
                 => aa_set_int_o(P,insert_int(X2,F4)) ) ) )
         => aa_set_int_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_8809_finite__ne__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
     => ( ( F3 != bot_bo228742789529271731at_nat )
       => ( ! [X2: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X2,bot_bo228742789529271731at_nat))
         => ( ! [X2: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
               => ( ( F4 != bot_bo228742789529271731at_nat )
                 => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X2),F4)
                   => ( aa_set5844364881195198285_nat_o(P,F4)
                     => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X2,F4)) ) ) ) )
           => aa_set5844364881195198285_nat_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_8810_finite__ne__induct,axiom,
    ! [F3: set_se6260736226359567993nt_int,P: fun_se9106460625052465826_int_o] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,F3)
     => ( ( F3 != bot_bo1488462491386950373nt_int )
       => ( ! [X2: set_Pr958786334691620121nt_int] : aa_set6820988702716878619_int_o(P,insert8897473484851387113nt_int(X2,bot_bo1488462491386950373nt_int))
         => ( ! [X2: set_Pr958786334691620121nt_int,F4: set_se6260736226359567993nt_int] :
                ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,F4)
               => ( ( F4 != bot_bo1488462491386950373nt_int )
                 => ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),F4)
                   => ( aa_set6820988702716878619_int_o(P,F4)
                     => aa_set6820988702716878619_int_o(P,insert8897473484851387113nt_int(X2,F4)) ) ) ) )
           => aa_set6820988702716878619_int_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_8811_finite__ne__induct,axiom,
    ! [F3: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( ( F3 != bot_bot_set_set_nat )
       => ( ! [X2: set_nat] : aa_set_set_nat_o(P,insert_set_nat(X2,bot_bot_set_set_nat))
         => ( ! [X2: set_nat,F4: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
               => ( ( F4 != bot_bot_set_set_nat )
                 => ( ~ aa_set_set_nat_o(member_set_nat(X2),F4)
                   => ( aa_set_set_nat_o(P,F4)
                     => aa_set_set_nat_o(P,insert_set_nat(X2,F4)) ) ) ) )
           => aa_set_set_nat_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_8812_finite__ne__induct,axiom,
    ! [F3: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
     => ( ( F3 != bot_bo8632255151191677115_nat_o )
       => ( ! [X2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(P,insert4923026906511862463_nat_o(X2,bot_bo8632255151191677115_nat_o))
         => ( ! [X2: fun_Pr6460944699146269935_nat_o,F4: set_fu3555266933746557007_nat_o] :
                ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F4)
               => ( ( F4 != bot_bo8632255151191677115_nat_o )
                 => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),F4)
                   => ( aa_set5091322549519198405at_o_o(P,F4)
                     => aa_set5091322549519198405at_o_o(P,insert4923026906511862463_nat_o(X2,F4)) ) ) ) )
           => aa_set5091322549519198405at_o_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_8813_finite__ne__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => ( ( F3 != bot_bo2099793752762293965at_nat )
       => ( ! [X2: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X2,bot_bo2099793752762293965at_nat))
         => ( ! [X2: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
               => ( ( F4 != bot_bo2099793752762293965at_nat )
                 => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),F4)
                   => ( aa_set3702741264012733747_nat_o(P,F4)
                     => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X2,F4)) ) ) ) )
           => aa_set3702741264012733747_nat_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_8814_finite__ne__induct,axiom,
    ! [F3: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( ( F3 != bot_bot_set_o )
       => ( ! [X2: $o] : aa_set_o_o(P,insert_o((X2),bot_bot_set_o))
         => ( ! [X2: $o,F4: set_o] :
                ( aa_set_o_o(finite_finite_o,F4)
               => ( ( F4 != bot_bot_set_o )
                 => ( ~ aa_set_o_o(member_o((X2)),F4)
                   => ( aa_set_o_o(P,F4)
                     => aa_set_o_o(P,insert_o((X2),F4)) ) ) ) )
           => aa_set_o_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_8815_finite__ne__induct,axiom,
    ! [F3: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( ( F3 != bot_bot_set_nat )
       => ( ! [X2: nat] : aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X2),bot_bot_set_nat))
         => ( ! [X2: nat,F4: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F4)
               => ( ( F4 != bot_bot_set_nat )
                 => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),F4)
                   => ( aa_set_nat_o(P,F4)
                     => aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X2),F4)) ) ) ) )
           => aa_set_nat_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_8816_finite__ne__induct,axiom,
    ! [F3: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( ( F3 != bot_bot_set_int )
       => ( ! [X2: int] : aa_set_int_o(P,insert_int(X2,bot_bot_set_int))
         => ( ! [X2: int,F4: set_int] :
                ( aa_set_int_o(finite_finite_int,F4)
               => ( ( F4 != bot_bot_set_int )
                 => ( ~ aa_set_int_o(member_int(X2),F4)
                   => ( aa_set_int_o(P,F4)
                     => aa_set_int_o(P,insert_int(X2,F4)) ) ) ) )
           => aa_set_int_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_8817_infinite__finite__induct,axiom,
    ! [P: fun_se7788542033685991124_nat_o,A2: set_Pr4329608150637261639at_nat] :
      ( ! [A6: set_Pr4329608150637261639at_nat] :
          ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A6)
         => aa_set5844364881195198285_nat_o(P,A6) )
     => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
       => ( ! [X2: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
             => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X2),F4)
               => ( aa_set5844364881195198285_nat_o(P,F4)
                 => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X2,F4)) ) ) )
         => aa_set5844364881195198285_nat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_8818_infinite__finite__induct,axiom,
    ! [P: fun_se9106460625052465826_int_o,A2: set_se6260736226359567993nt_int] :
      ( ! [A6: set_se6260736226359567993nt_int] :
          ( ~ aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,A6)
         => aa_set6820988702716878619_int_o(P,A6) )
     => ( aa_set6820988702716878619_int_o(P,bot_bo1488462491386950373nt_int)
       => ( ! [X2: set_Pr958786334691620121nt_int,F4: set_se6260736226359567993nt_int] :
              ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,F4)
             => ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X2),F4)
               => ( aa_set6820988702716878619_int_o(P,F4)
                 => aa_set6820988702716878619_int_o(P,insert8897473484851387113nt_int(X2,F4)) ) ) )
         => aa_set6820988702716878619_int_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_8819_infinite__finite__induct,axiom,
    ! [P: fun_set_set_nat_o,A2: set_set_nat] :
      ( ! [A6: set_set_nat] :
          ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A6)
         => aa_set_set_nat_o(P,A6) )
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [X2: set_nat,F4: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
             => ( ~ aa_set_set_nat_o(member_set_nat(X2),F4)
               => ( aa_set_set_nat_o(P,F4)
                 => aa_set_set_nat_o(P,insert_set_nat(X2,F4)) ) ) )
         => aa_set_set_nat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_8820_infinite__finite__induct,axiom,
    ! [P: fun_se7459991829846211660at_o_o,A2: set_fu3555266933746557007_nat_o] :
      ( ! [A6: set_fu3555266933746557007_nat_o] :
          ( ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A6)
         => aa_set5091322549519198405at_o_o(P,A6) )
     => ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
       => ( ! [X2: fun_Pr6460944699146269935_nat_o,F4: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F4)
             => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),F4)
               => ( aa_set5091322549519198405at_o_o(P,F4)
                 => aa_set5091322549519198405at_o_o(P,insert4923026906511862463_nat_o(X2,F4)) ) ) )
         => aa_set5091322549519198405at_o_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_8821_infinite__finite__induct,axiom,
    ! [P: fun_se4782206882984618170_nat_o,A2: set_Pr1261947904930325089at_nat] :
      ( ! [A6: set_Pr1261947904930325089at_nat] :
          ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A6)
         => aa_set3702741264012733747_nat_o(P,A6) )
     => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
       => ( ! [X2: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
             => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),F4)
               => ( aa_set3702741264012733747_nat_o(P,F4)
                 => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X2,F4)) ) ) )
         => aa_set3702741264012733747_nat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_8822_infinite__finite__induct,axiom,
    ! [P: fun_set_o_o,A2: set_o] :
      ( ! [A6: set_o] :
          ( ~ aa_set_o_o(finite_finite_o,A6)
         => aa_set_o_o(P,A6) )
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X2: $o,F4: set_o] :
              ( aa_set_o_o(finite_finite_o,F4)
             => ( ~ aa_set_o_o(member_o((X2)),F4)
               => ( aa_set_o_o(P,F4)
                 => aa_set_o_o(P,insert_o((X2),F4)) ) ) )
         => aa_set_o_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_8823_infinite__finite__induct,axiom,
    ! [P: fun_set_nat_o,A2: set_nat] :
      ( ! [A6: set_nat] :
          ( ~ aa_set_nat_o(finite_finite_nat,A6)
         => aa_set_nat_o(P,A6) )
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X2: nat,F4: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F4)
             => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),F4)
               => ( aa_set_nat_o(P,F4)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X2),F4)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_8824_infinite__finite__induct,axiom,
    ! [P: fun_set_int_o,A2: set_int] :
      ( ! [A6: set_int] :
          ( ~ aa_set_int_o(finite_finite_int,A6)
         => aa_set_int_o(P,A6) )
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X2: int,F4: set_int] :
              ( aa_set_int_o(finite_finite_int,F4)
             => ( ~ aa_set_int_o(member_int(X2),F4)
               => ( aa_set_int_o(P,F4)
                 => aa_set_int_o(P,insert_int(X2,F4)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_8825_finite__prod,axiom,
    ( finite6120865539452801872od_o_o(top_to7721136755696657239od_o_o)
  <=> ( aa_set_o_o(finite_finite_o,top_top_set_o)
      & aa_set_o_o(finite_finite_o,top_top_set_o) ) ) ).

% finite_prod
tff(fact_8826_finite__prod,axiom,
    ( finite7561617386449315510_o_rat(top_to642698169864860165_o_rat)
  <=> ( aa_set_o_o(finite_finite_o,top_top_set_o)
      & aa_set_rat_o(finite_finite_rat,top_top_set_rat) ) ) ).

% finite_prod
tff(fact_8827_finite__prod,axiom,
    ( finite1846473907987936430_o_nat(top_to7022684507342537725_o_nat)
  <=> ( aa_set_o_o(finite_finite_o,top_top_set_o)
      & aa_set_nat_o(finite_finite_nat,top_top_set_nat) ) ) ).

% finite_prod
tff(fact_8828_finite__prod,axiom,
    ( finite6891994925333515530_o_int(top_to4532601362410810969_o_int)
  <=> ( aa_set_o_o(finite_finite_o,top_top_set_o)
      & aa_set_int_o(finite_finite_int,top_top_set_int) ) ) ).

% finite_prod
tff(fact_8829_finite__prod,axiom,
    ( finite1617675504726861388_rat_o(top_to7320853134386005659_rat_o)
  <=> ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
      & aa_set_o_o(finite_finite_o,top_top_set_o) ) ) ).

% finite_prod
tff(fact_8830_finite__prod,axiom,
    ( finite2352427746407582394at_rat(top_to3112507417142880193at_rat)
  <=> ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
      & aa_set_rat_o(finite_finite_rat,top_top_set_rat) ) ) ).

% finite_prod
tff(fact_8831_finite__prod,axiom,
    ( finite5860656304800979122at_nat(top_to269121717765781945at_nat)
  <=> ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
      & aa_set_nat_o(finite_finite_nat,top_top_set_nat) ) ) ).

% finite_prod
tff(fact_8832_finite__prod,axiom,
    ( finite1682805285291782414at_int(top_to7002410609688830997at_int)
  <=> ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
      & aa_set_int_o(finite_finite_int,top_top_set_int) ) ) ).

% finite_prod
tff(fact_8833_finite__prod,axiom,
    ( finite5355008432043429460_nat_o(top_to8070287629520841379_nat_o)
  <=> ( aa_set_nat_o(finite_finite_nat,top_top_set_nat)
      & aa_set_o_o(finite_finite_o,top_top_set_o) ) ) ).

% finite_prod
tff(fact_8834_finite__prod,axiom,
    ( finite2668982390342448306at_rat(top_to7513191607651882425at_rat)
  <=> ( aa_set_nat_o(finite_finite_nat,top_top_set_nat)
      & aa_set_rat_o(finite_finite_rat,top_top_set_rat) ) ) ).

% finite_prod
tff(fact_8835_finite__Prod__UNIV,axiom,
    ( aa_set_o_o(finite_finite_o,top_top_set_o)
   => ( aa_set_o_o(finite_finite_o,top_top_set_o)
     => finite6120865539452801872od_o_o(top_to7721136755696657239od_o_o) ) ) ).

% finite_Prod_UNIV
tff(fact_8836_finite__Prod__UNIV,axiom,
    ( aa_set_o_o(finite_finite_o,top_top_set_o)
   => ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
     => finite7561617386449315510_o_rat(top_to642698169864860165_o_rat) ) ) ).

% finite_Prod_UNIV
tff(fact_8837_finite__Prod__UNIV,axiom,
    ( aa_set_o_o(finite_finite_o,top_top_set_o)
   => ( aa_set_nat_o(finite_finite_nat,top_top_set_nat)
     => finite1846473907987936430_o_nat(top_to7022684507342537725_o_nat) ) ) ).

% finite_Prod_UNIV
tff(fact_8838_finite__Prod__UNIV,axiom,
    ( aa_set_o_o(finite_finite_o,top_top_set_o)
   => ( aa_set_int_o(finite_finite_int,top_top_set_int)
     => finite6891994925333515530_o_int(top_to4532601362410810969_o_int) ) ) ).

% finite_Prod_UNIV
tff(fact_8839_finite__Prod__UNIV,axiom,
    ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
   => ( aa_set_o_o(finite_finite_o,top_top_set_o)
     => finite1617675504726861388_rat_o(top_to7320853134386005659_rat_o) ) ) ).

% finite_Prod_UNIV
tff(fact_8840_finite__Prod__UNIV,axiom,
    ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
   => ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
     => finite2352427746407582394at_rat(top_to3112507417142880193at_rat) ) ) ).

% finite_Prod_UNIV
tff(fact_8841_finite__Prod__UNIV,axiom,
    ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
   => ( aa_set_nat_o(finite_finite_nat,top_top_set_nat)
     => finite5860656304800979122at_nat(top_to269121717765781945at_nat) ) ) ).

% finite_Prod_UNIV
tff(fact_8842_finite__Prod__UNIV,axiom,
    ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
   => ( aa_set_int_o(finite_finite_int,top_top_set_int)
     => finite1682805285291782414at_int(top_to7002410609688830997at_int) ) ) ).

% finite_Prod_UNIV
tff(fact_8843_finite__Prod__UNIV,axiom,
    ( aa_set_nat_o(finite_finite_nat,top_top_set_nat)
   => ( aa_set_o_o(finite_finite_o,top_top_set_o)
     => finite5355008432043429460_nat_o(top_to8070287629520841379_nat_o) ) ) ).

% finite_Prod_UNIV
tff(fact_8844_finite__Prod__UNIV,axiom,
    ( aa_set_nat_o(finite_finite_nat,top_top_set_nat)
   => ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
     => finite2668982390342448306at_rat(top_to7513191607651882425at_rat) ) ) ).

% finite_Prod_UNIV
tff(fact_8845_Finite__Set_Ofinite__set,axiom,
    ( finite9047747110432174090at_nat(top_to7629004291339433233at_nat)
  <=> aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,top_to4669805908274784177at_nat) ) ).

% Finite_Set.finite_set
tff(fact_8846_Finite__Set_Ofinite__set,axiom,
    ( finite7047420756378620717st_nat(top_to1619013496918823676st_nat)
  <=> finite8100373058378681591st_nat(top_top_set_list_nat) ) ).

% Finite_Set.finite_set
tff(fact_8847_Finite__Set_Ofinite__set,axiom,
    ( finite_finite_set_o(top_top_set_set_o)
  <=> aa_set_o_o(finite_finite_o,top_top_set_o) ) ).

% Finite_Set.finite_set
tff(fact_8848_Finite__Set_Ofinite__set,axiom,
    ( finite6867581373910428453et_rat(top_top_set_set_rat)
  <=> aa_set_rat_o(finite_finite_rat,top_top_set_rat) ) ).

% Finite_Set.finite_set
tff(fact_8849_Finite__Set_Ofinite__set,axiom,
    ( aa_set_set_nat_o(finite1152437895449049373et_nat,top_top_set_set_nat)
  <=> aa_set_nat_o(finite_finite_nat,top_top_set_nat) ) ).

% Finite_Set.finite_set
tff(fact_8850_Finite__Set_Ofinite__set,axiom,
    ( finite6197958912794628473et_int(top_top_set_set_int)
  <=> aa_set_int_o(finite_finite_int,top_top_set_int) ) ).

% Finite_Set.finite_set
tff(fact_8851_finite__subset__induct_H,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F3),A2)
       => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
         => ( ! [A4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
               => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A4),A2)
                 => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F4),A2)
                   => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A4),F4)
                     => ( aa_set5844364881195198285_nat_o(P,F4)
                       => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(A4,F4)) ) ) ) ) )
           => aa_set5844364881195198285_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8852_finite__subset__induct_H,axiom,
    ! [F3: set_se6260736226359567993nt_int,A2: set_se6260736226359567993nt_int,P: fun_se9106460625052465826_int_o] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,F3)
     => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(F3),A2)
       => ( aa_set6820988702716878619_int_o(P,bot_bo1488462491386950373nt_int)
         => ( ! [A4: set_Pr958786334691620121nt_int,F4: set_se6260736226359567993nt_int] :
                ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,F4)
               => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A4),A2)
                 => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(F4),A2)
                   => ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A4),F4)
                     => ( aa_set6820988702716878619_int_o(P,F4)
                       => aa_set6820988702716878619_int_o(P,insert8897473484851387113nt_int(A4,F4)) ) ) ) ) )
           => aa_set6820988702716878619_int_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8853_finite__subset__induct_H,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F3),A2)
       => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
         => ( ! [A4: set_nat,F4: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
               => ( aa_set_set_nat_o(member_set_nat(A4),A2)
                 => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F4),A2)
                   => ( ~ aa_set_set_nat_o(member_set_nat(A4),F4)
                     => ( aa_set_set_nat_o(P,F4)
                       => aa_set_set_nat_o(P,insert_set_nat(A4,F4)) ) ) ) ) )
           => aa_set_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8854_finite__subset__induct_H,axiom,
    ! [F3: set_fu3555266933746557007_nat_o,A2: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(F3),A2)
       => ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
         => ( ! [A4: fun_Pr6460944699146269935_nat_o,F4: set_fu3555266933746557007_nat_o] :
                ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F4)
               => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A4),A2)
                 => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(F4),A2)
                   => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A4),F4)
                     => ( aa_set5091322549519198405at_o_o(P,F4)
                       => aa_set5091322549519198405at_o_o(P,insert4923026906511862463_nat_o(A4,F4)) ) ) ) ) )
           => aa_set5091322549519198405at_o_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8855_finite__subset__induct_H,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,F3),A2)
       => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
         => ( ! [A4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
               => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A4),A2)
                 => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,F4),A2)
                   => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A4),F4)
                     => ( aa_set3702741264012733747_nat_o(P,F4)
                       => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(A4,F4)) ) ) ) ) )
           => aa_set3702741264012733747_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8856_finite__subset__induct_H,axiom,
    ! [F3: set_o,A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,F3),A2)
       => ( aa_set_o_o(P,bot_bot_set_o)
         => ( ! [A4: $o,F4: set_o] :
                ( aa_set_o_o(finite_finite_o,F4)
               => ( aa_set_o_o(member_o((A4)),A2)
                 => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,F4),A2)
                   => ( ~ aa_set_o_o(member_o((A4)),F4)
                     => ( aa_set_o_o(P,F4)
                       => aa_set_o_o(P,insert_o((A4),F4)) ) ) ) ) )
           => aa_set_o_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8857_finite__subset__induct_H,axiom,
    ! [F3: set_int,A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,F3),A2)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A4: int,F4: set_int] :
                ( aa_set_int_o(finite_finite_int,F4)
               => ( aa_set_int_o(member_int(A4),A2)
                 => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,F4),A2)
                   => ( ~ aa_set_int_o(member_int(A4),F4)
                     => ( aa_set_int_o(P,F4)
                       => aa_set_int_o(P,insert_int(A4,F4)) ) ) ) ) )
           => aa_set_int_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8858_finite__subset__induct_H,axiom,
    ! [F3: set_nat,A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F3),A2)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A4: nat,F4: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F4)
               => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),A2)
                 => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F4),A2)
                   => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),F4)
                     => ( aa_set_nat_o(P,F4)
                       => aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(A4),F4)) ) ) ) ) )
           => aa_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8859_finite__subset__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F3),A2)
       => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
         => ( ! [A4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
               => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A4),A2)
                 => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A4),F4)
                   => ( aa_set5844364881195198285_nat_o(P,F4)
                     => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(A4,F4)) ) ) ) )
           => aa_set5844364881195198285_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_8860_finite__subset__induct,axiom,
    ! [F3: set_se6260736226359567993nt_int,A2: set_se6260736226359567993nt_int,P: fun_se9106460625052465826_int_o] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,F3)
     => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(F3),A2)
       => ( aa_set6820988702716878619_int_o(P,bot_bo1488462491386950373nt_int)
         => ( ! [A4: set_Pr958786334691620121nt_int,F4: set_se6260736226359567993nt_int] :
                ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,F4)
               => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A4),A2)
                 => ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A4),F4)
                   => ( aa_set6820988702716878619_int_o(P,F4)
                     => aa_set6820988702716878619_int_o(P,insert8897473484851387113nt_int(A4,F4)) ) ) ) )
           => aa_set6820988702716878619_int_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_8861_finite__subset__induct,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F3),A2)
       => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
         => ( ! [A4: set_nat,F4: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
               => ( aa_set_set_nat_o(member_set_nat(A4),A2)
                 => ( ~ aa_set_set_nat_o(member_set_nat(A4),F4)
                   => ( aa_set_set_nat_o(P,F4)
                     => aa_set_set_nat_o(P,insert_set_nat(A4,F4)) ) ) ) )
           => aa_set_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_8862_finite__subset__induct,axiom,
    ! [F3: set_fu3555266933746557007_nat_o,A2: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(F3),A2)
       => ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
         => ( ! [A4: fun_Pr6460944699146269935_nat_o,F4: set_fu3555266933746557007_nat_o] :
                ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F4)
               => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A4),A2)
                 => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A4),F4)
                   => ( aa_set5091322549519198405at_o_o(P,F4)
                     => aa_set5091322549519198405at_o_o(P,insert4923026906511862463_nat_o(A4,F4)) ) ) ) )
           => aa_set5091322549519198405at_o_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_8863_finite__subset__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,F3),A2)
       => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
         => ( ! [A4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
               => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A4),A2)
                 => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A4),F4)
                   => ( aa_set3702741264012733747_nat_o(P,F4)
                     => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(A4,F4)) ) ) ) )
           => aa_set3702741264012733747_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_8864_finite__subset__induct,axiom,
    ! [F3: set_o,A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,F3),A2)
       => ( aa_set_o_o(P,bot_bot_set_o)
         => ( ! [A4: $o,F4: set_o] :
                ( aa_set_o_o(finite_finite_o,F4)
               => ( aa_set_o_o(member_o((A4)),A2)
                 => ( ~ aa_set_o_o(member_o((A4)),F4)
                   => ( aa_set_o_o(P,F4)
                     => aa_set_o_o(P,insert_o((A4),F4)) ) ) ) )
           => aa_set_o_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_8865_finite__subset__induct,axiom,
    ! [F3: set_int,A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,F3),A2)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A4: int,F4: set_int] :
                ( aa_set_int_o(finite_finite_int,F4)
               => ( aa_set_int_o(member_int(A4),A2)
                 => ( ~ aa_set_int_o(member_int(A4),F4)
                   => ( aa_set_int_o(P,F4)
                     => aa_set_int_o(P,insert_int(A4,F4)) ) ) ) )
           => aa_set_int_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_8866_finite__subset__induct,axiom,
    ! [F3: set_nat,A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F3),A2)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A4: nat,F4: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F4)
               => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),A2)
                 => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),F4)
                   => ( aa_set_nat_o(P,F4)
                     => aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(A4),F4)) ) ) ) )
           => aa_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_8867_infinite__remove,axiom,
    ! [S: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,S)
     => ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,S),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) ) ).

% infinite_remove
tff(fact_8868_infinite__remove,axiom,
    ! [S: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
     => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,S),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) ) ).

% infinite_remove
tff(fact_8869_infinite__remove,axiom,
    ! [S: set_o,A: $o] :
      ( ~ aa_set_o_o(finite_finite_o,S)
     => ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(S,insert_o((A),bot_bot_set_o))) ) ).

% infinite_remove
tff(fact_8870_infinite__remove,axiom,
    ! [S: set_int,A: int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
     => ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S,insert_int(A,bot_bot_set_int))) ) ).

% infinite_remove
tff(fact_8871_infinite__remove,axiom,
    ! [S: set_nat,A: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) ) ).

% infinite_remove
tff(fact_8872_infinite__coinduct,axiom,
    ! [X5: fun_se7788542033685991124_nat_o,A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(X5,A2)
     => ( ! [A6: set_Pr4329608150637261639at_nat] :
            ( aa_set5844364881195198285_nat_o(X5,A6)
           => ? [X3: produc3843707927480180839at_nat] :
                ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),A6)
                & ( aa_set5844364881195198285_nat_o(X5,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A6),insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat)))
                  | ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A6),insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat))) ) ) )
       => ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A2) ) ) ).

% infinite_coinduct
tff(fact_8873_infinite__coinduct,axiom,
    ! [X5: fun_se4782206882984618170_nat_o,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(X5,A2)
     => ( ! [A6: set_Pr1261947904930325089at_nat] :
            ( aa_set3702741264012733747_nat_o(X5,A6)
           => ? [X3: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A6)
                & ( aa_set3702741264012733747_nat_o(X5,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A6),insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat)))
                  | ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A6),insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat))) ) ) )
       => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2) ) ) ).

% infinite_coinduct
tff(fact_8874_infinite__coinduct,axiom,
    ! [X5: fun_set_o_o,A2: set_o] :
      ( aa_set_o_o(X5,A2)
     => ( ! [A6: set_o] :
            ( aa_set_o_o(X5,A6)
           => ? [X3: $o] :
                ( aa_set_o_o(member_o((X3)),A6)
                & ( aa_set_o_o(X5,minus_minus_set_o(A6,insert_o((X3),bot_bot_set_o)))
                  | ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(A6,insert_o((X3),bot_bot_set_o))) ) ) )
       => ~ aa_set_o_o(finite_finite_o,A2) ) ) ).

% infinite_coinduct
tff(fact_8875_infinite__coinduct,axiom,
    ! [X5: fun_set_int_o,A2: set_int] :
      ( aa_set_int_o(X5,A2)
     => ( ! [A6: set_int] :
            ( aa_set_int_o(X5,A6)
           => ? [X3: int] :
                ( aa_set_int_o(member_int(X3),A6)
                & ( aa_set_int_o(X5,minus_minus_set_int(A6,insert_int(X3,bot_bot_set_int)))
                  | ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(A6,insert_int(X3,bot_bot_set_int))) ) ) )
       => ~ aa_set_int_o(finite_finite_int,A2) ) ) ).

% infinite_coinduct
tff(fact_8876_infinite__coinduct,axiom,
    ! [X5: fun_set_nat_o,A2: set_nat] :
      ( aa_set_nat_o(X5,A2)
     => ( ! [A6: set_nat] :
            ( aa_set_nat_o(X5,A6)
           => ? [X3: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A6)
                & ( aa_set_nat_o(X5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)))
                  | ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) ) ) )
       => ~ aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% infinite_coinduct
tff(fact_8877_finite__empty__induct,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A2)
     => ( aa_set5844364881195198285_nat_o(P,A2)
       => ( ! [A4: produc3843707927480180839at_nat,A6: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A6)
             => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A4),A6)
               => ( aa_set5844364881195198285_nat_o(P,A6)
                 => aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A6),insert9069300056098147895at_nat(A4,bot_bo228742789529271731at_nat))) ) ) )
         => aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat) ) ) ) ).

% finite_empty_induct
tff(fact_8878_finite__empty__induct,axiom,
    ! [A2: set_se6260736226359567993nt_int,P: fun_se9106460625052465826_int_o] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,A2)
     => ( aa_set6820988702716878619_int_o(P,A2)
       => ( ! [A4: set_Pr958786334691620121nt_int,A6: set_se6260736226359567993nt_int] :
              ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,A6)
             => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A4),A6)
               => ( aa_set6820988702716878619_int_o(P,A6)
                 => aa_set6820988702716878619_int_o(P,minus_2612819937483484256nt_int(A6,insert8897473484851387113nt_int(A4,bot_bo1488462491386950373nt_int))) ) ) )
         => aa_set6820988702716878619_int_o(P,bot_bo1488462491386950373nt_int) ) ) ) ).

% finite_empty_induct
tff(fact_8879_finite__empty__induct,axiom,
    ! [A2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(P,A2)
       => ( ! [A4: set_nat,A6: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A6)
             => ( aa_set_set_nat_o(member_set_nat(A4),A6)
               => ( aa_set_set_nat_o(P,A6)
                 => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A6,insert_set_nat(A4,bot_bot_set_set_nat))) ) ) )
         => aa_set_set_nat_o(P,bot_bot_set_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_8880_finite__empty__induct,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( aa_set5091322549519198405at_o_o(P,A2)
       => ( ! [A4: fun_Pr6460944699146269935_nat_o,A6: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A6)
             => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A4),A6)
               => ( aa_set5091322549519198405at_o_o(P,A6)
                 => aa_set5091322549519198405at_o_o(P,minus_6456595905716665654_nat_o(A6,insert4923026906511862463_nat_o(A4,bot_bo8632255151191677115_nat_o))) ) ) )
         => aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o) ) ) ) ).

% finite_empty_induct
tff(fact_8881_finite__empty__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(P,A2)
       => ( ! [A4: product_prod_nat_nat,A6: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A6)
             => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A4),A6)
               => ( aa_set3702741264012733747_nat_o(P,A6)
                 => aa_set3702741264012733747_nat_o(P,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A6),insert8211810215607154385at_nat(A4,bot_bo2099793752762293965at_nat))) ) ) )
         => aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat) ) ) ) ).

% finite_empty_induct
tff(fact_8882_finite__empty__induct,axiom,
    ! [A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(P,A2)
       => ( ! [A4: $o,A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( aa_set_o_o(member_o((A4)),A6)
               => ( aa_set_o_o(P,A6)
                 => aa_set_o_o(P,minus_minus_set_o(A6,insert_o((A4),bot_bot_set_o))) ) ) )
         => aa_set_o_o(P,bot_bot_set_o) ) ) ) ).

% finite_empty_induct
tff(fact_8883_finite__empty__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,A2)
       => ( ! [A4: int,A6: set_int] :
              ( aa_set_int_o(finite_finite_int,A6)
             => ( aa_set_int_o(member_int(A4),A6)
               => ( aa_set_int_o(P,A6)
                 => aa_set_int_o(P,minus_minus_set_int(A6,insert_int(A4,bot_bot_set_int))) ) ) )
         => aa_set_int_o(P,bot_bot_set_int) ) ) ) ).

% finite_empty_induct
tff(fact_8884_finite__empty__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,A2)
       => ( ! [A4: nat,A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),A6)
               => ( aa_set_nat_o(P,A6)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),aa_set_nat_set_nat(insert_nat(A4),bot_bot_set_nat))) ) ) )
         => aa_set_nat_o(P,bot_bot_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_8885_finite__remove__induct,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B2)
     => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
       => ( ! [A6: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A6)
             => ( ( A6 != bot_bo228742789529271731at_nat )
               => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A6),B2)
                 => ( ! [X3: produc3843707927480180839at_nat] :
                        ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),A6)
                       => aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A6),insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat))) )
                   => aa_set5844364881195198285_nat_o(P,A6) ) ) ) )
         => aa_set5844364881195198285_nat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_8886_finite__remove__induct,axiom,
    ! [B2: set_se6260736226359567993nt_int,P: fun_se9106460625052465826_int_o] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,B2)
     => ( aa_set6820988702716878619_int_o(P,bot_bo1488462491386950373nt_int)
       => ( ! [A6: set_se6260736226359567993nt_int] :
              ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,A6)
             => ( ( A6 != bot_bo1488462491386950373nt_int )
               => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A6),B2)
                 => ( ! [X3: set_Pr958786334691620121nt_int] :
                        ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),A6)
                       => aa_set6820988702716878619_int_o(P,minus_2612819937483484256nt_int(A6,insert8897473484851387113nt_int(X3,bot_bo1488462491386950373nt_int))) )
                   => aa_set6820988702716878619_int_o(P,A6) ) ) ) )
         => aa_set6820988702716878619_int_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_8887_finite__remove__induct,axiom,
    ! [B2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [A6: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A6)
             => ( ( A6 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A6),B2)
                 => ( ! [X3: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat(X3),A6)
                       => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A6,insert_set_nat(X3,bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(P,A6) ) ) ) )
         => aa_set_set_nat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_8888_finite__remove__induct,axiom,
    ! [B2: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,B2)
     => ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
       => ( ! [A6: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A6)
             => ( ( A6 != bot_bo8632255151191677115_nat_o )
               => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A6),B2)
                 => ( ! [X3: fun_Pr6460944699146269935_nat_o] :
                        ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A6)
                       => aa_set5091322549519198405at_o_o(P,minus_6456595905716665654_nat_o(A6,insert4923026906511862463_nat_o(X3,bot_bo8632255151191677115_nat_o))) )
                   => aa_set5091322549519198405at_o_o(P,A6) ) ) ) )
         => aa_set5091322549519198405at_o_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_8889_finite__remove__induct,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
     => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
       => ( ! [A6: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A6)
             => ( ( A6 != bot_bo2099793752762293965at_nat )
               => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A6),B2)
                 => ( ! [X3: product_prod_nat_nat] :
                        ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A6)
                       => aa_set3702741264012733747_nat_o(P,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A6),insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat))) )
                   => aa_set3702741264012733747_nat_o(P,A6) ) ) ) )
         => aa_set3702741264012733747_nat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_8890_finite__remove__induct,axiom,
    ! [B2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( ( A6 != bot_bot_set_o )
               => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A6),B2)
                 => ( ! [X3: $o] :
                        ( aa_set_o_o(member_o((X3)),A6)
                       => aa_set_o_o(P,minus_minus_set_o(A6,insert_o((X3),bot_bot_set_o))) )
                   => aa_set_o_o(P,A6) ) ) ) )
         => aa_set_o_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_8891_finite__remove__induct,axiom,
    ! [B2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [A6: set_int] :
              ( aa_set_int_o(finite_finite_int,A6)
             => ( ( A6 != bot_bot_set_int )
               => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A6),B2)
                 => ( ! [X3: int] :
                        ( aa_set_int_o(member_int(X3),A6)
                       => aa_set_int_o(P,minus_minus_set_int(A6,insert_int(X3,bot_bot_set_int))) )
                   => aa_set_int_o(P,A6) ) ) ) )
         => aa_set_int_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_8892_finite__remove__induct,axiom,
    ! [B2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( ( A6 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A6),B2)
                 => ( ! [X3: nat] :
                        ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A6)
                       => aa_set_nat_o(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A6) ) ) ) )
         => aa_set_nat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_8893_remove__induct,axiom,
    ! [P: fun_se7788542033685991124_nat_o,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
     => ( ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B2)
         => aa_set5844364881195198285_nat_o(P,B2) )
       => ( ! [A6: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A6)
             => ( ( A6 != bot_bo228742789529271731at_nat )
               => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A6),B2)
                 => ( ! [X3: produc3843707927480180839at_nat] :
                        ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),A6)
                       => aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A6),insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat))) )
                   => aa_set5844364881195198285_nat_o(P,A6) ) ) ) )
         => aa_set5844364881195198285_nat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_8894_remove__induct,axiom,
    ! [P: fun_se9106460625052465826_int_o,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(P,bot_bo1488462491386950373nt_int)
     => ( ( ~ aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,B2)
         => aa_set6820988702716878619_int_o(P,B2) )
       => ( ! [A6: set_se6260736226359567993nt_int] :
              ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,A6)
             => ( ( A6 != bot_bo1488462491386950373nt_int )
               => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A6),B2)
                 => ( ! [X3: set_Pr958786334691620121nt_int] :
                        ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),A6)
                       => aa_set6820988702716878619_int_o(P,minus_2612819937483484256nt_int(A6,insert8897473484851387113nt_int(X3,bot_bo1488462491386950373nt_int))) )
                   => aa_set6820988702716878619_int_o(P,A6) ) ) ) )
         => aa_set6820988702716878619_int_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_8895_remove__induct,axiom,
    ! [P: fun_set_set_nat_o,B2: set_set_nat] :
      ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
     => ( ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
         => aa_set_set_nat_o(P,B2) )
       => ( ! [A6: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A6)
             => ( ( A6 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A6),B2)
                 => ( ! [X3: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat(X3),A6)
                       => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A6,insert_set_nat(X3,bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(P,A6) ) ) ) )
         => aa_set_set_nat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_8896_remove__induct,axiom,
    ! [P: fun_se7459991829846211660at_o_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
     => ( ( ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,B2)
         => aa_set5091322549519198405at_o_o(P,B2) )
       => ( ! [A6: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A6)
             => ( ( A6 != bot_bo8632255151191677115_nat_o )
               => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A6),B2)
                 => ( ! [X3: fun_Pr6460944699146269935_nat_o] :
                        ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A6)
                       => aa_set5091322549519198405at_o_o(P,minus_6456595905716665654_nat_o(A6,insert4923026906511862463_nat_o(X3,bot_bo8632255151191677115_nat_o))) )
                   => aa_set5091322549519198405at_o_o(P,A6) ) ) ) )
         => aa_set5091322549519198405at_o_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_8897_remove__induct,axiom,
    ! [P: fun_se4782206882984618170_nat_o,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
     => ( ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
         => aa_set3702741264012733747_nat_o(P,B2) )
       => ( ! [A6: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A6)
             => ( ( A6 != bot_bo2099793752762293965at_nat )
               => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A6),B2)
                 => ( ! [X3: product_prod_nat_nat] :
                        ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A6)
                       => aa_set3702741264012733747_nat_o(P,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A6),insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat))) )
                   => aa_set3702741264012733747_nat_o(P,A6) ) ) ) )
         => aa_set3702741264012733747_nat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_8898_remove__induct,axiom,
    ! [P: fun_set_o_o,B2: set_o] :
      ( aa_set_o_o(P,bot_bot_set_o)
     => ( ( ~ aa_set_o_o(finite_finite_o,B2)
         => aa_set_o_o(P,B2) )
       => ( ! [A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( ( A6 != bot_bot_set_o )
               => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A6),B2)
                 => ( ! [X3: $o] :
                        ( aa_set_o_o(member_o((X3)),A6)
                       => aa_set_o_o(P,minus_minus_set_o(A6,insert_o((X3),bot_bot_set_o))) )
                   => aa_set_o_o(P,A6) ) ) ) )
         => aa_set_o_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_8899_remove__induct,axiom,
    ! [P: fun_set_int_o,B2: set_int] :
      ( aa_set_int_o(P,bot_bot_set_int)
     => ( ( ~ aa_set_int_o(finite_finite_int,B2)
         => aa_set_int_o(P,B2) )
       => ( ! [A6: set_int] :
              ( aa_set_int_o(finite_finite_int,A6)
             => ( ( A6 != bot_bot_set_int )
               => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A6),B2)
                 => ( ! [X3: int] :
                        ( aa_set_int_o(member_int(X3),A6)
                       => aa_set_int_o(P,minus_minus_set_int(A6,insert_int(X3,bot_bot_set_int))) )
                   => aa_set_int_o(P,A6) ) ) ) )
         => aa_set_int_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_8900_remove__induct,axiom,
    ! [P: fun_set_nat_o,B2: set_nat] :
      ( aa_set_nat_o(P,bot_bot_set_nat)
     => ( ( ~ aa_set_nat_o(finite_finite_nat,B2)
         => aa_set_nat_o(P,B2) )
       => ( ! [A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( ( A6 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A6),B2)
                 => ( ! [X3: nat] :
                        ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A6)
                       => aa_set_nat_o(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A6) ) ) ) )
         => aa_set_nat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_8901_finite__induct__select,axiom,
    ! [S: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,S)
     => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
       => ( ! [T4: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(T4),S)
             => ( aa_set5844364881195198285_nat_o(P,T4)
               => ? [X3: produc3843707927480180839at_nat] :
                    ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,S),T4))
                    & aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X3,T4)) ) ) )
         => aa_set5844364881195198285_nat_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_8902_finite__induct__select,axiom,
    ! [S: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
     => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
       => ( ! [T4: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,T4),S)
             => ( aa_set3702741264012733747_nat_o(P,T4)
               => ? [X3: product_prod_nat_nat] :
                    ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,S),T4))
                    & aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X3,T4)) ) ) )
         => aa_set3702741264012733747_nat_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_8903_finite__induct__select,axiom,
    ! [S: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [T4: set_o] :
              ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,T4),S)
             => ( aa_set_o_o(P,T4)
               => ? [X3: $o] :
                    ( aa_set_o_o(member_o((X3)),minus_minus_set_o(S,T4))
                    & aa_set_o_o(P,insert_o((X3),T4)) ) ) )
         => aa_set_o_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_8904_finite__induct__select,axiom,
    ! [S: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [T4: set_int] :
              ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,T4),S)
             => ( aa_set_int_o(P,T4)
               => ? [X3: int] :
                    ( aa_set_int_o(member_int(X3),minus_minus_set_int(S,T4))
                    & aa_set_int_o(P,insert_int(X3,T4)) ) ) )
         => aa_set_int_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_8905_finite__induct__select,axiom,
    ! [S: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [T4: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,T4),S)
             => ( aa_set_nat_o(P,T4)
               => ? [X3: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T4))
                    & aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X3),T4)) ) ) )
         => aa_set_nat_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_8906_finite__option__UNIV,axiom,
    ( finite6732403688824079472at_nat(top_to3251141154256563319at_nat)
  <=> aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,top_to4669805908274784177at_nat) ) ).

% finite_option_UNIV
tff(fact_8907_finite__option__UNIV,axiom,
    ( finite7857960318081769799st_nat(top_to1113147461230312982st_nat)
  <=> finite8100373058378681591st_nat(top_top_set_list_nat) ) ).

% finite_option_UNIV
tff(fact_8908_finite__option__UNIV,axiom,
    ( finite4093902646404507527tion_o(top_top_set_option_o)
  <=> aa_set_o_o(finite_finite_o,top_top_set_o) ) ).

% finite_option_UNIV
tff(fact_8909_finite__option__UNIV,axiom,
    ( finite2014924581280026175on_rat(top_to2540212048668676366on_rat)
  <=> aa_set_rat_o(finite_finite_rat,top_top_set_rat) ) ).

% finite_option_UNIV
tff(fact_8910_finite__option__UNIV,axiom,
    ( finite5523153139673422903on_nat(top_to8920198386146353926on_nat)
  <=> aa_set_nat_o(finite_finite_nat,top_top_set_nat) ) ).

% finite_option_UNIV
tff(fact_8911_finite__option__UNIV,axiom,
    ( finite1345302120164226195on_int(top_to6430115241214627170on_int)
  <=> aa_set_int_o(finite_finite_int,top_top_set_int) ) ).

% finite_option_UNIV
tff(fact_8912_diff__preserves__multiset,axiom,
    ! [M4: fun_se4101287982975402108nt_nat,N4: fun_se4101287982975402108nt_nat] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aTP_Lamm_aez(M4)))
     => aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aa_fun3785083049450726210_int_o(aTP_Lamm_afa(M4),N4))) ) ).

% diff_preserves_multiset
tff(fact_8913_diff__preserves__multiset,axiom,
    ! [M4: fun_set_nat_nat,N4: fun_set_nat_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_afb(M4)))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun1332161875654541722_nat_o(aTP_Lamm_afc(M4),N4))) ) ).

% diff_preserves_multiset
tff(fact_8914_diff__preserves__multiset,axiom,
    ! [M4: fun_fu3504661237974713170_o_nat,N4: fun_fu3504661237974713170_o_nat] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(aTP_Lamm_afd(M4)))
     => aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(aa_fun112519265078508482at_o_o(aTP_Lamm_afe(M4),N4))) ) ).

% diff_preserves_multiset
tff(fact_8915_diff__preserves__multiset,axiom,
    ! [M4: fun_nat_nat,N4: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_aff(M4)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_afg(M4),N4))) ) ).

% diff_preserves_multiset
tff(fact_8916_diff__preserves__multiset,axiom,
    ! [M4: fun_int_nat,N4: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_afh(M4)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_afi(M4),N4))) ) ).

% diff_preserves_multiset
tff(fact_8917_diff__preserves__multiset,axiom,
    ! [M4: fun_Pr2602117396471556196at_nat,N4: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aTP_Lamm_afj(M4)))
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun741873263805325634_nat_o(aTP_Lamm_afk(M4),N4))) ) ).

% diff_preserves_multiset
tff(fact_8918_add__mset__in__multiset,axiom,
    ! [M4: fun_se4101287982975402108nt_nat,A: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aTP_Lamm_aez(M4)))
     => aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aa_set4993530631398190431_int_o(aTP_Lamm_afl(M4),A))) ) ).

% add_mset_in_multiset
tff(fact_8919_add__mset__in__multiset,axiom,
    ! [M4: fun_set_nat_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_afb(M4)))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_afm(M4),A))) ) ).

% add_mset_in_multiset
tff(fact_8920_add__mset__in__multiset,axiom,
    ! [M4: fun_fu3504661237974713170_o_nat,A: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(aTP_Lamm_afd(M4)))
     => aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(aa_fun3380769697021226975at_o_o(aTP_Lamm_afn(M4),A))) ) ).

% add_mset_in_multiset
tff(fact_8921_add__mset__in__multiset,axiom,
    ! [M4: fun_nat_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_aff(M4)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_afo(M4),A))) ) ).

% add_mset_in_multiset
tff(fact_8922_add__mset__in__multiset,axiom,
    ! [M4: fun_int_nat,A: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_afh(M4)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_afp(M4),A))) ) ).

% add_mset_in_multiset
tff(fact_8923_add__mset__in__multiset,axiom,
    ! [M4: fun_Pr2602117396471556196at_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aTP_Lamm_afj(M4)))
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(aTP_Lamm_afq(M4),A))) ) ).

% add_mset_in_multiset
tff(fact_8924_finite__linorder__min__induct,axiom,
    ! [A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [B4: $o,A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( ! [X3: $o] :
                    ( aa_set_o_o(member_o((X3)),A6)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(B4)),(X3)) )
               => ( aa_set_o_o(P,A6)
                 => aa_set_o_o(P,insert_o((B4),A6)) ) ) )
         => aa_set_o_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8925_finite__linorder__min__induct,axiom,
    ! [A2: set_literal,P: fun_set_literal_o] :
      ( aa_set_literal_o(finite5847741373460823677iteral,A2)
     => ( aa_set_literal_o(P,bot_bot_set_literal)
       => ( ! [B4: literal,A6: set_literal] :
              ( aa_set_literal_o(finite5847741373460823677iteral,A6)
             => ( ! [X3: literal] :
                    ( aa_set_literal_o(member_literal(X3),A6)
                   => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,B4),X3) )
               => ( aa_set_literal_o(P,A6)
                 => aa_set_literal_o(P,insert_literal(B4,A6)) ) ) )
         => aa_set_literal_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8926_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B4: rat,A6: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A6)
             => ( ! [X3: rat] :
                    ( aa_set_rat_o(member_rat(X3),A6)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B4),X3) )
               => ( aa_set_rat_o(P,A6)
                 => aa_set_rat_o(P,insert_rat(B4,A6)) ) ) )
         => aa_set_rat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8927_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B4: nat,A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( ! [X3: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A6)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B4),X3) )
               => ( aa_set_nat_o(P,A6)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(B4),A6)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8928_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B4: int,A6: set_int] :
              ( aa_set_int_o(finite_finite_int,A6)
             => ( ! [X3: int] :
                    ( aa_set_int_o(member_int(X3),A6)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B4),X3) )
               => ( aa_set_int_o(P,A6)
                 => aa_set_int_o(P,insert_int(B4,A6)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8929_finite__linorder__max__induct,axiom,
    ! [A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [B4: $o,A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( ! [X3: $o] :
                    ( aa_set_o_o(member_o((X3)),A6)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(X3)),(B4)) )
               => ( aa_set_o_o(P,A6)
                 => aa_set_o_o(P,insert_o((B4),A6)) ) ) )
         => aa_set_o_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8930_finite__linorder__max__induct,axiom,
    ! [A2: set_literal,P: fun_set_literal_o] :
      ( aa_set_literal_o(finite5847741373460823677iteral,A2)
     => ( aa_set_literal_o(P,bot_bot_set_literal)
       => ( ! [B4: literal,A6: set_literal] :
              ( aa_set_literal_o(finite5847741373460823677iteral,A6)
             => ( ! [X3: literal] :
                    ( aa_set_literal_o(member_literal(X3),A6)
                   => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X3),B4) )
               => ( aa_set_literal_o(P,A6)
                 => aa_set_literal_o(P,insert_literal(B4,A6)) ) ) )
         => aa_set_literal_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8931_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B4: rat,A6: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A6)
             => ( ! [X3: rat] :
                    ( aa_set_rat_o(member_rat(X3),A6)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),B4) )
               => ( aa_set_rat_o(P,A6)
                 => aa_set_rat_o(P,insert_rat(B4,A6)) ) ) )
         => aa_set_rat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8932_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B4: nat,A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( ! [X3: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A6)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),B4) )
               => ( aa_set_nat_o(P,A6)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(B4),A6)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8933_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B4: int,A6: set_int] :
              ( aa_set_int_o(finite_finite_int,A6)
             => ( ! [X3: int] :
                    ( aa_set_int_o(member_int(X3),A6)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),B4) )
               => ( aa_set_int_o(P,A6)
                 => aa_set_int_o(P,insert_int(B4,A6)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8934_finite__ranking__induct,axiom,
    ! [S: set_o,P: fun_set_o_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X2: $o,S5: set_o] :
              ( aa_set_o_o(finite_finite_o,S5)
             => ( ! [Y4: $o] :
                    ( aa_set_o_o(member_o((Y4)),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(Y4))),aa_o_rat(F,(X2))) )
               => ( aa_set_o_o(P,S5)
                 => aa_set_o_o(P,insert_o((X2),S5)) ) ) )
         => aa_set_o_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8935_finite__ranking__induct,axiom,
    ! [S: set_nat,P: fun_set_nat_o,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X2: nat,S5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S5)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Y4)),aa_nat_rat(F,X2)) )
               => ( aa_set_nat_o(P,S5)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X2),S5)) ) ) )
         => aa_set_nat_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8936_finite__ranking__induct,axiom,
    ! [S: set_int,P: fun_set_int_o,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X2: int,S5: set_int] :
              ( aa_set_int_o(finite_finite_int,S5)
             => ( ! [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,Y4)),aa_int_rat(F,X2)) )
               => ( aa_set_int_o(P,S5)
                 => aa_set_int_o(P,insert_int(X2,S5)) ) ) )
         => aa_set_int_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8937_finite__ranking__induct,axiom,
    ! [S: set_o,P: fun_set_o_o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X2: $o,S5: set_o] :
              ( aa_set_o_o(finite_finite_o,S5)
             => ( ! [Y4: $o] :
                    ( aa_set_o_o(member_o((Y4)),S5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(Y4))),aa_o_nat(F,(X2))) )
               => ( aa_set_o_o(P,S5)
                 => aa_set_o_o(P,insert_o((X2),S5)) ) ) )
         => aa_set_o_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8938_finite__ranking__induct,axiom,
    ! [S: set_nat,P: fun_set_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X2: nat,S5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S5)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y4),S5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Y4)),aa_nat_nat(F,X2)) )
               => ( aa_set_nat_o(P,S5)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X2),S5)) ) ) )
         => aa_set_nat_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8939_finite__ranking__induct,axiom,
    ! [S: set_int,P: fun_set_int_o,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X2: int,S5: set_int] :
              ( aa_set_int_o(finite_finite_int,S5)
             => ( ! [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),S5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,Y4)),aa_int_nat(F,X2)) )
               => ( aa_set_int_o(P,S5)
                 => aa_set_int_o(P,insert_int(X2,S5)) ) ) )
         => aa_set_int_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8940_finite__ranking__induct,axiom,
    ! [S: set_o,P: fun_set_o_o,F: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X2: $o,S5: set_o] :
              ( aa_set_o_o(finite_finite_o,S5)
             => ( ! [Y4: $o] :
                    ( aa_set_o_o(member_o((Y4)),S5)
                   => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(Y4))),aa_o_int(F,(X2))) )
               => ( aa_set_o_o(P,S5)
                 => aa_set_o_o(P,insert_o((X2),S5)) ) ) )
         => aa_set_o_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8941_finite__ranking__induct,axiom,
    ! [S: set_nat,P: fun_set_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X2: nat,S5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S5)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y4),S5)
                   => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Y4)),aa_nat_int(F,X2)) )
               => ( aa_set_nat_o(P,S5)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X2),S5)) ) ) )
         => aa_set_nat_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8942_finite__ranking__induct,axiom,
    ! [S: set_int,P: fun_set_int_o,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X2: int,S5: set_int] :
              ( aa_set_int_o(finite_finite_int,S5)
             => ( ! [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),S5)
                   => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,Y4)),aa_int_int(F,X2)) )
               => ( aa_set_int_o(P,S5)
                 => aa_set_int_o(P,insert_int(X2,S5)) ) ) )
         => aa_set_int_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8943_finite__ranking__induct,axiom,
    ! [S: set_set_nat,P: fun_set_set_nat_o,F: fun_set_nat_rat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,S)
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [X2: set_nat,S5: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,S5)
             => ( ! [Y4: set_nat] :
                    ( aa_set_set_nat_o(member_set_nat(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(F,Y4)),aa_set_nat_rat(F,X2)) )
               => ( aa_set_set_nat_o(P,S5)
                 => aa_set_set_nat_o(P,insert_set_nat(X2,S5)) ) ) )
         => aa_set_set_nat_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8944_infinite__growing,axiom,
    ! [X5: set_o] :
      ( ( X5 != bot_bot_set_o )
     => ( ! [X2: $o] :
            ( aa_set_o_o(member_o((X2)),X5)
           => ? [Xa3: $o] :
                ( aa_set_o_o(member_o((Xa3)),X5)
                & aa_o_o(aa_o_fun_o_o(ord_less_o,(X2)),(Xa3)) ) )
       => ~ aa_set_o_o(finite_finite_o,X5) ) ) ).

% infinite_growing
tff(fact_8945_infinite__growing,axiom,
    ! [X5: set_literal] :
      ( ( X5 != bot_bot_set_literal )
     => ( ! [X2: literal] :
            ( aa_set_literal_o(member_literal(X2),X5)
           => ? [Xa3: literal] :
                ( aa_set_literal_o(member_literal(Xa3),X5)
                & aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X2),Xa3) ) )
       => ~ aa_set_literal_o(finite5847741373460823677iteral,X5) ) ) ).

% infinite_growing
tff(fact_8946_infinite__growing,axiom,
    ! [X5: set_rat] :
      ( ( X5 != bot_bot_set_rat )
     => ( ! [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),X5)
           => ? [Xa3: rat] :
                ( aa_set_rat_o(member_rat(Xa3),X5)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Xa3) ) )
       => ~ aa_set_rat_o(finite_finite_rat,X5) ) ) ).

% infinite_growing
tff(fact_8947_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),X5)
           => ? [Xa3: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),X5)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Xa3) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X5) ) ) ).

% infinite_growing
tff(fact_8948_infinite__growing,axiom,
    ! [X5: set_int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),X5)
           => ? [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),X5)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Xa3) ) )
       => ~ aa_set_int_o(finite_finite_int,X5) ) ) ).

% infinite_growing
tff(fact_8949_ex__min__if__finite,axiom,
    ! [S: set_o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ? [X2: $o] :
            ( aa_set_o_o(member_o((X2)),S)
            & ~ ? [Xa3: $o] :
                  ( aa_set_o_o(member_o((Xa3)),S)
                  & aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa3)),(X2)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8950_ex__min__if__finite,axiom,
    ! [S: set_literal] :
      ( aa_set_literal_o(finite5847741373460823677iteral,S)
     => ( ( S != bot_bot_set_literal )
       => ? [X2: literal] :
            ( aa_set_literal_o(member_literal(X2),S)
            & ~ ? [Xa3: literal] :
                  ( aa_set_literal_o(member_literal(Xa3),S)
                  & aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Xa3),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8951_ex__min__if__finite,axiom,
    ! [S: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S)
     => ( ( S != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),S)
            & ~ ? [Xa3: rat] :
                  ( aa_set_rat_o(member_rat(Xa3),S)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa3),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8952_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),S)
            & ~ ? [Xa3: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),S)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa3),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8953_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),S)
            & ~ ? [Xa3: int] :
                  ( aa_set_int_o(member_int(Xa3),S)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa3),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8954_filter__preserves__multiset,axiom,
    ! [M4: fun_se4101287982975402108nt_nat,P: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aTP_Lamm_aez(M4)))
     => aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_afr(M4),P))) ) ).

% filter_preserves_multiset
tff(fact_8955_filter__preserves__multiset,axiom,
    ! [M4: fun_set_nat_nat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_afb(M4)))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_afs(M4),P))) ) ).

% filter_preserves_multiset
tff(fact_8956_filter__preserves__multiset,axiom,
    ! [M4: fun_fu3504661237974713170_o_nat,P: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(aTP_Lamm_afd(M4)))
     => aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_aft(M4),P))) ) ).

% filter_preserves_multiset
tff(fact_8957_filter__preserves__multiset,axiom,
    ! [M4: fun_nat_nat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_aff(M4)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_afu(M4),P))) ) ).

% filter_preserves_multiset
tff(fact_8958_filter__preserves__multiset,axiom,
    ! [M4: fun_int_nat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_afh(M4)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_afv(M4),P))) ) ).

% filter_preserves_multiset
tff(fact_8959_filter__preserves__multiset,axiom,
    ! [M4: fun_Pr2602117396471556196at_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aTP_Lamm_afj(M4)))
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_afw(M4),P))) ) ).

% filter_preserves_multiset
tff(fact_8960_divide__int__def,axiom,
    ! [K: int,L: int] :
      ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = $ite(
          L = zero_zero_int,
          zero_zero_int,
          $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ) ).

% divide_int_def
tff(fact_8961_sum__diff1_H__aux,axiom,
    ! [F3: set_o,I4: set_o,F: fun_o_Code_integer,I: $o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,collect_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_afx(I4),F))),F3)
       => ( groups1402912129352969042nteger(F,minus_minus_set_o(I4,insert_o((I),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((I)),I4),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(groups1402912129352969042nteger(F,I4)),aa_o_Code_integer(F,(I))),groups1402912129352969042nteger(F,I4)) ) ) ) ).

% sum_diff1'_aux
tff(fact_8962_sum__diff1_H__aux,axiom,
    ! [F3: set_int,I4: set_int,F: fun_int_Code_integer,I: int] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_xq(I4),F))),F3)
       => ( groups926780983652909934nteger(F,minus_minus_set_int(I4,insert_int(I,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(I),I4),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(groups926780983652909934nteger(F,I4)),aa_int_Code_integer(F,I)),groups926780983652909934nteger(F,I4)) ) ) ) ).

% sum_diff1'_aux
tff(fact_8963_sum__diff1_H__aux,axiom,
    ! [F3: set_o,I4: set_o,F: fun_o_rat,I: $o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,collect_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_afy(I4),F))),F3)
       => ( groups3921277224699582669_o_rat(F,minus_minus_set_o(I4,insert_o((I),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((I)),I4),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups3921277224699582669_o_rat(F,I4)),aa_o_rat(F,(I))),groups3921277224699582669_o_rat(F,I4)) ) ) ) ).

% sum_diff1'_aux
tff(fact_8964_sum__diff1_H__aux,axiom,
    ! [F3: set_int,I4: set_int,F: fun_int_rat,I: int] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_xu(I4),F))),F3)
       => ( groups2350640619554545897nt_rat(F,minus_minus_set_int(I4,insert_int(I,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(I),I4),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups2350640619554545897nt_rat(F,I4)),aa_int_rat(F,I)),groups2350640619554545897nt_rat(F,I4)) ) ) ) ).

% sum_diff1'_aux
tff(fact_8965_sum__diff1_H__aux,axiom,
    ! [F3: set_o,I4: set_o,F: fun_o_int,I: $o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,collect_o(aa_fun_o_int_fun_o_o(aTP_Lamm_afz(I4),F))),F3)
       => ( groups4553916814277028129_o_int(F,minus_minus_set_o(I4,insert_o((I),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((I)),I4),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4553916814277028129_o_int(F,I4)),aa_o_int(F,(I))),groups4553916814277028129_o_int(F,I4)) ) ) ) ).

% sum_diff1'_aux
tff(fact_8966_sum__diff1_H__aux,axiom,
    ! [F3: set_int,I4: set_int,F: fun_int_int,I: int] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(aa_fun8155382727227203058_int_o(aTP_Lamm_yc(I4),F))),F3)
       => ( groups2983280209131991357nt_int(F,minus_minus_set_int(I4,insert_int(I,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(I),I4),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups2983280209131991357nt_int(F,I4)),aa_int_int(F,I)),groups2983280209131991357nt_int(F,I4)) ) ) ) ).

% sum_diff1'_aux
tff(fact_8967_sum__diff1_H__aux,axiom,
    ! [F3: set_nat,I4: set_nat,F: fun_nat_Code_integer,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_xo(I4),F))),F3)
       => ( aa_set871585252387937313nteger(groups555127423416065298nteger(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I4),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat))) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I4),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_set871585252387937313nteger(groups555127423416065298nteger(F),I4)),aa_nat_Code_integer(F,I)),aa_set871585252387937313nteger(groups555127423416065298nteger(F),I4)) ) ) ) ).

% sum_diff1'_aux
tff(fact_8968_sum__diff1_H__aux,axiom,
    ! [F3: set_nat,I4: set_nat,F: fun_nat_rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_xs(I4),F))),F3)
       => ( aa_set_nat_rat(groups1351286907653491341at_rat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I4),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat))) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I4),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_set_nat_rat(groups1351286907653491341at_rat(F),I4)),aa_nat_rat(F,I)),aa_set_nat_rat(groups1351286907653491341at_rat(F),I4)) ) ) ) ).

% sum_diff1'_aux
tff(fact_8969_sum__diff1_H__aux,axiom,
    ! [F3: set_nat,I4: set_nat,F: fun_nat_int,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun2180201864171524266_nat_o(aTP_Lamm_ya(I4),F))),F3)
       => ( aa_set_nat_int(groups1983926497230936801at_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I4),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat))) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I4),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups1983926497230936801at_int(F),I4)),aa_nat_int(F,I)),aa_set_nat_int(groups1983926497230936801at_int(F),I4)) ) ) ) ).

% sum_diff1'_aux
tff(fact_8970_sum__diff1_H__aux,axiom,
    ! [F3: set_set_nat,I4: set_set_nat,F: fun_se7586849273513198688nteger,I: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun5627232717963835469_nat_o(aTP_Lamm_ye(I4),F))),F3)
       => ( groups9221925307898054856nteger(F,minus_2163939370556025621et_nat(I4,insert_set_nat(I,bot_bot_set_set_nat))) = $ite(aa_set_set_nat_o(member_set_nat(I),I4),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(groups9221925307898054856nteger(F,I4)),aa_set871585252387937313nteger(F,I)),groups9221925307898054856nteger(F,I4)) ) ) ) ).

% sum_diff1'_aux
tff(fact_8971_even__sum__iff,axiom,
    ! [A2: set_o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),groups8507830703676809646_o_nat(F,A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),finite_card_o(collect_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_aga(A2),F)))) ) ) ).

% even_sum_iff
tff(fact_8972_even__sum__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),groups4541462559716669496nt_nat(F,A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),finite_card_int(collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_agb(A2),F)))) ) ) ).

% even_sum_iff
tff(fact_8973_even__sum__iff,axiom,
    ! [A2: set_o,F: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups8505340233167759370_o_int(F,A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),finite_card_o(collect_o(aa_fun_o_int_fun_o_o(aTP_Lamm_agc(A2),F)))) ) ) ).

% even_sum_iff
tff(fact_8974_even__sum__iff,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(F),A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_set_nat_nat(finite_card_nat,collect_nat(aa_fun2180201864171524266_nat_o(aTP_Lamm_agd(A2),F)))) ) ) ).

% even_sum_iff
tff(fact_8975_even__sum__iff,axiom,
    ! [A2: set_o,F: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups4406642042086082107nteger(F,A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),finite_card_o(collect_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_age(A2),F)))) ) ) ).

% even_sum_iff
tff(fact_8976_even__sum__iff,axiom,
    ! [A2: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_set_nat_nat(finite_card_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_agf(A2),F)))) ) ) ).

% even_sum_iff
tff(fact_8977_even__sum__iff,axiom,
    ! [A2: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7873554091576472773nteger(F,A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),finite_card_int(collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_agg(A2),F)))) ) ) ).

% even_sum_iff
tff(fact_8978_even__sum__iff,axiom,
    ! [A2: set_o,F: fun_o_Code_natural] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),groups3230237193842799622atural(F,A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),finite_card_o(collect_o(aa_fun2872501519293664346un_o_o(aTP_Lamm_agh(A2),F)))) ) ) ).

% even_sum_iff
tff(fact_8979_even__sum__iff,axiom,
    ! [A2: set_nat,F: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),groups6325495683096345652atural(F,A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_set_nat_nat(finite_card_nat,collect_nat(aa_fun440078198722963374_nat_o(aTP_Lamm_agi(A2),F)))) ) ) ).

% even_sum_iff
tff(fact_8980_even__sum__iff,axiom,
    ! [A2: set_int,F: fun_int_Code_natural] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),groups6697149243333190288atural(F,A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),finite_card_int(collect_int(aa_fun296549085072024822_int_o(aTP_Lamm_agj(A2),F)))) ) ) ).

% even_sum_iff
tff(fact_8981_nat__not__finite,axiom,
    ~ aa_set_nat_o(finite_finite_nat,top_top_set_nat) ).

% nat_not_finite
tff(fact_8982_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_8983_card__Collect__less__nat,axiom,
    ! [N: nat] : ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gg,N))) = N ) ).

% card_Collect_less_nat
tff(fact_8984_card__eq__UNIV,axiom,
    ! [S: set_o] :
      ( ( finite_card_o(S) = finite_card_o(top_top_set_o) )
    <=> ( S = top_top_set_o ) ) ).

% card_eq_UNIV
tff(fact_8985_card__eq__UNIV2,axiom,
    ! [S: set_o] :
      ( ( finite_card_o(top_top_set_o) = finite_card_o(S) )
    <=> ( S = top_top_set_o ) ) ).

% card_eq_UNIV2
tff(fact_8986_card__Collect__le__nat,axiom,
    ! [N: nat] : ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gf,N))) = aa_nat_nat(suc,N) ) ).

% card_Collect_le_nat
tff(fact_8987_card__UNIV__bool,axiom,
    finite_card_o(top_top_set_o) = numeral_numeral_nat(aa_num_num(bit0,one)) ).

% card_UNIV_bool
tff(fact_8988_card_Oempty,axiom,
    finite_card_list_nat(bot_bot_set_list_nat) = zero_zero_nat ).

% card.empty
tff(fact_8989_card_Oempty,axiom,
    finite_card_set_nat(bot_bot_set_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_8990_card_Oempty,axiom,
    finite711546835091564841at_nat(bot_bo2099793752762293965at_nat) = zero_zero_nat ).

% card.empty
tff(fact_8991_card_Oempty,axiom,
    finite_card_o(bot_bot_set_o) = zero_zero_nat ).

% card.empty
tff(fact_8992_card_Oempty,axiom,
    aa_set_nat_nat(finite_card_nat,bot_bot_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_8993_card_Oempty,axiom,
    finite_card_int(bot_bot_set_int) = zero_zero_nat ).

% card.empty
tff(fact_8994_card__ge__UNIV,axiom,
    ! [S: set_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(top_top_set_o)),finite_card_o(S))
    <=> ( S = top_top_set_o ) ) ).

% card_ge_UNIV
tff(fact_8995_sum_Oempty_H,axiom,
    ! [P3: fun_o_Code_integer] : ( groups1402912129352969042nteger(P3,bot_bot_set_o) = zero_z3403309356797280102nteger ) ).

% sum.empty'
tff(fact_8996_sum_Oempty_H,axiom,
    ! [P3: fun_o_rat] : ( groups3921277224699582669_o_rat(P3,bot_bot_set_o) = zero_zero_rat ) ).

% sum.empty'
tff(fact_8997_sum_Oempty_H,axiom,
    ! [P3: fun_o_nat] : ( groups4556407284786078405_o_nat(P3,bot_bot_set_o) = zero_zero_nat ) ).

% sum.empty'
tff(fact_8998_sum_Oempty_H,axiom,
    ! [P3: fun_o_int] : ( groups4553916814277028129_o_int(P3,bot_bot_set_o) = zero_zero_int ) ).

% sum.empty'
tff(fact_8999_sum_Oempty_H,axiom,
    ! [P3: fun_nat_Code_integer] : ( aa_set871585252387937313nteger(groups555127423416065298nteger(P3),bot_bot_set_nat) = zero_z3403309356797280102nteger ) ).

% sum.empty'
tff(fact_9000_sum_Oempty_H,axiom,
    ! [P3: fun_nat_rat] : ( aa_set_nat_rat(groups1351286907653491341at_rat(P3),bot_bot_set_nat) = zero_zero_rat ) ).

% sum.empty'
tff(fact_9001_sum_Oempty_H,axiom,
    ! [P3: fun_nat_nat] : ( aa_set_nat_nat(groups1986416967739987077at_nat(P3),bot_bot_set_nat) = zero_zero_nat ) ).

% sum.empty'
tff(fact_9002_sum_Oempty_H,axiom,
    ! [P3: fun_nat_int] : ( aa_set_nat_int(groups1983926497230936801at_int(P3),bot_bot_set_nat) = zero_zero_int ) ).

% sum.empty'
tff(fact_9003_sum_Oempty_H,axiom,
    ! [P3: fun_int_Code_integer] : ( groups926780983652909934nteger(P3,bot_bot_set_int) = zero_z3403309356797280102nteger ) ).

% sum.empty'
tff(fact_9004_sum_Oempty_H,axiom,
    ! [P3: fun_int_rat] : ( groups2350640619554545897nt_rat(P3,bot_bot_set_int) = zero_zero_rat ) ).

% sum.empty'
tff(fact_9005_prod__constant,axiom,
    ! [Y: int,A2: set_list_nat] : ( groups2905156660866384563at_int(aTP_Lamm_ew(Y),A2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),finite_card_list_nat(A2)) ) ).

% prod_constant
tff(fact_9006_prod__constant,axiom,
    ! [Y: int,A2: set_o] : ( groups3502327434004483295_o_int(aTP_Lamm_eo(Y),A2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),finite_card_o(A2)) ) ).

% prod_constant
tff(fact_9007_prod__constant,axiom,
    ! [Y: int,A2: set_set_nat] : ( groups4246057289670975065at_int(aa_int2448287375063835929at_int(aTP_Lamm_agk,Y),A2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),finite_card_set_nat(A2)) ) ).

% prod_constant
tff(fact_9008_prod__constant,axiom,
    ! [Y: nat,A2: set_list_nat] : ( groups2907647131375434839at_nat(aTP_Lamm_ev(Y),A2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),finite_card_list_nat(A2)) ) ).

% prod_constant
tff(fact_9009_prod__constant,axiom,
    ! [Y: nat,A2: set_o] : ( groups3504817904513533571_o_nat(aTP_Lamm_en(Y),A2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),finite_card_o(A2)) ) ).

% prod_constant
tff(fact_9010_prod__constant,axiom,
    ! [Y: nat,A2: set_set_nat] : ( groups4248547760180025341at_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_agl,Y),A2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),finite_card_set_nat(A2)) ) ).

% prod_constant
tff(fact_9011_prod__constant,axiom,
    ! [Y: nat,A2: set_int] : ( groups1707563613775114915nt_nat(aa_nat_fun_int_nat(aTP_Lamm_et,Y),A2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),finite_card_int(A2)) ) ).

% prod_constant
tff(fact_9012_prod__constant,axiom,
    ! [Y: nat,A2: set_nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_er,Y)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_set_nat_nat(finite_card_nat,A2)) ) ).

% prod_constant
tff(fact_9013_prod__constant,axiom,
    ! [Y: int,A2: set_nat] : ( aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_es,Y)),A2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_set_nat_nat(finite_card_nat,A2)) ) ).

% prod_constant
tff(fact_9014_prod__constant,axiom,
    ! [Y: int,A2: set_int] : ( groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_eu,Y),A2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),finite_card_int(A2)) ) ).

% prod_constant
tff(fact_9015_card__0__eq,axiom,
    ! [A2: set_list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( ( finite_card_list_nat(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_list_nat ) ) ) ).

% card_0_eq
tff(fact_9016_card__0__eq,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( finite_card_set_nat(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_set_nat ) ) ) ).

% card_0_eq
tff(fact_9017_card__0__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( ( finite711546835091564841at_nat(A2) = zero_zero_nat )
      <=> ( A2 = bot_bo2099793752762293965at_nat ) ) ) ).

% card_0_eq
tff(fact_9018_card__0__eq,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( finite_card_o(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_o ) ) ) ).

% card_0_eq
tff(fact_9019_card__0__eq,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_set_nat_nat(finite_card_nat,A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_nat ) ) ) ).

% card_0_eq
tff(fact_9020_card__0__eq,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( finite_card_int(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_int ) ) ) ).

% card_0_eq
tff(fact_9021_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_9022_nat__neg__numeral,axiom,
    ! [K: num] : ( aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ) ).

% nat_neg_numeral
tff(fact_9023_nat__zminus__int,axiom,
    ! [N: nat] : ( aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_nat ) ).

% nat_zminus_int
tff(fact_9024_sum__constant,axiom,
    ! [Y: code_integer,A2: set_o] : ( groups4406642042086082107nteger(aTP_Lamm_agm(Y),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_o(A2))),Y) ) ).

% sum_constant
tff(fact_9025_sum__constant,axiom,
    ! [Y: code_integer,A2: set_nat] : ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_agn(Y)),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(finite_card_nat,A2))),Y) ) ).

% sum_constant
tff(fact_9026_sum__constant,axiom,
    ! [Y: code_integer,A2: set_int] : ( groups7873554091576472773nteger(aTP_Lamm_ago(Y),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_int(A2))),Y) ) ).

% sum_constant
tff(fact_9027_sum__constant,axiom,
    ! [Y: int,A2: set_o] : ( groups8505340233167759370_o_int(aTP_Lamm_eo(Y),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_o(A2))),Y) ) ).

% sum_constant
tff(fact_9028_sum__constant,axiom,
    ! [Y: int,A2: set_nat] : ( aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_es,Y)),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,A2))),Y) ) ).

% sum_constant
tff(fact_9029_sum__constant,axiom,
    ! [Y: nat,A2: set_o] : ( groups8507830703676809646_o_nat(aTP_Lamm_en(Y),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_o(A2))),Y) ) ).

% sum_constant
tff(fact_9030_sum__constant,axiom,
    ! [Y: nat,A2: set_int] : ( groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_et,Y),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_int(A2))),Y) ) ).

% sum_constant
tff(fact_9031_sum__constant,axiom,
    ! [Y: rat,A2: set_o] : ( groups7872700643590313910_o_rat(aTP_Lamm_agp(Y),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_o(A2))),Y) ) ).

% sum_constant
tff(fact_9032_sum__constant,axiom,
    ! [Y: rat,A2: set_nat] : ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_agq(Y)),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A2))),Y) ) ).

% sum_constant
tff(fact_9033_sum__constant,axiom,
    ! [Y: rat,A2: set_int] : ( groups3906332499630173760nt_rat(aTP_Lamm_agr(Y),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A2))),Y) ) ).

% sum_constant
tff(fact_9034_card__Diff__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A2)
     => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B2)
       => ( finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),insert9069300056098147895at_nat(A,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A2),B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_9035_card__Diff__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2)
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B2)
       => ( finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),insert8211810215607154385at_nat(A,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_9036_card__Diff__insert,axiom,
    ! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A2)
     => ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),B2)
       => ( finite4053189226111446337nt_int(minus_2612819937483484256nt_int(A2,insert8897473484851387113nt_int(A,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite4053189226111446337nt_int(minus_2612819937483484256nt_int(A2,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_9037_card__Diff__insert,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),B2)
       => ( finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(A,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A2,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_9038_card__Diff__insert,axiom,
    ! [A: list_nat,A2: set_list_nat,B2: set_list_nat] :
      ( member_list_nat(A,A2)
     => ( ~ member_list_nat(A,B2)
       => ( finite_card_list_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(A,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A2,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_9039_card__Diff__insert,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ( ~ aa_set_o_o(member_o((A)),B2)
       => ( finite_card_o(minus_minus_set_o(A2,insert_o((A),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_o(minus_minus_set_o(A2,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_9040_card__Diff__insert,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ( ~ aa_set_set_nat_o(member_set_nat(A),B2)
       => ( finite_card_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(A,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A2,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_9041_card__Diff__insert,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( ~ aa_set_int_o(member_int(A),B2)
       => ( finite_card_int(minus_minus_set_int(A2,insert_int(A,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(minus_minus_set_int(A2,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_9042_card__Diff__insert,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A2)
     => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),B2)
       => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_9043_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ) ).

% card_atLeastAtMost_int
tff(fact_9044_sum_Oinsert_H,axiom,
    ! [I4: set_o,P3: fun_o_Code_integer,I: $o] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_afx(I4),P3)))
     => ( groups1402912129352969042nteger(P3,insert_o((I),I4)) = $ite(aa_set_o_o(member_o((I)),I4),groups1402912129352969042nteger(P3,I4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(P3,(I))),groups1402912129352969042nteger(P3,I4))) ) ) ).

% sum.insert'
tff(fact_9045_sum_Oinsert_H,axiom,
    ! [I4: set_nat,P3: fun_nat_Code_integer,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_xo(I4),P3)))
     => ( aa_set871585252387937313nteger(groups555127423416065298nteger(P3),aa_set_nat_set_nat(insert_nat(I),I4)) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I4),aa_set871585252387937313nteger(groups555127423416065298nteger(P3),I4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(P3,I)),aa_set871585252387937313nteger(groups555127423416065298nteger(P3),I4))) ) ) ).

% sum.insert'
tff(fact_9046_sum_Oinsert_H,axiom,
    ! [I4: set_int,P3: fun_int_Code_integer,I: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_xq(I4),P3)))
     => ( groups926780983652909934nteger(P3,insert_int(I,I4)) = $ite(aa_set_int_o(member_int(I),I4),groups926780983652909934nteger(P3,I4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_int_Code_integer(P3,I)),groups926780983652909934nteger(P3,I4))) ) ) ).

% sum.insert'
tff(fact_9047_sum_Oinsert_H,axiom,
    ! [I4: set_o,P3: fun_o_rat,I: $o] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_afy(I4),P3)))
     => ( groups3921277224699582669_o_rat(P3,insert_o((I),I4)) = $ite(aa_set_o_o(member_o((I)),I4),groups3921277224699582669_o_rat(P3,I4),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(P3,(I))),groups3921277224699582669_o_rat(P3,I4))) ) ) ).

% sum.insert'
tff(fact_9048_sum_Oinsert_H,axiom,
    ! [I4: set_nat,P3: fun_nat_rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_xs(I4),P3)))
     => ( aa_set_nat_rat(groups1351286907653491341at_rat(P3),aa_set_nat_set_nat(insert_nat(I),I4)) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I4),aa_set_nat_rat(groups1351286907653491341at_rat(P3),I4),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(P3,I)),aa_set_nat_rat(groups1351286907653491341at_rat(P3),I4))) ) ) ).

% sum.insert'
tff(fact_9049_sum_Oinsert_H,axiom,
    ! [I4: set_int,P3: fun_int_rat,I: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_xu(I4),P3)))
     => ( groups2350640619554545897nt_rat(P3,insert_int(I,I4)) = $ite(aa_set_int_o(member_int(I),I4),groups2350640619554545897nt_rat(P3,I4),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(P3,I)),groups2350640619554545897nt_rat(P3,I4))) ) ) ).

% sum.insert'
tff(fact_9050_sum_Oinsert_H,axiom,
    ! [I4: set_o,P3: fun_o_nat,I: $o] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_ags(I4),P3)))
     => ( groups4556407284786078405_o_nat(P3,insert_o((I),I4)) = $ite(aa_set_o_o(member_o((I)),I4),groups4556407284786078405_o_nat(P3,I4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(P3,(I))),groups4556407284786078405_o_nat(P3,I4))) ) ) ).

% sum.insert'
tff(fact_9051_sum_Oinsert_H,axiom,
    ! [I4: set_nat,P3: fun_nat_nat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_xw(I4),P3)))
     => ( aa_set_nat_nat(groups1986416967739987077at_nat(P3),aa_set_nat_set_nat(insert_nat(I),I4)) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I4),aa_set_nat_nat(groups1986416967739987077at_nat(P3),I4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(P3,I)),aa_set_nat_nat(groups1986416967739987077at_nat(P3),I4))) ) ) ).

% sum.insert'
tff(fact_9052_sum_Oinsert_H,axiom,
    ! [I4: set_int,P3: fun_int_nat,I: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_xy(I4),P3)))
     => ( groups2985770679641041633nt_nat(P3,insert_int(I,I4)) = $ite(aa_set_int_o(member_int(I),I4),groups2985770679641041633nt_nat(P3,I4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(P3,I)),groups2985770679641041633nt_nat(P3,I4))) ) ) ).

% sum.insert'
tff(fact_9053_sum_Oinsert_H,axiom,
    ! [I4: set_o,P3: fun_o_int,I: $o] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_int_fun_o_o(aTP_Lamm_afz(I4),P3)))
     => ( groups4553916814277028129_o_int(P3,insert_o((I),I4)) = $ite(aa_set_o_o(member_o((I)),I4),groups4553916814277028129_o_int(P3,I4),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(P3,(I))),groups4553916814277028129_o_int(P3,I4))) ) ) ).

% sum.insert'
tff(fact_9054_card__doubleton__eq__2__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( finite3771342082235030671at_nat(insert9069300056098147895at_nat(A,insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat))) = numeral_numeral_nat(aa_num_num(bit0,one)) )
    <=> ( A != B ) ) ).

% card_doubleton_eq_2_iff
tff(fact_9055_card__doubleton__eq__2__iff,axiom,
    ! [A: list_nat,B: list_nat] :
      ( ( finite_card_list_nat(insert_list_nat(A,insert_list_nat(B,bot_bot_set_list_nat))) = numeral_numeral_nat(aa_num_num(bit0,one)) )
    <=> ( A != B ) ) ).

% card_doubleton_eq_2_iff
tff(fact_9056_card__doubleton__eq__2__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_card_set_nat(insert_set_nat(A,insert_set_nat(B,bot_bot_set_set_nat))) = numeral_numeral_nat(aa_num_num(bit0,one)) )
    <=> ( A != B ) ) ).

% card_doubleton_eq_2_iff
tff(fact_9057_card__doubleton__eq__2__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( finite711546835091564841at_nat(insert8211810215607154385at_nat(A,insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat))) = numeral_numeral_nat(aa_num_num(bit0,one)) )
    <=> ( A != B ) ) ).

% card_doubleton_eq_2_iff
tff(fact_9058_card__doubleton__eq__2__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( finite_card_o(insert_o((A),insert_o((B),bot_bot_set_o))) = numeral_numeral_nat(aa_num_num(bit0,one)) )
    <=> ~ ( (A)
        <=> (B) ) ) ).

% card_doubleton_eq_2_iff
tff(fact_9059_card__doubleton__eq__2__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat))) = numeral_numeral_nat(aa_num_num(bit0,one)) )
    <=> ( A != B ) ) ).

% card_doubleton_eq_2_iff
tff(fact_9060_card__doubleton__eq__2__iff,axiom,
    ! [A: int,B: int] :
      ( ( finite_card_int(insert_int(A,insert_int(B,bot_bot_set_int))) = numeral_numeral_nat(aa_num_num(bit0,one)) )
    <=> ( A != B ) ) ).

% card_doubleton_eq_2_iff
tff(fact_9061_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_9062_sum__of__bool__eq,axiom,
    ! [A2: set_o,P: fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,A2)
       => ( groups7872700643590313910_o_rat(aTP_Lamm_agt(P),A2) = aa_nat_rat(semiri681578069525770553at_rat,finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),collect_o(P)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9063_sum__of__bool__eq,axiom,
    ! [A2: set_int,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups3906332499630173760nt_rat(aTP_Lamm_agu(P),A2) = aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9064_sum__of__bool__eq,axiom,
    ! [A2: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_agv(P)),A2) = aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9065_sum__of__bool__eq,axiom,
    ! [A2: set_o,P: fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,A2)
       => ( groups8505340233167759370_o_int(aTP_Lamm_agw(P),A2) = aa_nat_int(semiri1314217659103216013at_int,finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),collect_o(P)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9066_sum__of__bool__eq,axiom,
    ! [A2: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_agx(P)),A2) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9067_sum__of__bool__eq,axiom,
    ! [A2: set_o,P: fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,A2)
       => ( groups8507830703676809646_o_nat(aTP_Lamm_agy(P),A2) = aa_nat_nat(semiri1316708129612266289at_nat,finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),collect_o(P)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9068_sum__of__bool__eq,axiom,
    ! [A2: set_int,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups4541462559716669496nt_nat(aTP_Lamm_agz(P),A2) = aa_nat_nat(semiri1316708129612266289at_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9069_sum__of__bool__eq,axiom,
    ! [A2: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aha(P)),A2) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9070_sum__of__bool__eq,axiom,
    ! [A2: set_int,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups4538972089207619220nt_int(aTP_Lamm_ahb(P),A2) = aa_nat_int(semiri1314217659103216013at_int,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9071_sum__of__bool__eq,axiom,
    ! [A2: set_list_nat,P: fun_list_nat_o] :
      ( finite8100373058378681591st_nat(A2)
     => ( finite8100373058378681591st_nat(A2)
       => ( groups3760926236672600436at_rat(aTP_Lamm_ahc(P),A2) = aa_nat_rat(semiri681578069525770553at_rat,finite_card_list_nat(aa_set1363491528093000170st_nat(aa_set1164110704340113021st_nat(inf_inf_set_list_nat,A2),collect_list_nat(P)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9072_nat__numeral__diff__1,axiom,
    ! [V: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,numeral_numeral_nat(V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),one_one_int)) ) ).

% nat_numeral_diff_1
tff(fact_9073_n__subsets,axiom,
    ! [A2: set_Pr958786334691620121nt_int,K: nat] :
      ( aa_set6170482117830564987_int_o(finite2998713641127702882nt_int,A2)
     => ( finite4053189226111446337nt_int(collec5210948495886036740nt_int(aa_nat4194932087953772816_int_o(aTP_Lamm_ahd(A2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_set4753215920113914285nt_nat(finite6756421564338198497nt_int,A2)),K) ) ) ).

% n_subsets
tff(fact_9074_n__subsets,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( finite2364142230527598318st_nat(collect_set_list_nat(aa_nat7721107121239158477_nat_o(aTP_Lamm_ahe(A2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_list_nat(A2)),K) ) ) ).

% n_subsets
tff(fact_9075_n__subsets,axiom,
    ! [A2: set_o,K: nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( finite_card_set_o(collect_set_o(aa_nat_fun_set_o_o(aTP_Lamm_ahf(A2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_o(A2)),K) ) ) ).

% n_subsets
tff(fact_9076_n__subsets,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( finite1149291290879098388et_nat(collect_set_set_nat(aa_nat5160323483289836711_nat_o(aTP_Lamm_ahg(A2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_set_nat(A2)),K) ) ) ).

% n_subsets
tff(fact_9077_n__subsets,axiom,
    ! [A2: set_int,K: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( finite_card_set_int(collect_set_int(aa_nat_fun_set_int_o(aTP_Lamm_ahh(A2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_int(A2)),K) ) ) ).

% n_subsets
tff(fact_9078_n__subsets,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,K: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( finite4356350796350151305at_nat(collec5514110066124741708at_nat(aa_nat1727191234135941576_nat_o(aTP_Lamm_ahi(A2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite711546835091564841at_nat(A2)),K) ) ) ).

% n_subsets
tff(fact_9079_n__subsets,axiom,
    ! [A2: set_nat,K: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_ahj(A2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_set_nat_nat(finite_card_nat,A2)),K) ) ) ).

% n_subsets
tff(fact_9080_sum_Onon__neutral_H,axiom,
    ! [G: fun_nat_Code_integer,I4: set_nat] : ( aa_set871585252387937313nteger(groups555127423416065298nteger(G),collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ahk(G),I4))) = aa_set871585252387937313nteger(groups555127423416065298nteger(G),I4) ) ).

% sum.non_neutral'
tff(fact_9081_sum_Onon__neutral_H,axiom,
    ! [G: fun_int_Code_integer,I4: set_int] : ( groups926780983652909934nteger(G,collect_int(aa_set_int_fun_int_o(aTP_Lamm_ahl(G),I4))) = groups926780983652909934nteger(G,I4) ) ).

% sum.non_neutral'
tff(fact_9082_sum_Onon__neutral_H,axiom,
    ! [G: fun_nat_rat,I4: set_nat] : ( aa_set_nat_rat(groups1351286907653491341at_rat(G),collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ahm(G),I4))) = aa_set_nat_rat(groups1351286907653491341at_rat(G),I4) ) ).

% sum.non_neutral'
tff(fact_9083_sum_Onon__neutral_H,axiom,
    ! [G: fun_int_rat,I4: set_int] : ( groups2350640619554545897nt_rat(G,collect_int(aa_set_int_fun_int_o(aTP_Lamm_ahn(G),I4))) = groups2350640619554545897nt_rat(G,I4) ) ).

% sum.non_neutral'
tff(fact_9084_sum_Onon__neutral_H,axiom,
    ! [G: fun_nat_nat,I4: set_nat] : ( aa_set_nat_nat(groups1986416967739987077at_nat(G),collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_aho(G),I4))) = aa_set_nat_nat(groups1986416967739987077at_nat(G),I4) ) ).

% sum.non_neutral'
tff(fact_9085_sum_Onon__neutral_H,axiom,
    ! [G: fun_int_nat,I4: set_int] : ( groups2985770679641041633nt_nat(G,collect_int(aa_set_int_fun_int_o(aTP_Lamm_ahp(G),I4))) = groups2985770679641041633nt_nat(G,I4) ) ).

% sum.non_neutral'
tff(fact_9086_sum_Onon__neutral_H,axiom,
    ! [G: fun_nat_int,I4: set_nat] : ( aa_set_nat_int(groups1983926497230936801at_int(G),collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ahq(G),I4))) = aa_set_nat_int(groups1983926497230936801at_int(G),I4) ) ).

% sum.non_neutral'
tff(fact_9087_sum_Onon__neutral_H,axiom,
    ! [G: fun_int_int,I4: set_int] : ( groups2983280209131991357nt_int(G,collect_int(aa_set_int_fun_int_o(aTP_Lamm_ahr(G),I4))) = groups2983280209131991357nt_int(G,I4) ) ).

% sum.non_neutral'
tff(fact_9088_sum_Onon__neutral_H,axiom,
    ! [G: fun_se7586849273513198688nteger,I4: set_set_nat] : ( groups9221925307898054856nteger(G,collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_ahs(G),I4))) = groups9221925307898054856nteger(G,I4) ) ).

% sum.non_neutral'
tff(fact_9089_sum_Onon__neutral_H,axiom,
    ! [G: fun_set_nat_rat,I4: set_set_nat] : ( groups713094340420062787at_rat(G,collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_aht(G),I4))) = groups713094340420062787at_rat(G,I4) ) ).

% sum.non_neutral'
tff(fact_9090_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,top_top_set_set_nat)
     => ( ( finite_card_set_nat(A2) = finite_card_set_nat(top_top_set_set_nat) )
       => ( A2 = top_top_set_set_nat ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
tff(fact_9091_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,top_to4669805908274784177at_nat)
     => ( ( finite711546835091564841at_nat(A2) = finite711546835091564841at_nat(top_to4669805908274784177at_nat) )
       => ( A2 = top_to4669805908274784177at_nat ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
tff(fact_9092_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A2: set_list_nat] :
      ( finite8100373058378681591st_nat(top_top_set_list_nat)
     => ( ( finite_card_list_nat(A2) = finite_card_list_nat(top_top_set_list_nat) )
       => ( A2 = top_top_set_list_nat ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
tff(fact_9093_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,top_top_set_o)
     => ( ( finite_card_o(A2) = finite_card_o(top_top_set_o) )
       => ( A2 = top_top_set_o ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
tff(fact_9094_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
     => ( ( finite_card_rat(A2) = finite_card_rat(top_top_set_rat) )
       => ( A2 = top_top_set_rat ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
tff(fact_9095_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,top_top_set_nat)
     => ( ( aa_set_nat_nat(finite_card_nat,A2) = aa_set_nat_nat(finite_card_nat,top_top_set_nat) )
       => ( A2 = top_top_set_nat ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
tff(fact_9096_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,top_top_set_int)
     => ( ( finite_card_int(A2) = finite_card_int(top_top_set_int) )
       => ( A2 = top_top_set_int ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
tff(fact_9097_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_9098_sum__multicount__gen,axiom,
    ! [S2: set_o,T2: set_nat,R: fun_o_fun_nat_o,K: fun_nat_nat] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),T2)
             => ( finite_card_o(collect_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_ahu(S2),R),X2))) = aa_nat_nat(K,X2) ) )
         => ( groups8507830703676809646_o_nat(aa_fun6229755271245457521_o_nat(aTP_Lamm_ahw(T2),R),S2) = aa_set_nat_nat(groups3542108847815614940at_nat(K),T2) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9099_sum__multicount__gen,axiom,
    ! [S2: set_int,T2: set_nat,R: fun_int_fun_nat_o,K: fun_nat_nat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),T2)
             => ( finite_card_int(collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_vb(S2),R),X2))) = aa_nat_nat(K,X2) ) )
         => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aTP_Lamm_ahx(T2),R),S2) = aa_set_nat_nat(groups3542108847815614940at_nat(K),T2) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9100_sum__multicount__gen,axiom,
    ! [S2: set_nat,T2: set_o,R: fun_nat_fun_o_o,K: fun_o_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_o_o(finite_finite_o,T2)
       => ( ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),T2)
             => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_o_fun_nat_o(aa_fun2937962278134185242_nat_o(aTP_Lamm_ahy(S2),R),(X2)))) = aa_o_nat(K,(X2)) ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4178193321845537899at_nat(aTP_Lamm_aia(T2),R)),S2) = groups8507830703676809646_o_nat(K,T2) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9101_sum__multicount__gen,axiom,
    ! [S2: set_nat,T2: set_int,R: fun_nat_fun_int_o,K: fun_int_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_int_o(finite_finite_int,T2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),T2)
             => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_vf(S2),R),X2))) = aa_int_nat(K,X2) ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8666973529875159599at_nat(aTP_Lamm_aib(T2),R)),S2) = groups4541462559716669496nt_nat(K,T2) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9102_sum__multicount__gen,axiom,
    ! [S2: set_nat,T2: set_nat,R: fun_nat_fun_nat_o,K: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),T2)
             => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_vm(S2),R),X2))) = aa_nat_nat(K,X2) ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8857679445107423627at_nat(aTP_Lamm_aic(T2),R)),S2) = aa_set_nat_nat(groups3542108847815614940at_nat(K),T2) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9103_sum__multicount__gen,axiom,
    ! [S2: set_list_nat,T2: set_nat,R: fun_li2421049240157457050_nat_o,K: fun_nat_nat] :
      ( finite8100373058378681591st_nat(S2)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),T2)
             => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aa_fun1865438607270172220_nat_o(aTP_Lamm_aid(S2),R),X2))) = aa_nat_nat(K,X2) ) )
         => ( groups4396056296759096172at_nat(aa_fun8348502103087546635at_nat(aTP_Lamm_aif(T2),R),S2) = aa_set_nat_nat(groups3542108847815614940at_nat(K),T2) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9104_sum__multicount__gen,axiom,
    ! [S2: set_set_nat,T2: set_nat,R: fun_se382663106265794036_nat_o,K: fun_nat_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,S2)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),T2)
             => ( finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aa_fun9105140013458882876_nat_o(aTP_Lamm_vx(S2),R),X2))) = aa_nat_nat(K,X2) ) )
         => ( groups8294997508430121362at_nat(aa_fun6238879321244136459at_nat(aTP_Lamm_aig(T2),R),S2) = aa_set_nat_nat(groups3542108847815614940at_nat(K),T2) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9105_sum__multicount__gen,axiom,
    ! [S2: set_nat,T2: set_list_nat,R: fun_na7234455742697417626_nat_o,K: fun_list_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( finite8100373058378681591st_nat(T2)
       => ( ! [X2: list_nat] :
              ( member_list_nat(X2,T2)
             => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_lis1686035672126679939_nat_o(aa_fun1932512389758407740_nat_o(aTP_Lamm_aih(S2),R),X2))) = aa_list_nat_nat(K,X2) ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5158605793559895163at_nat(aTP_Lamm_aij(T2),R)),S2) = groups4396056296759096172at_nat(K,T2) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9106_sum__multicount__gen,axiom,
    ! [S2: set_nat,T2: set_set_nat,R: fun_na8563630686737515764_nat_o,K: fun_set_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,T2)
       => ( ! [X2: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(X2),T2)
             => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_set_nat_fun_nat_o(aa_fun2835368149466334012_nat_o(aTP_Lamm_wh(S2),R),X2))) = aa_set_nat_nat(K,X2) ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7187098123162309461at_nat(aTP_Lamm_aik(T2),R)),S2) = groups8294997508430121362at_nat(K,T2) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9107_sum__multicount__gen,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,T2: set_nat,R: fun_Pr6984107991879650211_nat_o,K: fun_nat_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S2)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),T2)
             => ( finite711546835091564841at_nat(collec3392354462482085612at_nat(aa_nat7695548938202568488_nat_o(aa_fun2489310642977492936_nat_o(aTP_Lamm_aex(S2),R),X2))) = aa_nat_nat(K,X2) ) )
         => ( groups977919841031483927at_nat(aa_fun2860604400581938115at_nat(aTP_Lamm_aim(T2),R),S2) = aa_set_nat_nat(groups3542108847815614940at_nat(K),T2) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9108_card__less,axiom,
    ! [M4: set_nat,I: nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),M4)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ain(M4),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9109_card__less__Suc,axiom,
    ! [M4: set_nat,I: nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),M4)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aio(M4),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ain(M4),I))) ) ) ).

% card_less_Suc
tff(fact_9110_card__less__Suc2,axiom,
    ! [M4: set_nat,I: nat] :
      ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),M4)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aio(M4),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ain(M4),I))) ) ) ).

% card_less_Suc2
tff(fact_9111_nat__plus__as__int,axiom,
    ! [X3: nat,Xa3: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ) ).

% nat_plus_as_int
tff(fact_9112_nat__times__as__int,axiom,
    ! [X3: nat,Xa3: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ) ).

% nat_times_as_int
tff(fact_9113_nat__minus__as__int,axiom,
    ! [X3: nat,Xa3: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ) ).

% nat_minus_as_int
tff(fact_9114_nat__div__as__int,axiom,
    ! [X3: nat,Xa3: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ) ).

% nat_div_as_int
tff(fact_9115_nat__mod__as__int,axiom,
    ! [X3: nat,Xa3: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ) ).

% nat_mod_as_int
tff(fact_9116_Suc__as__int,axiom,
    ! [X3: nat] : ( aa_nat_nat(suc,X3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),one_one_int)) ) ).

% Suc_as_int
tff(fact_9117_card__sum__le__nat__sum,axiom,
    ! [S: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_da),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_set_nat_nat(finite_card_nat,S)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_da),S)) ).

% card_sum_le_nat_sum
tff(fact_9118_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_9119_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z6))) ) ) ).

% nat_mult_distrib_neg
tff(fact_9120_diff__nat__eq__if,axiom,
    ! [Z: int,Z6: int] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) = $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,Z6),zero_zero_int),
          aa_int_nat(nat2,Z),
          $let(
            d: int,
            d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z6),
            $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ) ).

% diff_nat_eq_if
tff(fact_9121_card__UNIV__char,axiom,
    finite_card_char(top_top_set_char) = numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))) ).

% card_UNIV_char
tff(fact_9122_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      ( code_divmod_integer(K,L) = $ite(
          K = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
            aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
              $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K,L),aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_aip(L)),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_9123_or__minus__numerals_I2_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ) ).

% or_minus_numerals(2)
tff(fact_9124_or__minus__numerals_I6_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ) ).

% or_minus_numerals(6)
tff(fact_9125_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,N)),one_one_int))) ) ).

% or_minus_minus_numerals
tff(fact_9126_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,N)),one_one_int))) ) ).

% and_minus_minus_numerals
tff(fact_9127_UNIV__bool,axiom,
    top_top_set_o = insert_o($false,insert_o($true,bot_bot_set_o)) ).

% UNIV_bool
tff(fact_9128_set__bit__int__def,axiom,
    ! [N: nat,K: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) ) ).

% set_bit_int_def
tff(fact_9129_set__bit__nat__def,axiom,
    ! [M: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ) ).

% set_bit_nat_def
tff(fact_9130_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_9131_or__not__numerals_I2_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ) ).

% or_not_numerals(2)
tff(fact_9132_or__not__numerals_I4_J,axiom,
    ! [M: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ) ).

% or_not_numerals(4)
tff(fact_9133_or__not__numerals_I3_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ) ).

% or_not_numerals(3)
tff(fact_9134_or__not__numerals_I7_J,axiom,
    ! [M: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ) ).

% or_not_numerals(7)
tff(fact_9135_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = 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,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ) ).

% or_not_numerals(5)
tff(fact_9136_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = 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,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ) ).

% or_not_numerals(9)
tff(fact_9137_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = 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,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ) ).

% or_not_numerals(8)
tff(fact_9138_or__int__unfold,axiom,
    ! [K: int,L: int] :
      ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = $ite(
          ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
          | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
          aa_int_int(uminus_uminus_int,one_one_int),
          $ite(
            K = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ).

% or_int_unfold
tff(fact_9139_or__minus__numerals_I5_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(one,bitM(N)))) ) ).

% or_minus_numerals(5)
tff(fact_9140_or__minus__numerals_I1_J,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(one,bitM(N)))) ) ).

% or_minus_numerals(1)
tff(fact_9141_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int))) ) ).

% card_greaterThanLessThan_int
tff(fact_9142_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,aa_num_num(bit0,N)))) ) ).

% or_minus_numerals(8)
tff(fact_9143_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,aa_num_num(bit0,N)))) ) ).

% or_minus_numerals(4)
tff(fact_9144_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,bitM(N)))) ) ).

% or_minus_numerals(3)
tff(fact_9145_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,bitM(N)))) ) ).

% or_minus_numerals(7)
tff(fact_9146_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_9147_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : ( set_or4662586982721622107an_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or5832277885323065728an_int(L,U) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_9148_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,N))) ) ).

% int_numeral_or_not_num_neg
tff(fact_9149_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(N,M))) ) ).

% int_numeral_not_or_num_neg
tff(fact_9150_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] : ( aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,N)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ) ).

% numeral_or_not_num_eq
tff(fact_9151_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9152_max__nat_Osemilattice__neutr__axioms,axiom,
    aa_nat_o(semila9081495762789891438tr_nat(ord_max_nat),zero_zero_nat) ).

% max_nat.semilattice_neutr_axioms
tff(fact_9153_max__nat_Ocomm__monoid__axioms,axiom,
    aa_nat_o(comm_monoid_nat(ord_max_nat),zero_zero_nat) ).

% max_nat.comm_monoid_axioms
tff(fact_9154_max__nat_Omonoid__axioms,axiom,
    aa_nat_o(monoid_nat(ord_max_nat),zero_zero_nat) ).

% max_nat.monoid_axioms
tff(fact_9155_max__Suc2,axiom,
    ! [M: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_aiq,N)),M) ) ).

% max_Suc2
tff(fact_9156_max__Suc1,axiom,
    ! [N: nat,M: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_air,N)),M) ) ).

% max_Suc1
tff(fact_9157_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_gf,aTP_Lamm_gg) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9158_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( set_or8404916559141939852nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,L),one_one_Code_integer),U) = set_or4266950643985792945nteger(L,U) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_9159_Code__Numeral_Onegative__def,axiom,
    code_negative = comp_C3531382070062128313er_num(uminus1351360451143612070nteger,numera6620942414471956472nteger) ).

% Code_Numeral.negative_def
tff(fact_9160_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9161_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_9162_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_9163_Inf__int__def,axiom,
    ! [X5: set_int] : ( complete_Inf_Inf_int(X5) = aa_int_int(uminus_uminus_int,complete_Sup_Sup_int(image_int_int(uminus_uminus_int,X5))) ) ).

% Inf_int_def
tff(fact_9164_Inf__nat__def1,axiom,
    ! [K5: set_nat] :
      ( ( K5 != bot_bot_set_nat )
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_set_nat_nat(complete_Inf_Inf_nat,K5)),K5) ) ).

% Inf_nat_def1
tff(fact_9165_int__in__range__abs,axiom,
    ! [N: nat] : aa_set_int_o(member_int(aa_nat_int(semiri1314217659103216013at_int,N)),image_int_int(abs_abs_int,top_top_set_int)) ).

% int_in_range_abs
tff(fact_9166_image__Suc__atMost,axiom,
    ! [N: nat] : ( aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,N)) ) ).

% image_Suc_atMost
tff(fact_9167_image__Suc__lessThan,axiom,
    ! [N: nat] : ( aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N) ) ).

% image_Suc_lessThan
tff(fact_9168_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : ( image_int_int(aa_int_fun_int_int(aTP_Lamm_ais,L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = set_or4662586982721622107an_int(L,U) ) ).

% image_add_int_atLeastLessThan
tff(fact_9169_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : ( image_4470545334726330049nteger(aTP_Lamm_ait(L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(U),L))) = set_or8404916559141939852nteger(L,U) ) ).

% image_add_integer_atLeastLessThan
tff(fact_9170_range__mod,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiu,N)),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ) ) ).

% range_mod
tff(fact_9171_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,X: nat,Y: nat] :
      ( aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiv,C)),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
          aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat),bot_bot_set_nat) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9172_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% UNIV_char_of_nat
tff(fact_9173_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% range_nat_of_char
tff(fact_9174_pred__def,axiom,
    ! [Nat: nat] : ( aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_da),Nat) ) ).

% pred_def
tff(fact_9175_range__abs__Nats,axiom,
    image_int_int(abs_abs_int,top_top_set_int) = semiring_1_Nats_int ).

% range_abs_Nats
tff(fact_9176_binomial__def,axiom,
    ! [N: nat,K: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_aiw(N),K))) ) ).

% binomial_def
tff(fact_9177_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_aiy),Xa),X)) ) ).

% times_int.abs_eq
tff(fact_9178_rat__minus__code,axiom,
    ! [P3: rat,Q3: rat] : ( quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,P3),Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aja(Q3)),quotient_of(P3)) ) ).

% rat_minus_code
tff(fact_9179_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_9180_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: 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,W)))),aa_nat_nat(suc,N))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),N) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_9181_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,N))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),N) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9182_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_9183_quotient__of__number_I3_J,axiom,
    ! [K: num] : ( quotient_of(numeral_numeral_rat(K)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int) ) ).

% quotient_of_number(3)
tff(fact_9184_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_9185_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),numeral_numeral_nat(N))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),pred_numeral(N)) ) ).

% bit_minus_numeral_int(1)
tff(fact_9186_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),numeral_numeral_nat(N))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),pred_numeral(N)) ) ).

% bit_minus_numeral_int(2)
tff(fact_9187_quotient__of__number_I5_J,axiom,
    ! [K: num] : ( quotient_of(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(K))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ) ).

% quotient_of_number(5)
tff(fact_9188_diff__rat__def,axiom,
    ! [Q3: rat,R2: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Q3),R2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Q3),aa_rat_rat(uminus_uminus_rat,R2)) ) ).

% diff_rat_def
tff(fact_9189_quotient__of__div,axiom,
    ! [R2: rat,N: int,D2: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,N),D2) )
     => ( R2 = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_int_rat(ring_1_of_int_rat,N)),aa_int_rat(ring_1_of_int_rat,D2)) ) ) ).

% quotient_of_div
tff(fact_9190_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X2: nat,Y2: nat] : ( Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2)) ) ).

% eq_Abs_Integ
tff(fact_9191_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int)),N)
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) ).

% bit_not_int_iff'
tff(fact_9192_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q3: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P3),Q3) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3) ) ).

% quotient_of_denom_pos
tff(fact_9193_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_9194_int__def,axiom,
    ! [N: nat] : ( aa_nat_int(semiri1314217659103216013at_int,N) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),zero_zero_nat)) ) ).

% int_def
tff(fact_9195_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,K)),N)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))),N) ) ).

% bit_minus_int_iff
tff(fact_9196_rat__floor__code,axiom,
    ! [P3: rat] : ( aa_rat_int(archim3151403230148437115or_rat,P3) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(divide_divide_int),quotient_of(P3)) ) ).

% rat_floor_code
tff(fact_9197_rat__uminus__code,axiom,
    ! [P3: rat] : ( quotient_of(aa_rat_rat(uminus_uminus_rat,P3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ajb),quotient_of(P3)) ) ).

% rat_uminus_code
tff(fact_9198_rat__abs__code,axiom,
    ! [P3: rat] : ( quotient_of(aa_rat_rat(abs_abs_rat,P3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ajc),quotient_of(P3)) ) ).

% rat_abs_code
tff(fact_9199_rat__less__eq__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P3),Q3)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_aje(Q3)),quotient_of(P3)) ) ).

% rat_less_eq_code
tff(fact_9200_rat__less__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P3),Q3)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_ajg(Q3)),quotient_of(P3)) ) ).

% rat_less_code
tff(fact_9201_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : ( aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ajh),X)) ) ).

% uminus_int.abs_eq
tff(fact_9202_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N3: nat] :
          ( ! [M5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),M5)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),M5)
              <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N3) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N3),one_one_nat))
              <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N3) ) ) ) ).

% int_bit_bound
tff(fact_9203_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_9204_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_ajj),Xa),X) ) ).

% less_int.abs_eq
tff(fact_9205_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_ajl),Xa),X) ) ).

% less_eq_int.abs_eq
tff(fact_9206_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_ajn),Xa),X)) ) ).

% plus_int.abs_eq
tff(fact_9207_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_ajp),Xa),X)) ) ).

% minus_int.abs_eq
tff(fact_9208_rat__divide__code,axiom,
    ! [P3: rat,Q3: rat] : ( quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ajr(Q3)),quotient_of(P3)) ) ).

% rat_divide_code
tff(fact_9209_rat__times__code,axiom,
    ! [P3: rat,Q3: rat] : ( quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P3),Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ajt(Q3)),quotient_of(P3)) ) ).

% rat_times_code
tff(fact_9210_rat__plus__code,axiom,
    ! [P3: rat,Q3: rat] : ( quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P3),Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ajv(Q3)),quotient_of(P3)) ) ).

% rat_plus_code
tff(fact_9211_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_9212_rat__inverse__code,axiom,
    ! [P3: rat] : ( quotient_of(aa_rat_rat(inverse_inverse_rat,P3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ajw),quotient_of(P3)) ) ).

% rat_inverse_code
tff(fact_9213_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_9214_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_9215_char__of__integer__code,axiom,
    ! [K: code_integer] : ( char_of_integer(K) = produc4188289175737317920o_char(aTP_Lamm_ake,code_bit_cut_integer(K)) ) ).

% char_of_integer_code
tff(fact_9216_Frct__code__post_I5_J,axiom,
    ! [K: num] : ( frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),numeral_numeral_rat(K)) ) ).

% Frct_code_post(5)
tff(fact_9217_floor__rat__def,axiom,
    ! [X: rat] : ( aa_rat_int(archim3151403230148437115or_rat,X) = the_int(aTP_Lamm_akf(X)) ) ).

% floor_rat_def
tff(fact_9218_Frct__code__post_I9_J,axiom,
    ! [Q3: product_prod_int_int] : ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,frct(Q3))) = frct(Q3) ) ).

% Frct_code_post(9)
tff(fact_9219_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_9220_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_9221_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_9222_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_9223_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_9224_Frct__code__post_I4_J,axiom,
    ! [K: num] : ( frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int)) = numeral_numeral_rat(K) ) ).

% Frct_code_post(4)
tff(fact_9225_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : ( frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,numeral_numeral_rat(K)),numeral_numeral_rat(L)) ) ).

% Frct_code_post(6)
tff(fact_9226_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_ajl),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa)) ) ).

% less_eq_int.rep_eq
tff(fact_9227_less__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Xa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_ajj),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa)) ) ).

% less_int.rep_eq
tff(fact_9228_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_ajh)) ).

% uminus_int_def
tff(fact_9229_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_akg)) ).

% pred_nat_def
tff(fact_9230_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N)),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% less_eq
tff(fact_9231_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N)),transi2905341329935302413cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% pred_nat_trancl_eq_le
tff(fact_9232_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_aiy)) ).

% times_int_def
tff(fact_9233_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat)) ).

% natLess_def
tff(fact_9234_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_ajp)) ).

% minus_int_def
tff(fact_9235_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_ajn)) ).

% plus_int_def
tff(fact_9236_Gcd__remove0__nat,axiom,
    ! [M4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M4)
     => ( aa_set_nat_nat(gcd_Gcd_nat,M4) = aa_set_nat_nat(gcd_Gcd_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,M4),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9237_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,numeral_numeral_nat(M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = case_option_int_num(zero_zero_int,aTP_Lamm_akh(M),bit_take_bit_num(numeral_numeral_nat(M),N)) ) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9238_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(aa_int_fun_int_rat(fract,A),B))
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(aa_int_fun_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_9239_minus__rat__cancel,axiom,
    ! [A: int,B: int] : ( aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ).

% minus_rat_cancel
tff(fact_9240_minus__rat,axiom,
    ! [A: int,B: int] : ( aa_rat_rat(uminus_uminus_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,A)),B) ) ).

% minus_rat
tff(fact_9241_Gcd__nat__eq__one,axiom,
    ! [N4: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,one_one_nat),N4)
     => ( aa_set_nat_nat(gcd_Gcd_nat,N4) = one_one_nat ) ) ).

% Gcd_nat_eq_one
tff(fact_9242_eq__rat_I2_J,axiom,
    ! [A: int] : ( aa_int_rat(aa_int_fun_int_rat(fract,A),zero_zero_int) = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),one_one_int) ) ).

% eq_rat(2)
tff(fact_9243_Fract__of__nat__eq,axiom,
    ! [K: nat] : ( aa_int_rat(aa_int_fun_int_rat(fract,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) = aa_nat_rat(semiri681578069525770553at_rat,K) ) ).

% Fract_of_nat_eq
tff(fact_9244_quotient__of__eq,axiom,
    ! [A: int,B: int,P3: int,Q3: int] :
      ( ( quotient_of(aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P3),Q3) )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,P3),Q3) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).

% quotient_of_eq
tff(fact_9245_One__rat__def,axiom,
    one_one_rat = aa_int_rat(aa_int_fun_int_rat(fract,one_one_int),one_one_int) ).

% One_rat_def
tff(fact_9246_Fract__of__int__eq,axiom,
    ! [K: int] : ( aa_int_rat(aa_int_fun_int_rat(fract,K),one_one_int) = aa_int_rat(ring_1_of_int_rat,K) ) ).

% Fract_of_int_eq
tff(fact_9247_normalize__eq,axiom,
    ! [A: int,B: int,P3: 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,P3),Q3) )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,P3),Q3) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).

% normalize_eq
tff(fact_9248_Zero__rat__def,axiom,
    zero_zero_rat = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).

% Zero_rat_def
tff(fact_9249_rat__number__expand_I3_J,axiom,
    ! [K: num] : ( numeral_numeral_rat(K) = aa_int_rat(aa_int_fun_int_rat(fract,aa_num_int(numeral_numeral_int,K)),one_one_int) ) ).

% rat_number_expand(3)
tff(fact_9250_rat__number__collapse_I3_J,axiom,
    ! [W: num] : ( aa_int_rat(aa_int_fun_int_rat(fract,aa_num_int(numeral_numeral_int,W)),one_one_int) = numeral_numeral_rat(W) ) ).

% rat_number_collapse(3)
tff(fact_9251_quotient__of__Fract,axiom,
    ! [A: int,B: int] : ( quotient_of(aa_int_rat(aa_int_fun_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_9252_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(aa_int_fun_int_rat(fract,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% one_less_Fract_iff
tff(fact_9253_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(aa_int_fun_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_9254_rat__number__collapse_I5_J,axiom,
    aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% rat_number_collapse(5)
tff(fact_9255_Fract__add__one,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(plus_plus_int,M),N)),N) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(aa_int_fun_int_rat(fract,M),N)),one_one_rat) ) ) ).

% Fract_add_one
tff(fact_9256_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(aa_int_fun_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_9257_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(aa_int_fun_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_9258_rat__number__collapse_I4_J,axiom,
    ! [W: num] : ( aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),one_one_int) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) ) ).

% rat_number_collapse(4)
tff(fact_9259_rat__number__expand_I5_J,axiom,
    ! [K: num] : ( aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(K)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ) ).

% rat_number_expand(5)
tff(fact_9260_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bitM(N))) ) ).

% and_minus_numerals(3)
tff(fact_9261_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bitM(N))) ) ).

% and_minus_numerals(7)
tff(fact_9262_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,N))) ) ).

% and_minus_numerals(8)
tff(fact_9263_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,N))) ) ).

% and_minus_numerals(4)
tff(fact_9264_Gcd__int__eq,axiom,
    ! [N4: set_nat] : ( gcd_Gcd_int(image_nat_int(semiri1314217659103216013at_int,N4)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,N4)) ) ).

% Gcd_int_eq
tff(fact_9265_Gcd__abs__eq,axiom,
    ! [K5: set_int] : ( gcd_Gcd_int(image_int_int(abs_abs_int,K5)) = gcd_Gcd_int(K5) ) ).

% Gcd_abs_eq
tff(fact_9266_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] : ( aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(aTP_Lamm_aki,K5)) = aa_int_nat(nat2,gcd_Gcd_int(K5)) ) ).

% Gcd_nat_abs_eq
tff(fact_9267_Gcd__int__def,axiom,
    ! [K5: set_int] : ( gcd_Gcd_int(K5) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),K5))) ) ).

% Gcd_int_def
tff(fact_9268_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,N)) ) ).

% int_numeral_and_not_num
tff(fact_9269_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(N,M)) ) ).

% int_numeral_not_and_num
tff(fact_9270_Gcd__eq__Max,axiom,
    ! [M4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M4)
     => ( ( M4 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),M4)
         => ( aa_set_nat_nat(gcd_Gcd_nat,M4) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_akj,M4))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9271_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] : ( bit_take_bit_num(numeral_numeral_nat(R2),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(pred_numeral(R2),M))) ) ).

% take_bit_num_simps(7)
tff(fact_9272_set__decode__def,axiom,
    ! [X: nat] : ( aa_nat_set_nat(nat_set_decode,X) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_akk,X)) ) ).

% set_decode_def
tff(fact_9273_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_9274_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_if,N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_9275_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : ( bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(N,M))) ) ).

% take_bit_num_simps(4)
tff(fact_9276_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K2: nat,M3: nat] : ( X != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K2),M3) ) ).

% prod_decode_aux.cases
tff(fact_9277_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_9278_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : ( bit_and_not_num(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(some_num(one),aTP_Lamm_akl),bit_and_not_num(M,N)) ) ).

% and_not_num.simps(8)
tff(fact_9279_divide__nat__def,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = $ite(N = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_akm(M),N)))) ) ).

% divide_nat_def
tff(fact_9280_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_id,N))) = aa_int_int(abs_abs_int,N) ) ) ).

% Max_divisors_self_int
tff(fact_9281_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] : ( bit_take_bit_num(numeral_numeral_nat(R2),aa_num_num(bit0,M)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_akn),bit_take_bit_num(pred_numeral(R2),M)) ) ).

% take_bit_num_simps(6)
tff(fact_9282_surj__int__encode,axiom,
    image_int_nat(nat_int_encode,top_top_set_int) = top_top_set_nat ).

% surj_int_encode
tff(fact_9283_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X),Xa) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa)),aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(suc,X)))) ) ) ).

% prod_decode_aux.elims
tff(fact_9284_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : ( bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit0,M)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_akn),bit_take_bit_num(N,M)) ) ).

% take_bit_num_simps(3)
tff(fact_9285_prod__decode__aux_Osimps,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),K),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)),aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,K)))) ) ).

% prod_decode_aux.simps
tff(fact_9286_take__bit__num__code,axiom,
    ! [N: nat,M: num] : ( bit_take_bit_num(N,M) = produc478579273971653890on_num(aTP_Lamm_akr,product_Pair_nat_num(N,M)) ) ).

% take_bit_num_code
tff(fact_9287_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_9288_greaterThan__0,axiom,
    aa_nat_set_nat(set_or1210151606488870762an_nat,zero_zero_nat) = aa_set_nat_set_nat(image_nat_nat(suc),top_top_set_nat) ).

% greaterThan_0
tff(fact_9289_greaterThan__Suc,axiom,
    ! [K: nat] : ( aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or1210151606488870762an_nat,K)),aa_set_nat_set_nat(insert_nat(aa_nat_nat(suc,K)),bot_bot_set_nat)) ) ).

% greaterThan_Suc
tff(fact_9290_and__not__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( bit_and_not_num(X,Xa) = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ( ? [N3: num] : ( Xa = aa_num_num(bit0,N3) )
             => ( Y != some_num(one) ) ) )
         => ( ( ( X = one )
             => ( ? [N3: num] : ( Xa = aa_num_num(bit1,N3) )
               => ( Y != none_num ) ) )
           => ( ! [M3: num] :
                  ( ( X = aa_num_num(bit0,M3) )
                 => ( ( Xa = one )
                   => ( Y != some_num(aa_num_num(bit0,M3)) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa_num_num(bit0,M3) )
                   => ! [N3: num] :
                        ( ( Xa = aa_num_num(bit0,N3) )
                       => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),bit_and_not_num(M3,N3)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa_num_num(bit0,M3) )
                     => ! [N3: num] :
                          ( ( Xa = aa_num_num(bit1,N3) )
                         => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),bit_and_not_num(M3,N3)) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa_num_num(bit1,M3) )
                       => ( ( Xa = one )
                         => ( Y != some_num(aa_num_num(bit0,M3)) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa_num_num(bit1,M3) )
                         => ! [N3: num] :
                              ( ( Xa = aa_num_num(bit0,N3) )
                             => ( Y != aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(some_num(one),aTP_Lamm_akl),bit_and_not_num(M3,N3)) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X = aa_num_num(bit1,M3) )
                           => ! [N3: num] :
                                ( ( Xa = aa_num_num(bit1,N3) )
                               => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),bit_and_not_num(M3,N3)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_9291_surj__int__decode,axiom,
    image_nat_int(nat_int_decode,top_top_set_nat) = top_top_set_int ).

% surj_int_decode
tff(fact_9292_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),K)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aks,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_9293_and__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( bit_un7362597486090784418nd_num(X,Xa) = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != some_num(one) ) ) )
       => ( ( ( X = one )
           => ( ? [N3: num] : ( Xa = aa_num_num(bit0,N3) )
             => ( Y != none_num ) ) )
         => ( ( ( X = one )
             => ( ? [N3: num] : ( Xa = aa_num_num(bit1,N3) )
               => ( Y != some_num(one) ) ) )
           => ( ( ? [M3: num] : ( X = aa_num_num(bit0,M3) )
               => ( ( Xa = one )
                 => ( Y != none_num ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa_num_num(bit0,M3) )
                   => ! [N3: num] :
                        ( ( Xa = aa_num_num(bit0,N3) )
                       => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),bit_un7362597486090784418nd_num(M3,N3)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa_num_num(bit0,M3) )
                     => ! [N3: num] :
                          ( ( Xa = aa_num_num(bit1,N3) )
                         => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),bit_un7362597486090784418nd_num(M3,N3)) ) ) )
                 => ( ( ? [M3: num] : ( X = aa_num_num(bit1,M3) )
                     => ( ( Xa = one )
                       => ( Y != some_num(one) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa_num_num(bit1,M3) )
                         => ! [N3: num] :
                              ( ( Xa = aa_num_num(bit0,N3) )
                             => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),bit_un7362597486090784418nd_num(M3,N3)) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X = aa_num_num(bit1,M3) )
                           => ! [N3: num] :
                                ( ( Xa = aa_num_num(bit1,N3) )
                               => ( Y != aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(some_num(one),aTP_Lamm_akl),bit_un7362597486090784418nd_num(M3,N3)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_9294_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_9295_less__int__def,axiom,
    ord_less_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),produc6395135316500328296_nat_o(aTP_Lamm_ajj)) ).

% less_int_def
tff(fact_9296_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),produc6395135316500328296_nat_o(aTP_Lamm_ajl)) ).

% less_eq_int_def
tff(fact_9297_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : ( bit_un7362597486090784418nd_num(aa_num_num(bit1,M),aa_num_num(bit1,N)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(some_num(one),aTP_Lamm_akl),bit_un7362597486090784418nd_num(M,N)) ) ).

% and_num.simps(9)
tff(fact_9298_uminus__integer__def,axiom,
    uminus1351360451143612070nteger = map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int,uminus_uminus_int) ).

% uminus_integer_def
tff(fact_9299_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9300_min__Suc2,axiom,
    ! [M: nat,N: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_akt,N)),M) ) ).

% min_Suc2
tff(fact_9301_min__Suc1,axiom,
    ! [N: nat,M: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_aku,N)),M) ) ).

% min_Suc1
tff(fact_9302_Code__Numeral_Odup__def,axiom,
    code_dup = map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int,aTP_Lamm_akv) ).

% Code_Numeral.dup_def
tff(fact_9303_Code__Numeral_Osub__code_I9_J,axiom,
    ! [M: num,N: num] : ( aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(code_dup,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),N))),one_one_Code_integer) ) ).

% Code_Numeral.sub_code(9)
tff(fact_9304_Code__Numeral_Osub__code_I8_J,axiom,
    ! [M: num,N: num] : ( aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(code_dup,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),N))),one_one_Code_integer) ) ).

% Code_Numeral.sub_code(8)
tff(fact_9305_Code__Numeral_Osub__def,axiom,
    code_sub = map_fu8540202848259802501nteger(id_num,map_fu8638147718074629079nteger(id_num,code_integer_of_int),aTP_Lamm_akw) ).

% Code_Numeral.sub_def
tff(fact_9306_and__not__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( bit_and_not_num(X,Xa) = Y )
     => ( accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(X,Xa))
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = none_num )
               => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N3: num] :
                  ( ( Xa = aa_num_num(bit0,N3) )
                 => ( ( Y = some_num(one) )
                   => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit0,N3))) ) ) )
           => ( ( ( X = one )
               => ! [N3: num] :
                    ( ( Xa = aa_num_num(bit1,N3) )
                   => ( ( Y = none_num )
                     => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit1,N3))) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa_num_num(bit0,M3) )
                   => ( ( Xa = one )
                     => ( ( Y = some_num(aa_num_num(bit0,M3)) )
                       => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M3),one)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa_num_num(bit0,M3) )
                     => ! [N3: num] :
                          ( ( Xa = aa_num_num(bit0,N3) )
                         => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),bit_and_not_num(M3,N3)) )
                           => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N3))) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa_num_num(bit0,M3) )
                       => ! [N3: num] :
                            ( ( Xa = aa_num_num(bit1,N3) )
                           => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),bit_and_not_num(M3,N3)) )
                             => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N3))) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa_num_num(bit1,M3) )
                         => ( ( Xa = one )
                           => ( ( Y = some_num(aa_num_num(bit0,M3)) )
                             => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M3),one)) ) ) )
                     => ( ! [M3: num] :
                            ( ( X = aa_num_num(bit1,M3) )
                           => ! [N3: num] :
                                ( ( Xa = aa_num_num(bit0,N3) )
                               => ( ( Y = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(some_num(one),aTP_Lamm_akl),bit_and_not_num(M3,N3)) )
                                 => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N3))) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X = aa_num_num(bit1,M3) )
                             => ! [N3: num] :
                                  ( ( Xa = aa_num_num(bit1,N3) )
                                 => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),bit_and_not_num(M3,N3)) )
                                   => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_9307_and__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( bit_un7362597486090784418nd_num(X,Xa) = Y )
     => ( accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(X,Xa))
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = some_num(one) )
               => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N3: num] :
                  ( ( Xa = aa_num_num(bit0,N3) )
                 => ( ( Y = none_num )
                   => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit0,N3))) ) ) )
           => ( ( ( X = one )
               => ! [N3: num] :
                    ( ( Xa = aa_num_num(bit1,N3) )
                   => ( ( Y = some_num(one) )
                     => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit1,N3))) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa_num_num(bit0,M3) )
                   => ( ( Xa = one )
                     => ( ( Y = none_num )
                       => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M3),one)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa_num_num(bit0,M3) )
                     => ! [N3: num] :
                          ( ( Xa = aa_num_num(bit0,N3) )
                         => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),bit_un7362597486090784418nd_num(M3,N3)) )
                           => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N3))) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa_num_num(bit0,M3) )
                       => ! [N3: num] :
                            ( ( Xa = aa_num_num(bit1,N3) )
                           => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),bit_un7362597486090784418nd_num(M3,N3)) )
                             => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N3))) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa_num_num(bit1,M3) )
                         => ( ( Xa = one )
                           => ( ( Y = some_num(one) )
                             => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M3),one)) ) ) )
                     => ( ! [M3: num] :
                            ( ( X = aa_num_num(bit1,M3) )
                           => ! [N3: num] :
                                ( ( Xa = aa_num_num(bit0,N3) )
                               => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),bit_un7362597486090784418nd_num(M3,N3)) )
                                 => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N3))) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X = aa_num_num(bit1,M3) )
                             => ! [N3: num] :
                                  ( ( Xa = aa_num_num(bit1,N3) )
                                 => ( ( Y = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(some_num(one),aTP_Lamm_akl),bit_un7362597486090784418nd_num(M3,N3)) )
                                   => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_9308_or__not__num__neg_Opelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( bit_or_not_num_neg(X,Xa) = Y )
     => ( accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(X,Xa))
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = one )
               => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [M3: num] :
                  ( ( Xa = aa_num_num(bit0,M3) )
                 => ( ( Y = aa_num_num(bit1,M3) )
                   => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,aa_num_num(bit0,M3))) ) ) )
           => ( ( ( X = one )
               => ! [M3: num] :
                    ( ( Xa = aa_num_num(bit1,M3) )
                   => ( ( Y = aa_num_num(bit1,M3) )
                     => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,aa_num_num(bit1,M3))) ) ) )
             => ( ! [N3: num] :
                    ( ( X = aa_num_num(bit0,N3) )
                   => ( ( Xa = one )
                     => ( ( Y = aa_num_num(bit0,one) )
                       => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N3),one)) ) ) )
               => ( ! [N3: num] :
                      ( ( X = aa_num_num(bit0,N3) )
                     => ! [M3: num] :
                          ( ( Xa = aa_num_num(bit0,M3) )
                         => ( ( Y = bitM(bit_or_not_num_neg(N3,M3)) )
                           => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N3),aa_num_num(bit0,M3))) ) ) )
                 => ( ! [N3: num] :
                        ( ( X = aa_num_num(bit0,N3) )
                       => ! [M3: num] :
                            ( ( Xa = aa_num_num(bit1,M3) )
                           => ( ( Y = aa_num_num(bit0,bit_or_not_num_neg(N3,M3)) )
                             => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N3),aa_num_num(bit1,M3))) ) ) )
                   => ( ! [N3: num] :
                          ( ( X = aa_num_num(bit1,N3) )
                         => ( ( Xa = one )
                           => ( ( Y = one )
                             => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N3),one)) ) ) )
                     => ( ! [N3: num] :
                            ( ( X = aa_num_num(bit1,N3) )
                           => ! [M3: num] :
                                ( ( Xa = aa_num_num(bit0,M3) )
                               => ( ( Y = bitM(bit_or_not_num_neg(N3,M3)) )
                                 => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N3),aa_num_num(bit0,M3))) ) ) )
                       => ~ ! [N3: num] :
                              ( ( X = aa_num_num(bit1,N3) )
                             => ! [M3: num] :
                                  ( ( Xa = aa_num_num(bit1,M3) )
                                 => ( ( Y = bitM(bit_or_not_num_neg(N3,M3)) )
                                   => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N3),aa_num_num(bit1,M3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_9309_xor__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( bit_un2480387367778600638or_num(X,Xa) = Y )
     => ( accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(X,Xa))
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = none_num )
               => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N3: num] :
                  ( ( Xa = aa_num_num(bit0,N3) )
                 => ( ( Y = some_num(aa_num_num(bit1,N3)) )
                   => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,aa_num_num(bit0,N3))) ) ) )
           => ( ( ( X = one )
               => ! [N3: num] :
                    ( ( Xa = aa_num_num(bit1,N3) )
                   => ( ( Y = some_num(aa_num_num(bit0,N3)) )
                     => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,aa_num_num(bit1,N3))) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa_num_num(bit0,M3) )
                   => ( ( Xa = one )
                     => ( ( Y = some_num(aa_num_num(bit1,M3)) )
                       => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M3),one)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa_num_num(bit0,M3) )
                     => ! [N3: num] :
                          ( ( Xa = aa_num_num(bit0,N3) )
                         => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),bit_un2480387367778600638or_num(M3,N3)) )
                           => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N3))) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa_num_num(bit0,M3) )
                       => ! [N3: num] :
                            ( ( Xa = aa_num_num(bit1,N3) )
                           => ( ( Y = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M3,N3))) )
                             => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N3))) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa_num_num(bit1,M3) )
                         => ( ( Xa = one )
                           => ( ( Y = some_num(aa_num_num(bit0,M3)) )
                             => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M3),one)) ) ) )
                     => ( ! [M3: num] :
                            ( ( X = aa_num_num(bit1,M3) )
                           => ! [N3: num] :
                                ( ( Xa = aa_num_num(bit0,N3) )
                               => ( ( Y = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M3,N3))) )
                                 => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N3))) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X = aa_num_num(bit1,M3) )
                             => ! [N3: num] :
                                  ( ( Xa = aa_num_num(bit1,N3) )
                                 => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),bit_un2480387367778600638or_num(M3,N3)) )
                                   => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_9310_Field__natLeq__on,axiom,
    ! [N: nat] : ( field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_akx(N)))) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gg,N)) ) ).

% Field_natLeq_on
tff(fact_9311_range__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_set_nat(image_nat_nat(aa_set6537093552120046177at_nat(infini8530281810654367211te_nat,S)),top_top_set_nat) = S ) ) ).

% range_enumerate
tff(fact_9312_Inf__nat__def,axiom,
    ! [X5: set_nat] : ( aa_set_nat_nat(complete_Inf_Inf_nat,X5) = ord_Least_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ai,X5)) ) ).

% Inf_nat_def
tff(fact_9313_wf__less,axiom,
    aa_set3702741264012733747_nat_o(wf_nat,collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat))) ).

% wf_less
tff(fact_9314_Least__Suc,axiom,
    ! [P: fun_nat_o,N: nat] :
      ( aa_nat_o(P,N)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_aky(P))) ) ) ) ).

% Least_Suc
tff(fact_9315_rat__sgn__code,axiom,
    ! [P3: rat] : ( quotient_of(aa_rat_rat(sgn_sgn_rat,P3)) = 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(P3)))),one_one_int) ) ).

% rat_sgn_code
tff(fact_9316_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa) = Y )
     => ( Y = $ite(Xa = 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(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa)))))) ) ) ).

% bezw.elims
tff(fact_9317_bezw_Osimps,axiom,
    ! [X: nat,Y: nat] :
      ( bezw(X,Y) = $ite(Y = 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(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)))))) ) ).

% bezw.simps
tff(fact_9318_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
     => ( bezw(X,Y) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ).

% bezw_non_0
tff(fact_9319_one__mod__minus__numeral,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,N)))) ) ).

% one_mod_minus_numeral
tff(fact_9320_minus__one__mod__numeral,axiom,
    ! [N: num] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,N))) ) ).

% minus_one_mod_numeral
tff(fact_9321_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(M,N))) ) ).

% minus_numeral_mod_numeral
tff(fact_9322_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(M,N)))) ) ).

% numeral_mod_minus_numeral
tff(fact_9323_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))
       => ~ ( ( Y = $ite(Xa = 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(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)) ) ) ) ).

% bezw.pelims
tff(fact_9324_normalize__def,axiom,
    ! [P3: product_prod_int_int] :
      ( aa_Pro6970504127699319548nt_int(normalize,P3) = $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,P3)),
          $let(
            a: int,
            a:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P3)),
            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,P3)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P3)),a)) ),
          $ite(
            aa_Pro6799258434465214121nt_int(product_snd_int_int,P3) = 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,P3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P3))),
              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,P3)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P3)),a)) ) ) ) ) ).

% normalize_def
tff(fact_9325_gcd__1__int,axiom,
    ! [M: int] : ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),one_one_int) = one_one_int ) ).

% gcd_1_int
tff(fact_9326_gcd__neg1__int,axiom,
    ! [X: int,Y: int] : ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ) ).

% gcd_neg1_int
tff(fact_9327_gcd__neg2__int,axiom,
    ! [X: int,Y: int] : ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ) ).

% gcd_neg2_int
tff(fact_9328_gcd__neg__numeral__2__int,axiom,
    ! [X: int,N: num] : ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_num_int(numeral_numeral_int,N)) ) ).

% gcd_neg_numeral_2_int
tff(fact_9329_gcd__neg__numeral__1__int,axiom,
    ! [N: num,X: int] : ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),X) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_num_int(numeral_numeral_int,N)),X) ) ).

% gcd_neg_numeral_1_int
tff(fact_9330_dup_Orsp,axiom,
    bNF_re4712519889275205905nt_int(fequal_int,fequal_int,aTP_Lamm_akv,aTP_Lamm_akv) ).

% dup.rsp
tff(fact_9331_sub_Orsp,axiom,
    bNF_re6378542964492772123um_int(fequal_num,bNF_re1822329894187522285nt_int(fequal_num,fequal_int),aTP_Lamm_akw,aTP_Lamm_akw) ).

% sub.rsp
tff(fact_9332_uminus__integer_Orsp,axiom,
    bNF_re4712519889275205905nt_int(fequal_int,fequal_int,uminus_uminus_int,uminus_uminus_int) ).

% uminus_integer.rsp
tff(fact_9333_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,Y))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),Y)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),aa_int_int(uminus_uminus_int,Y))) ) )
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_9334_gcd__is__Max__divisors__int,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_akz(N),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_9335_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X),Xa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa)),aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(suc,X)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_9336_plus__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_ala,plus_plus_rat) ).

% plus_rat.transfer
tff(fact_9337_gcd__1__nat,axiom,
    ! [M: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),one_one_nat) = one_one_nat ) ).

% gcd_1_nat
tff(fact_9338_Gcd__in,axiom,
    ! [A2: set_nat] :
      ( ! [A4: nat,B4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A4),A2)
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B4),A2)
           => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A4),B4)),A2) ) )
     => ( ( A2 != bot_bot_set_nat )
       => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_set_nat_nat(gcd_Gcd_nat,A2)),A2) ) ) ).

% Gcd_in
tff(fact_9339_gcd__nat_Osemilattice__neutr__axioms,axiom,
    aa_nat_o(semila9081495762789891438tr_nat(gcd_gcd_nat),zero_zero_nat) ).

% gcd_nat.semilattice_neutr_axioms
tff(fact_9340_gcd__nat_Ocomm__monoid__axioms,axiom,
    aa_nat_o(comm_monoid_nat(gcd_gcd_nat),zero_zero_nat) ).

% gcd_nat.comm_monoid_axioms
tff(fact_9341_gcd__nat_Omonoid__axioms,axiom,
    aa_nat_o(monoid_nat(gcd_gcd_nat),zero_zero_nat) ).

% gcd_nat.monoid_axioms
tff(fact_9342_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_ig) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9343_one__rat_Otransfer,axiom,
    aa_rat_o(aa_Pro2298929076809959388_rat_o(pcr_rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)),one_one_rat) ).

% one_rat.transfer
tff(fact_9344_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_alb(N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9345_zero__rat_Otransfer,axiom,
    aa_rat_o(aa_Pro2298929076809959388_rat_o(pcr_rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)),zero_zero_rat) ).

% zero_rat.transfer
tff(fact_9346_Fract_Otransfer,axiom,
    bNF_re2486238095283335024nt_rat(fequal_int,bNF_re2214769303045360666nt_rat(fequal_int,pcr_rat),aTP_Lamm_alc,fract) ).

% Fract.transfer
tff(fact_9347_uminus__rat_Otransfer,axiom,
    aa_fun_rat_rat_o(aa_fun382765352565706371_rat_o(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_ald),uminus_uminus_rat) ).

% uminus_rat.transfer
tff(fact_9348_times__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_ale,times_times_rat) ).

% times_rat.transfer
tff(fact_9349_inverse__rat_Otransfer,axiom,
    aa_fun_rat_rat_o(aa_fun382765352565706371_rat_o(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_alf),inverse_inverse_rat) ).

% inverse_rat.transfer
tff(fact_9350_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))
       => ~ ( ( Y = $ite(Xa = zero_zero_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9351_positive_Otransfer,axiom,
    bNF_re1494630372529172596at_o_o(pcr_rat,fequal_o,aTP_Lamm_alg,positive) ).

% positive.transfer
tff(fact_9352_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_aiy),times_times_int) ).

% times_int.transfer
tff(fact_9353_zero__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),zero_zero_int) ).

% zero_int.transfer
tff(fact_9354_positive__minus,axiom,
    ! [X: rat] :
      ( ~ aa_rat_o(positive,X)
     => ( ( X != zero_zero_rat )
       => aa_rat_o(positive,aa_rat_rat(uminus_uminus_rat,X)) ) ) ).

% positive_minus
tff(fact_9355_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_alh,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_9356_uminus__int_Otransfer,axiom,
    aa_fun_int_int_o(aa_fun856177901615811547_int_o(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_ajh)),uminus_uminus_int) ).

% uminus_int.transfer
tff(fact_9357_nat_Otransfer,axiom,
    bNF_re4555766996558763186at_nat(pcr_int,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),nat2) ).

% nat.transfer
tff(fact_9358_one__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int) ).

% one_int.transfer
tff(fact_9359_less__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_ajj),ord_less_int) ).

% less_int.transfer
tff(fact_9360_less__eq__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_ajl),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_9361_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_ajn),plus_plus_int) ).

% plus_int.transfer
tff(fact_9362_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_ajp),minus_minus_int) ).

% minus_int.transfer
tff(fact_9363_positive__def,axiom,
    positive = map_fu898904425404107465nt_o_o(rep_Rat,id_o,aTP_Lamm_alg) ).

% positive_def
tff(fact_9364_plus__rat__def,axiom,
    plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_ala) ).

% plus_rat_def
tff(fact_9365_inverse__rat__def,axiom,
    inverse_inverse_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_alf) ).

% inverse_rat_def
tff(fact_9366_times__rat__def,axiom,
    times_times_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_ale) ).

% times_rat_def
tff(fact_9367_uminus__rat__def,axiom,
    uminus_uminus_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_ald) ).

% uminus_rat_def
tff(fact_9368_one__rat__def,axiom,
    one_one_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) ).

% one_rat_def
tff(fact_9369_Fract_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( aa_int_rat(aa_int_fun_int_rat(fract,Xa),X) = aa_Pro6166618844887768661nt_rat(abs_Rat,
          $ite(X = 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,Xa),X))) ) ).

% Fract.abs_eq
tff(fact_9370_zero__rat__def,axiom,
    zero_zero_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)) ).

% zero_rat_def
tff(fact_9371_Fract__def,axiom,
    fract = map_fu7945985449245211511nt_rat(id_int,map_fu3424225382358772769nt_rat(id_int,abs_Rat),aTP_Lamm_alc) ).

% Fract_def
tff(fact_9372_plus__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa),Xa)
     => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa)),aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,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_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,X)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_9373_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
     => ( aa_rat_rat(inverse_inverse_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,
            $ite(aa_Pro6799258434465214121nt_int(product_fst_int_int,X) = 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_Pro6799258434465214121nt_int(product_snd_int_int,X)),aa_Pro6799258434465214121nt_int(product_fst_int_int,X)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_9374_times__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa),Xa)
     => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa)),aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_9375_one__rat_Orsp,axiom,
    aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) ).

% one_rat.rsp
tff(fact_9376_zero__rat_Orsp,axiom,
    aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,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,zero_zero_int),one_one_int)) ).

% zero_rat.rsp
tff(fact_9377_Fract_Orsp,axiom,
    bNF_re4808202430149235895nt_int(fequal_int,bNF_re6250860962936578807nt_int(fequal_int,ratrel),aTP_Lamm_alc,aTP_Lamm_alc) ).

% Fract.rsp
tff(fact_9378_ratrel__def,axiom,
    ! [X3: product_prod_int_int,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X3),Xa3)
    <=> ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,X3) != zero_zero_int )
        & ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa3) != zero_zero_int )
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa3)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3)) ) ) ) ).

% ratrel_def
tff(fact_9379_times__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_ale,aTP_Lamm_ale) ).

% times_rat.rsp
tff(fact_9380_uminus__rat_Orsp,axiom,
    aa_fun2650603632653508759_int_o(aa_fun3734617511155415725_int_o(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_ald),aTP_Lamm_ald) ).

% uminus_rat.rsp
tff(fact_9381_positive_Orsp,axiom,
    bNF_re8699439704749558557nt_o_o(ratrel,fequal_o,aTP_Lamm_alg,aTP_Lamm_alg) ).

% positive.rsp
tff(fact_9382_inverse__rat_Orsp,axiom,
    aa_fun2650603632653508759_int_o(aa_fun3734617511155415725_int_o(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_alf),aTP_Lamm_alf) ).

% inverse_rat.rsp
tff(fact_9383_plus__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_ala,aTP_Lamm_ala) ).

% plus_rat.rsp
tff(fact_9384_uminus__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
     => ( aa_rat_rat(uminus_uminus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),aa_Pro6799258434465214121nt_int(product_snd_int_int,X))) ) ) ).

% uminus_rat.abs_eq
tff(fact_9385_cr__rat__def,axiom,
    ! [X3: product_prod_int_int,Xa3: rat] :
      ( aa_rat_o(cr_rat(X3),Xa3)
    <=> ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X3),X3)
        & ( aa_Pro6166618844887768661nt_rat(abs_Rat,X3) = Xa3 ) ) ) ).

% cr_rat_def
tff(fact_9386_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_aiy),produc4528391901023682385at_nat(aTP_Lamm_aiy)) ).

% times_int.rsp
tff(fact_9387_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),V) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y) ) ) ).

% intrel_iff
tff(fact_9388_zero__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int.rsp
tff(fact_9389_uminus__int_Orsp,axiom,
    aa_fun3538214411680374423_nat_o(aa_fun6920661532540581549_nat_o(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_ajh)),produc2626176000494625587at_nat(aTP_Lamm_ajh)) ).

% uminus_int.rsp
tff(fact_9390_nat_Orsp,axiom,
    bNF_re8246922863344978751at_nat(intrel,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat.rsp
tff(fact_9391_one__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int.rsp
tff(fact_9392_intrel__def,axiom,
    intrel = produc6395135316500328296_nat_o(aTP_Lamm_alj) ).

% intrel_def
tff(fact_9393_less__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_ajj),produc6395135316500328296_nat_o(aTP_Lamm_ajj)) ).

% less_int.rsp
tff(fact_9394_less__eq__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_ajl),produc6395135316500328296_nat_o(aTP_Lamm_ajl)) ).

% less_eq_int.rsp
tff(fact_9395_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_ajn),produc4528391901023682385at_nat(aTP_Lamm_ajn)) ).

% plus_int.rsp
tff(fact_9396_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_ajp),produc4528391901023682385at_nat(aTP_Lamm_ajp)) ).

% minus_int.rsp
tff(fact_9397_quotient__of__def,axiom,
    ! [X: rat] : ( quotient_of(X) = the_Pr4378521158711661632nt_int(aTP_Lamm_alk(X)) ) ).

% quotient_of_def
tff(fact_9398_signed__take__bit__eq__concat__bit,axiom,
    ! [N: nat,K: int] : ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),aa_int_int(uminus_uminus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N)))) ) ).

% signed_take_bit_eq_concat_bit
tff(fact_9399_normalize__stable,axiom,
    ! [Q3: int,P3: 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,P3),Q3)
       => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P3),Q3)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P3),Q3) ) ) ) ).

% normalize_stable
tff(fact_9400_quotient__of__coprime,axiom,
    ! [R2: rat,P3: int,Q3: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P3),Q3) )
     => aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P3),Q3) ) ).

% quotient_of_coprime
tff(fact_9401_normalize__coprime,axiom,
    ! [R2: product_prod_int_int,P3: int,Q3: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P3),Q3) )
     => aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P3),Q3) ) ).

% normalize_coprime
tff(fact_9402_coprime__common__divisor__int,axiom,
    ! [A: int,B: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),B)
         => ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ) ) ).

% coprime_common_divisor_int
tff(fact_9403_coprime__common__divisor__nat,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),B)
         => ( X = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_9404_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N) ) ).

% coprime_diff_one_left_nat
tff(fact_9405_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ).

% coprime_diff_one_right_nat
tff(fact_9406_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_9407_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_9408_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( upto(X,Xa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa)),nil_int) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa)) ) ) ) ).

% upto.pelims
tff(fact_9409_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))),nil_int) ) ).

% upto_rec_numeral(4)
tff(fact_9410_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_num_int(numeral_numeral_int,N))),nil_int) ) ).

% upto_rec_numeral(3)
tff(fact_9411_length__upto,axiom,
    ! [I: int,J: int] : ( 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_9412_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)),cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_num_int(numeral_numeral_int,N))),nil_int) ) ).

% upto_rec_numeral(1)
tff(fact_9413_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))),nil_int) ) ).

% upto_rec_numeral(2)
tff(fact_9414_atLeastLessThan__upto,axiom,
    ! [I: int,J: 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_9415_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( upto(X,Xa) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa)),nil_int) ) ) ).

% upto.elims
tff(fact_9416_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_9417_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_9418_greaterThanLessThan__upto,axiom,
    ! [I: int,J: 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_9419_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9420_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9421_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9422_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_9423_sort__upto,axiom,
    ! [I: int,J: int] : ( linord1735203802627413978nt_int(aTP_Lamm_db,upto(I,J)) = upto(I,J) ) ).

% sort_upto
tff(fact_9424_sort__upt,axiom,
    ! [M: nat,N: nat] : ( linord738340561235409698at_nat(aTP_Lamm_da,upt(M,N)) = upt(M,N) ) ).

% sort_upt
tff(fact_9425_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] : ( map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_all,Ofs),upt(A,B)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ofs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ofs)) ) ).

% map_add_upt'
tff(fact_9426_map__add__upt,axiom,
    ! [N: nat,M: nat] : ( map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_all,N),upt(zero_zero_nat,M)) = upt(N,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ) ).

% map_add_upt
tff(fact_9427_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(X,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = X )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9428_upt__filter__extend,axiom,
    ! [U: nat,U2: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),U2)
     => ( ! [I2: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),I2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),U2) )
           => ~ aa_nat_o(P,I2) )
       => ( filter_nat2(P,upt(zero_zero_nat,U)) = filter_nat2(P,upt(zero_zero_nat,U2)) ) ) ) ).

% upt_filter_extend
tff(fact_9429_map__decr__upt,axiom,
    ! [M: nat,N: nat] : ( map_nat_nat(aTP_Lamm_alm,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ) ).

% map_decr_upt
tff(fact_9430_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_list_nat_nat(groups4561878855575611511st_nat,upt(M,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_da),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) ) ).

% sum_list_upt
tff(fact_9431_card__length__sum__list__rec,axiom,
    ! [M: nat,N4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aln(M),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_alo(M),N4)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_alp(M),N4)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9432_card__length__sum__list,axiom,
    ! [M: nat,N4: nat] : ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aln(M),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),M)),one_one_nat)),N4) ) ).

% card_length_sum_list
tff(fact_9433_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K: nat] :
      ( ! [N3: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,N3),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N3) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiv,K)),N4) ) ).

% inj_on_diff_nat
tff(fact_9434_rat__denum,axiom,
    ? [F2: fun_nat_rat] : ( image_nat_rat(F2,top_top_set_nat) = top_top_set_rat ) ).

% rat_denum
tff(fact_9435_surj__nat__to__rat__surj,axiom,
    image_nat_rat(nat_to_rat_surj,top_top_set_nat) = top_top_set_rat ).

% surj_nat_to_rat_surj
tff(fact_9436_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_list_nat_nat(last_nat,upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_9437_Sup__int__def,axiom,
    ! [X5: set_int] : ( complete_Sup_Sup_int(X5) = the_int(aTP_Lamm_alq(X5)) ) ).

% Sup_int_def
tff(fact_9438_Rats__eq__range__nat__to__rat__surj,axiom,
    field_6020823756834552118ts_rat = image_nat_rat(nat_to_rat_surj,top_top_set_nat) ).

% Rats_eq_range_nat_to_rat_surj
tff(fact_9439_nat__to__rat__surj__def,axiom,
    ! [N: nat] : ( aa_nat_rat(nat_to_rat_surj,N) = aa_Pro3608043709957902237at_rat(produc6207742614233964070at_rat(aTP_Lamm_alr),aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ) ).

% nat_to_rat_surj_def
tff(fact_9440_surj__prod__decode,axiom,
    image_5846123807819985514at_nat(nat_prod_decode,top_top_set_nat) = top_to4669805908274784177at_nat ).

% surj_prod_decode
tff(fact_9441_list__decode_Oelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( aa_nat_list_nat(nat_list_decode,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N3: nat] :
              ( ( X = aa_nat_nat(suc,N3) )
             => ( Y != produc2761476792215241774st_nat(aTP_Lamm_als,aa_nat7603221163087113075at_nat(nat_prod_decode,N3)) ) ) ) ) ).

% list_decode.elims
tff(fact_9442_surj__list__decode,axiom,
    image_nat_list_nat(nat_list_decode,top_top_set_nat) = top_top_set_list_nat ).

% surj_list_decode
tff(fact_9443_list__decode_Osimps_I2_J,axiom,
    ! [N: nat] : ( aa_nat_list_nat(nat_list_decode,aa_nat_nat(suc,N)) = produc2761476792215241774st_nat(aTP_Lamm_als,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ) ).

% list_decode.simps(2)
tff(fact_9444_list__decode_Opelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( aa_nat_list_nat(nat_list_decode,X) = Y )
     => ( aa_nat_o(accp_nat(nat_list_decode_rel),X)
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat) ) )
         => ~ ! [N3: nat] :
                ( ( X = aa_nat_nat(suc,N3) )
               => ( ( Y = produc2761476792215241774st_nat(aTP_Lamm_als,aa_nat7603221163087113075at_nat(nat_prod_decode,N3)) )
                 => ~ aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N3)) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_9445_list__decode_Opinduct,axiom,
    ! [A0: nat,P: fun_nat_o] :
      ( aa_nat_o(accp_nat(nat_list_decode_rel),A0)
     => ( ( aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat)
         => aa_nat_o(P,zero_zero_nat) )
       => ( ! [N3: nat] :
              ( aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N3))
             => ( ! [X3: nat,Y4: nat] :
                    ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y4) = aa_nat7603221163087113075at_nat(nat_prod_decode,N3) )
                   => aa_nat_o(P,Y4) )
               => aa_nat_o(P,aa_nat_nat(suc,N3)) ) )
         => aa_nat_o(P,A0) ) ) ) ).

% list_decode.pinduct
tff(fact_9446_list__decode_Opsimps_I2_J,axiom,
    ! [N: nat] :
      ( aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N))
     => ( aa_nat_list_nat(nat_list_decode,aa_nat_nat(suc,N)) = produc2761476792215241774st_nat(aTP_Lamm_als,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ) ) ).

% list_decode.psimps(2)
tff(fact_9447_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_9448_bij__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(bij_betw_nat_nat(aa_set6537093552120046177at_nat(infini8530281810654367211te_nat,S)),top_top_set_nat),S) ) ).

% bij_enumerate
tff(fact_9449_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_9450_bij__prod__decode,axiom,
    aa_set3702741264012733747_nat_o(bij_be8693218025023041337at_nat(nat_prod_decode,top_top_set_nat),top_to4669805908274784177at_nat) ).

% bij_prod_decode
tff(fact_9451_bij__list__decode,axiom,
    bij_be6293887246118711976st_nat(nat_list_decode,top_top_set_nat,top_top_set_list_nat) ).

% bij_list_decode
tff(fact_9452_Rep__unit,axiom,
    ! [X: product_unit] : aa_set_o_o(member_o(aa_Product_unit_o(product_Rep_unit,X)),insert_o($true,bot_bot_set_o)) ).

% Rep_unit
tff(fact_9453_Rep__unit__induct,axiom,
    ! [Y: $o,P: fun_o_o] :
      ( aa_set_o_o(member_o((Y)),insert_o($true,bot_bot_set_o))
     => ( ! [X2: product_unit] : aa_o_o(P,aa_Product_unit_o(product_Rep_unit,X2))
       => aa_o_o(P,(Y)) ) ) ).

% Rep_unit_induct
tff(fact_9454_Rep__unit__cases,axiom,
    ! [Y: $o] :
      ( aa_set_o_o(member_o((Y)),insert_o($true,bot_bot_set_o))
     => ~ ! [X2: product_unit] :
            ( (Y)
          <=> ~ aa_Product_unit_o(product_Rep_unit,X2) ) ) ).

% Rep_unit_cases
tff(fact_9455_type__definition__unit,axiom,
    aa_set_o_o(type_d6188575255521822967unit_o(product_Rep_unit,product_Abs_unit),insert_o($true,bot_bot_set_o)) ).

% type_definition_unit
tff(fact_9456_Abs__unit__inverse,axiom,
    ! [Y: $o] :
      ( aa_set_o_o(member_o((Y)),insert_o($true,bot_bot_set_o))
     => ( aa_Product_unit_o(product_Rep_unit,aa_o_Product_unit(product_Abs_unit,(Y)))
      <=> (Y) ) ) ).

% Abs_unit_inverse
tff(fact_9457_Abs__unit__inject,axiom,
    ! [X: $o,Y: $o] :
      ( aa_set_o_o(member_o((X)),insert_o($true,bot_bot_set_o))
     => ( aa_set_o_o(member_o((Y)),insert_o($true,bot_bot_set_o))
       => ( ( aa_o_Product_unit(product_Abs_unit,(X)) = aa_o_Product_unit(product_Abs_unit,(Y)) )
        <=> ( (X)
          <=> (Y) ) ) ) ) ).

% Abs_unit_inject
tff(fact_9458_Abs__unit__cases,axiom,
    ! [X: product_unit] :
      ~ ! [Y2: $o] :
          ( ( X = aa_o_Product_unit(product_Abs_unit,(Y2)) )
         => ~ aa_set_o_o(member_o((Y2)),insert_o($true,bot_bot_set_o)) ) ).

% Abs_unit_cases
tff(fact_9459_Abs__unit__induct,axiom,
    ! [P: fun_Product_unit_o,X: product_unit] :
      ( ! [Y2: $o] :
          ( aa_set_o_o(member_o((Y2)),insert_o($true,bot_bot_set_o))
         => aa_Product_unit_o(P,aa_o_Product_unit(product_Abs_unit,(Y2))) )
     => aa_Product_unit_o(P,X) ) ).

% Abs_unit_induct
tff(fact_9460_natLeq__on__wo__rel,axiom,
    ! [N: nat] : aa_set3702741264012733747_nat_o(bNF_We3818239936649020644el_nat,collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_akx(N)))) ).

% natLeq_on_wo_rel
tff(fact_9461_pair__lessI2,axiom,
    ! [A: nat,B: nat,S2: nat,T2: 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,S2),T2)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T2)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_9462_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z)),fun_pair_less)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ).

% pair_less_iff1
tff(fact_9463_butlast__upt,axiom,
    ! [M: nat,N: nat] : ( butlast_nat(upt(M,N)) = upt(M,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ).

% butlast_upt
tff(fact_9464_pair__lessI1,axiom,
    ! [A: nat,B: nat,S2: nat,T2: 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),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T2)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_9465_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),XS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_min_strict)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS))),fun_min_strict) ) ) ) ).

% smin_insertI
tff(fact_9466_smin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( ( X5 != bot_bo2099793752762293965at_nat )
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,bot_bo2099793752762293965at_nat)),fun_min_strict) ) ).

% smin_emptyI
tff(fact_9467_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y5: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),Y5)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Y5)),fun_max_strict)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,X5),Y5)),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_9468_pair__leqI2,axiom,
    ! [A: nat,B: nat,S2: nat,T2: 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,S2),T2)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T2)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_9469_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_aky(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9470_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_alt(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9471_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,F3),at_top_nat)
    <=> ! [N5: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N5),F3) ) ).

% le_sequentially
tff(fact_9472_pair__leq__def,axiom,
    fun_pair_leq = sup_su718114333110466843at_nat(fun_pair_less,id_Pro2258643101195443293at_nat) ).

% pair_leq_def
tff(fact_9473_smax__emptyI,axiom,
    ! [Y5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Y5)
     => ( ( Y5 != bot_bo2099793752762293965at_nat )
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,Y5)),fun_max_strict) ) ) ).

% smax_emptyI
tff(fact_9474_filterlim__atMost__at__top,axiom,
    filter3212408913953519116et_nat(set_ord_atMost_nat,finite3254316476582989077op_nat(top_top_set_nat),at_top_nat) ).

% filterlim_atMost_at_top
tff(fact_9475_filterlim__lessThan__at__top,axiom,
    filter3212408913953519116et_nat(set_ord_lessThan_nat,finite3254316476582989077op_nat(top_top_set_nat),at_top_nat) ).

% filterlim_lessThan_at_top
tff(fact_9476_pair__leqI1,axiom,
    ! [A: nat,B: nat,S2: nat,T2: 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),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T2)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_9477_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),XS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_min_weak)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS))),fun_min_weak) ) ) ) ).

% wmin_insertI
tff(fact_9478_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),YS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_max_weak)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,XS),YS)),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_9479_max__weak__def,axiom,
    fun_max_weak = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,max_ex8135407076693332796at_nat(fun_pair_leq)),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% max_weak_def
tff(fact_9480_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_9481_eventually__False__sequentially,axiom,
    ~ eventually_nat(aTP_Lamm_bj,at_top_nat) ).

% eventually_False_sequentially
tff(fact_9482_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_max_strict,fun_max_weak)) ).

% max_rpair_set
tff(fact_9483_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_min_strict,fun_min_weak)) ).

% min_rpair_set
tff(fact_9484_wmax__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,X5)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,X5)),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_9485_wmin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,bot_bo2099793752762293965at_nat)),fun_min_weak) ).

% wmin_emptyI
tff(fact_9486_min__weak__def,axiom,
    fun_min_weak = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,min_ex6901939911449802026at_nat(fun_pair_leq)),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% min_weak_def
tff(fact_9487_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_9488_atLeast__Suc,axiom,
    ! [K: nat] : ( aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atLeast_nat,K)),aa_set_nat_set_nat(insert_nat(K),bot_bot_set_nat)) ) ).

% atLeast_Suc
tff(fact_9489_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_9490_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : ( set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or6656581121297822940st_int(L,U) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_9491_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( set_or189985376899183464nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,L),one_one_Code_integer),U) = set_or2715278749043346189nteger(L,U) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_9492_greaterThanAtMost__upto,axiom,
    ! [I: int,J: 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_9493_abs__division__segment,axiom,
    ! [K: int] : ( aa_int_int(abs_abs_int,aa_int_int(euclid3395696857347342551nt_int,K)) = one_one_int ) ).

% abs_division_segment
tff(fact_9494_division__segment__nat__def,axiom,
    ! [N: nat] : ( aa_nat_nat(euclid3398187327856392827nt_nat,N) = one_one_nat ) ).

% division_segment_nat_def
tff(fact_9495_division__segment__int__def,axiom,
    ! [K: int] :
      ( aa_int_int(euclid3395696857347342551nt_int,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% division_segment_int_def
tff(fact_9496_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),less_than)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).

% less_than_iff
tff(fact_9497_total__less__than,axiom,
    aa_set3702741264012733747_nat_o(total_on_nat(top_top_set_nat),less_than) ).

% total_less_than
tff(fact_9498_natLeq__on__well__order__on,axiom,
    ! [N: nat] : aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gg,N))),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_akx(N)))) ).

% natLeq_on_well_order_on
tff(fact_9499_natLeq__on__Well__order,axiom,
    ! [N: nat] : aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_akx(N))))),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_akx(N)))) ).

% natLeq_on_Well_order
tff(fact_9500_ctwo__Cnotzero,axiom,
    ( ~ member444158400953824016od_o_o(produc763777882069021527od_o_o(bNF_Cardinal_ctwo,bNF_Cardinal_czero_o),bNF_We2654380646378065620so_o_o)
    & bNF_Ca8331644756375544342r_on_o(field_o(bNF_Cardinal_ctwo),bNF_Cardinal_ctwo) ) ).

% ctwo_Cnotzero
tff(fact_9501_ctwo__def,axiom,
    bNF_Cardinal_ctwo = bNF_Ca3172636760949973492d_of_o(top_top_set_o) ).

% ctwo_def
tff(fact_9502_ctwo__ordLess__natLeq,axiom,
    member4095101504841534314at_nat(produc8517790099723286449at_nat(bNF_Cardinal_ctwo,bNF_Ca8665028551170535155natLeq),bNF_We8182288985678559134_o_nat) ).

% ctwo_ordLess_natLeq
tff(fact_9503_natLeq__card__order,axiom,
    aa_set3702741264012733747_nat_o(bNF_Ca1281551314933786834on_nat(top_top_set_nat),bNF_Ca8665028551170535155natLeq) ).

% natLeq_card_order
tff(fact_9504_natLeq__underS__less,axiom,
    ! [N: nat] : ( aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gg,N)) ) ).

% natLeq_underS_less
tff(fact_9505_Field__natLeq,axiom,
    field_nat(bNF_Ca8665028551170535155natLeq) = top_top_set_nat ).

% Field_natLeq
tff(fact_9506_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_9507_Restr__natLeq2,axiom,
    ! [N: nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N),aTP_Lamm_alu(N))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_akx(N))) ) ).

% Restr_natLeq2
tff(fact_9508_Restr__natLeq,axiom,
    ! [N: nat] : ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gg,N)),aTP_Lamm_alv(N))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_akx(N))) ) ).

% Restr_natLeq
tff(fact_9509_card__of__nat,axiom,
    aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bNF_Ca3793111618940312692of_nat(top_top_set_nat),bNF_Ca8665028551170535155natLeq)),bNF_We5258908940166488438at_nat) ).

% card_of_nat
tff(fact_9510_card__of__Field__natLeq,axiom,
    aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bNF_Ca3793111618940312692of_nat(field_nat(bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq)),bNF_We5258908940166488438at_nat) ).

% card_of_Field_natLeq
tff(fact_9511_num__of__nat__One,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),one_one_nat)
     => ( num_of_nat(N) = one ) ) ).

% num_of_nat_One
tff(fact_9512_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N5: nat] :
        ! [M2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),M2)
         => ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N2)
             => aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N2),M2)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_9513_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_9514_int_Oid__abs__transfer,axiom,
    bNF_re7934895593101944656at_int(basic_5328504652464829177at_nat(fequal_nat,fequal_nat),pcr_int,aTP_Lamm_alw,abs_Integ) ).

% int.id_abs_transfer
tff(fact_9515_smsI,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,Z4: multis2468970476368604999at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A2),set_ms8126754132646256062at_nat(B2))),fun_max_strict)
     => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z4),A2),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z4),B2)),ms_strict) ) ).

% smsI
tff(fact_9516_wmsI,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,Z4: multis2468970476368604999at_nat] :
      ( ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A2),set_ms8126754132646256062at_nat(B2))),fun_max_strict)
        | ( ( A2 = zero_z1048942125864253310at_nat )
          & ( B2 = zero_z1048942125864253310at_nat ) ) )
     => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z4),A2),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z4),B2)),ms_weak) ) ).

% wmsI
tff(fact_9517_ms__weak__def,axiom,
    ms_weak = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,ms_strict),id_mul2649389997224486051at_nat) ).

% ms_weak_def
tff(fact_9518_ms__reduction__pair,axiom,
    fun_re7357418987779152907at_nat(produc5245064249948416855at_nat(ms_strict,ms_weak)) ).

% ms_reduction_pair
tff(fact_9519_ms__strictI,axiom,
    ! [Z4: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( pw_leq(Z4,Z7)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A2),set_ms8126754132646256062at_nat(B2))),fun_max_strict)
       => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z4),A2),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z7),B2)),ms_strict) ) ) ).

% ms_strictI
tff(fact_9520_ms__weakI1,axiom,
    ! [Z4: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( pw_leq(Z4,Z7)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A2),set_ms8126754132646256062at_nat(B2))),fun_max_strict)
       => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z4),A2),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z7),B2)),ms_weak) ) ) ).

% ms_weakI1
tff(fact_9521_pw__leq__lstep,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
     => pw_leq(add_ms2612439473150266591at_nat(X,zero_z1048942125864253310at_nat),add_ms2612439473150266591at_nat(Y,zero_z1048942125864253310at_nat)) ) ).

% pw_leq_lstep
tff(fact_9522_pw__leq_Ocases,axiom,
    ! [A1: multis2468970476368604999at_nat,A22: multis2468970476368604999at_nat] :
      ( pw_leq(A1,A22)
     => ( ( ( A1 = zero_z1048942125864253310at_nat )
         => ( A22 != zero_z1048942125864253310at_nat ) )
       => ~ ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat,X6: multis2468970476368604999at_nat] :
              ( ( A1 = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,add_ms2612439473150266591at_nat(X2,zero_z1048942125864253310at_nat)),X6) )
             => ! [Y6: multis2468970476368604999at_nat] :
                  ( ( A22 = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,add_ms2612439473150266591at_nat(Y2,zero_z1048942125864253310at_nat)),Y6) )
                 => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X2,Y2),fun_pair_leq)
                   => ~ pw_leq(X6,Y6) ) ) ) ) ) ).

% pw_leq.cases
tff(fact_9523_pw__leq_Osimps,axiom,
    ! [A1: multis2468970476368604999at_nat,A22: multis2468970476368604999at_nat] :
      ( pw_leq(A1,A22)
    <=> ( ( ( A1 = zero_z1048942125864253310at_nat )
          & ( A22 = zero_z1048942125864253310at_nat ) )
        | ? [X4: product_prod_nat_nat,Y3: product_prod_nat_nat,X7: multis2468970476368604999at_nat,Y7: multis2468970476368604999at_nat] :
            ( ( A1 = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,add_ms2612439473150266591at_nat(X4,zero_z1048942125864253310at_nat)),X7) )
            & ( A22 = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,add_ms2612439473150266591at_nat(Y3,zero_z1048942125864253310at_nat)),Y7) )
            & member8206827879206165904at_nat(produc6161850002892822231at_nat(X4,Y3),fun_pair_leq)
            & pw_leq(X7,Y7) ) ) ) ).

% pw_leq.simps
tff(fact_9524_pw__leq__step,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,X5: multis2468970476368604999at_nat,Y5: multis2468970476368604999at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
     => ( pw_leq(X5,Y5)
       => pw_leq(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,add_ms2612439473150266591at_nat(X,zero_z1048942125864253310at_nat)),X5),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,add_ms2612439473150266591at_nat(Y,zero_z1048942125864253310at_nat)),Y5)) ) ) ).

% pw_leq_step
tff(fact_9525_ms__weakI2,axiom,
    ! [Z4: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat] :
      ( pw_leq(Z4,Z7)
     => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z4),zero_z1048942125864253310at_nat),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z7),zero_z1048942125864253310at_nat)),ms_weak) ) ).

% ms_weakI2
tff(fact_9526_pw__leq__split,axiom,
    ! [X5: multis2468970476368604999at_nat,Y5: multis2468970476368604999at_nat] :
      ( pw_leq(X5,Y5)
     => ? [A6: multis2468970476368604999at_nat,B8: multis2468970476368604999at_nat,Z8: multis2468970476368604999at_nat] :
          ( ( X5 = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A6),Z8) )
          & ( Y5 = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B8),Z8) )
          & ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A6),set_ms8126754132646256062at_nat(B8))),fun_max_strict)
            | ( ( B8 = zero_z1048942125864253310at_nat )
              & ( A6 = zero_z1048942125864253310at_nat ) ) ) ) ) ).

% pw_leq_split
tff(fact_9527_rat_Odomain,axiom,
    ! [X3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(domain9213661015745956397nt_rat(pcr_rat),X3)
    <=> ? [Y3: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(basic_4387203522000727145nt_int(fequal_int,fequal_int),X3),Y3)
          & aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Y3),Y3) ) ) ).

% rat.domain
tff(fact_9528_rat_Odomain__par,axiom,
    ! [DR1: fun_int_o,DR2: fun_int_o,P22: fun_Pr3238374608701176162_int_o] :
      ( ( domainp_int_int(fequal_int) = DR1 )
     => ( ( domainp_int_int(fequal_int) = DR2 )
       => ( bNF_re8699439704749558557nt_o_o(basic_4387203522000727145nt_int(fequal_int,fequal_int),fequal_o,P22,aTP_Lamm_alx)
         => ( domain9213661015745956397nt_rat(pcr_rat) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,basic_1567116559311922317nt_int(DR1,DR2)),P22) ) ) ) ) ).

% rat.domain_par
tff(fact_9529_rat_Odomain__par__left__total,axiom,
    ! [P4: fun_Pr3238374608701176162_int_o] :
      ( left_t3131394472396969446nt_int(basic_4387203522000727145nt_int(fequal_int,fequal_int))
     => ( bNF_re8699439704749558557nt_o_o(basic_4387203522000727145nt_int(fequal_int,fequal_int),fequal_o,P4,aTP_Lamm_alx)
       => ( domain9213661015745956397nt_rat(pcr_rat) = P4 ) ) ) ).

% rat.domain_par_left_total
tff(fact_9530_Abs__rat__inject,axiom,
    ! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),collec5210948495886036740nt_int(aTP_Lamm_aly))
     => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y),collec5210948495886036740nt_int(aTP_Lamm_aly))
       => ( ( aa_set4118085860027418549nt_rat(abs_rat,X) = aa_set4118085860027418549nt_rat(abs_rat,Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_rat_inject
tff(fact_9531_Abs__rat__induct,axiom,
    ! [P: fun_rat_o,X: rat] :
      ( ! [Y2: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y2),collec5210948495886036740nt_int(aTP_Lamm_aly))
         => aa_rat_o(P,aa_set4118085860027418549nt_rat(abs_rat,Y2)) )
     => aa_rat_o(P,X) ) ).

% Abs_rat_induct
tff(fact_9532_Abs__rat__cases,axiom,
    ! [X: rat] :
      ~ ! [Y2: set_Pr958786334691620121nt_int] :
          ( ( X = aa_set4118085860027418549nt_rat(abs_rat,Y2) )
         => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y2),collec5210948495886036740nt_int(aTP_Lamm_aly)) ) ).

% Abs_rat_cases
tff(fact_9533_type__definition__rat,axiom,
    aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(rep_rat,abs_rat),collec5210948495886036740nt_int(aTP_Lamm_aly)) ).

% type_definition_rat
tff(fact_9534_Abs__rat__inverse,axiom,
    ! [Y: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y),collec5210948495886036740nt_int(aTP_Lamm_aly))
     => ( aa_rat8624586542854104963nt_int(rep_rat,aa_set4118085860027418549nt_rat(abs_rat,Y)) = Y ) ) ).

% Abs_rat_inverse
tff(fact_9535_Rep__rat__induct,axiom,
    ! [Y: set_Pr958786334691620121nt_int,P: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y),collec5210948495886036740nt_int(aTP_Lamm_aly))
     => ( ! [X2: rat] : aa_set6170482117830564987_int_o(P,aa_rat8624586542854104963nt_int(rep_rat,X2))
       => aa_set6170482117830564987_int_o(P,Y) ) ) ).

% Rep_rat_induct
tff(fact_9536_Rep__rat__cases,axiom,
    ! [Y: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y),collec5210948495886036740nt_int(aTP_Lamm_aly))
     => ~ ! [X2: rat] : ( Y != aa_rat8624586542854104963nt_int(rep_rat,X2) ) ) ).

% Rep_rat_cases
tff(fact_9537_Rep__rat,axiom,
    ! [X: rat] : aa_set6820988702716878619_int_o(member2340774599025711042nt_int(aa_rat8624586542854104963nt_int(rep_rat,X)),collec5210948495886036740nt_int(aTP_Lamm_aly)) ).

% Rep_rat
tff(fact_9538_cr__int__def,axiom,
    ! [X3: product_prod_nat_nat] : ( aa_Pro2912763917461432128_int_o(cr_int,X3) = aa_int_fun_int_o(fequal_int,aa_Pro4240683299535347697at_int(abs_Integ,X3)) ) ).

% cr_int_def
tff(fact_9539_in__range_Osimps,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As))
    <=> ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),As)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(H2)) ) ) ).

% in_range.simps
tff(fact_9540_in__range_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(in_range,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As4) )
           => ! [X2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X2),As4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),lim_Product_unit(H)) ) ) ) ).

% in_range.elims(3)
tff(fact_9541_in__range_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As4) )
           => ~ ! [X3: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),As4)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),lim_Product_unit(H)) ) ) ) ).

% in_range.elims(2)
tff(fact_9542_in__range_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(in_range,X)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As4) )
           => ( (Y)
            <=> ~ ! [X4: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),As4)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(H)) ) ) ) ) ).

% in_range.elims(1)
tff(fact_9543_surj__prod__encode,axiom,
    image_2486076414777270412at_nat(nat_prod_encode,top_to4669805908274784177at_nat) = top_top_set_nat ).

% surj_prod_encode
tff(fact_9544_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_9545_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_9546_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_9547_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X) = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X2: nat,Xs2: list_nat] :
              ( ( X = cons_nat(X2,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),aa_list_nat_nat(nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_9548_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_alz) ).

% prod_encode_def
tff(fact_9549_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_9550_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_9551_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] : ( aa_list_nat_nat(nat_list_encode,cons_nat(X,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_list_nat_nat(nat_list_encode,Xs)))) ) ).

% list_encode.simps(2)
tff(fact_9552_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X) = Y )
     => ( aa_list_nat_o(accp_list_nat(nat_list_encode_rel),X)
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),nil_nat) ) )
         => ~ ! [X2: nat,Xs2: list_nat] :
                ( ( X = cons_nat(X2,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),aa_list_nat_nat(nat_list_encode,Xs2)))) )
                 => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),cons_nat(X2,Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_9553_String_Oless__literal__def,axiom,
    ord_less_literal = map_fu6041574301308154232eral_o(explode,map_fu6698795530259886495ar_o_o(explode,id_o),lexordp_char(aTP_Lamm_ama)) ).

% String.less_literal_def
tff(fact_9554_less__literal_Orep__eq,axiom,
    ! [X: literal,Xa: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,X),Xa)
    <=> aa_list_char_o(aa_lis7814493268959572977char_o(lexordp_char(aTP_Lamm_ama),aa_literal_list_char(explode,X)),aa_literal_list_char(explode,Xa)) ) ).

% less_literal.rep_eq
tff(fact_9555_String_Oless__eq__literal__def,axiom,
    ord_less_eq_literal = map_fu6041574301308154232eral_o(explode,map_fu6698795530259886495ar_o_o(explode,id_o),lexordp_eq_char(aTP_Lamm_ama)) ).

% String.less_eq_literal_def
tff(fact_9556_less__literal_Otransfer,axiom,
    bNF_re9191414151699148995eral_o(pcr_literal,bNF_re8730477052009823892al_o_o(pcr_literal,fequal_o),lexordp_char(aTP_Lamm_ama),ord_less_literal) ).

% less_literal.transfer
tff(fact_9557_literal_Orep__transfer,axiom,
    bNF_re4991257583289732696t_char(pcr_literal,list_all2_char_char(fequal_char),aTP_Lamm_amb,explode) ).

% literal.rep_transfer
tff(fact_9558_less__eq__literal_Otransfer,axiom,
    bNF_re9191414151699148995eral_o(pcr_literal,bNF_re8730477052009823892al_o_o(pcr_literal,fequal_o),lexordp_eq_char(aTP_Lamm_ama),ord_less_eq_literal) ).

% less_eq_literal.transfer
tff(fact_9559_less__eq__literal_Orep__eq,axiom,
    ! [X: literal,Xa: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,X),Xa)
    <=> aa_list_char_o(aa_lis7814493268959572977char_o(lexordp_eq_char(aTP_Lamm_ama),aa_literal_list_char(explode,X)),aa_literal_list_char(explode,Xa)) ) ).

% less_eq_literal.rep_eq
tff(fact_9560_String_Ocr__literal__def,axiom,
    ! [X3: list_char,Xa3: literal] :
      ( aa_literal_o(aa_lis3557855670089359654eral_o(cr_literal,X3),Xa3)
    <=> ( X3 = aa_literal_list_char(explode,Xa3) ) ) ).

% String.cr_literal_def
tff(fact_9561_MOST__SucD,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_aky(P),cofinite_nat)
     => eventually_nat(P,cofinite_nat) ) ).

% MOST_SucD
tff(fact_9562_MOST__SucI,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,cofinite_nat)
     => eventually_nat(aTP_Lamm_aky(P),cofinite_nat) ) ).

% MOST_SucI
tff(fact_9563_MOST__Suc__iff,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_aky(P),cofinite_nat)
    <=> eventually_nat(P,cofinite_nat) ) ).

% MOST_Suc_iff
tff(fact_9564_MOST__nat,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,cofinite_nat)
    <=> ? [M2: nat] :
        ! [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N2)
         => aa_nat_o(P,N2) ) ) ).

% MOST_nat
tff(fact_9565_MOST__ge__nat,axiom,
    ! [M: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,M),cofinite_nat) ).

% MOST_ge_nat
tff(fact_9566_MOST__nat__le,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,cofinite_nat)
    <=> ? [M2: nat] :
        ! [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N2)
         => aa_nat_o(P,N2) ) ) ).

% MOST_nat_le
tff(fact_9567_less__literal_Orsp,axiom,
    bNF_re6109064546783844641char_o(bNF_eq_onp_list_char(aTP_Lamm_amc),bNF_re6668474094832959905ar_o_o(bNF_eq_onp_list_char(aTP_Lamm_amc),fequal_o),lexordp_char(aTP_Lamm_ama),lexordp_char(aTP_Lamm_ama)) ).

% less_literal.rsp
tff(fact_9568_plus__literal_Orsp,axiom,
    bNF_re1514760611062797473t_char(bNF_eq_onp_list_char(aTP_Lamm_amc),bNF_re1407854171455731685t_char(bNF_eq_onp_list_char(aTP_Lamm_amc),bNF_eq_onp_list_char(aTP_Lamm_amc)),append_char,append_char) ).

% plus_literal.rsp
tff(fact_9569_asciis__of__literal_Orsp,axiom,
    bNF_re901197953848981245nteger(bNF_eq_onp_list_char(aTP_Lamm_amc),fequal5224421480335545304nteger,map_ch5843194146513361140nteger(comm_s4049598766089966537nteger),map_ch5843194146513361140nteger(comm_s4049598766089966537nteger)) ).

% asciis_of_literal.rsp
tff(fact_9570_size__literal_Orsp,axiom,
    bNF_re76688982720252355at_nat(bNF_eq_onp_list_char(aTP_Lamm_amc),fequal_nat,size_size_list_char,size_size_list_char) ).

% size_literal.rsp
tff(fact_9571_Literal_Orsp,axiom,
    bNF_re5337651037176608865t_char(fequal_o,bNF_re8108790427976964573t_char(fequal_o,bNF_re2075097589415080885t_char(fequal_o,bNF_re8572817166782503389t_char(fequal_o,bNF_re7288172596490189065t_char(fequal_o,bNF_re8717008942347199453t_char(fequal_o,bNF_re7974392315488110685t_char(fequal_o,bNF_re1407854171455731685t_char(bNF_eq_onp_list_char(aTP_Lamm_amc),bNF_eq_onp_list_char(aTP_Lamm_amc)))))))),aTP_Lamm_amd,aTP_Lamm_amd) ).

% Literal.rsp
tff(fact_9572_literal_Oexplode,axiom,
    ! [X: literal] : member_list_char(aa_literal_list_char(explode,X),collect_list_char(aTP_Lamm_amc)) ).

% literal.explode
tff(fact_9573_literal_Oexplode__cases,axiom,
    ! [Y: list_char] :
      ( member_list_char(Y,collect_list_char(aTP_Lamm_amc))
     => ~ ! [X2: literal] : ( Y != aa_literal_list_char(explode,X2) ) ) ).

% literal.explode_cases
tff(fact_9574_literal_Oexplode__induct,axiom,
    ! [Y: list_char,P: fun_list_char_o] :
      ( member_list_char(Y,collect_list_char(aTP_Lamm_amc))
     => ( ! [X2: literal] : aa_list_char_o(P,aa_literal_list_char(explode,X2))
       => aa_list_char_o(P,Y) ) ) ).

% literal.explode_induct
tff(fact_9575_zero__literal_Orsp,axiom,
    aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_amc),nil_char),nil_char) ).

% zero_literal.rsp
tff(fact_9576_literal_Odomain__eq,axiom,
    ! [X3: list_char] :
      ( aa_list_char_o(domain7888411570377768909iteral(pcr_literal),X3)
    <=> ! [Xa2: char] :
          ( member_char(Xa2,set_char2(X3))
         => ~ aa_char_o(digit7,Xa2) ) ) ).

% literal.domain_eq
tff(fact_9577_literal_Odomain,axiom,
    ! [X3: list_char] :
      ( aa_list_char_o(domain7888411570377768909iteral(pcr_literal),X3)
    <=> ? [Y3: list_char] :
          ( aa_list_char_o(aa_lis7814493268959572977char_o(list_all2_char_char(fequal_char),X3),Y3)
          & ! [Xa2: char] :
              ( member_char(Xa2,set_char2(Y3))
             => ~ aa_char_o(digit7,Xa2) ) ) ) ).

% literal.domain
tff(fact_9578_less__eq__literal_Orsp,axiom,
    bNF_re6109064546783844641char_o(bNF_eq_onp_list_char(aTP_Lamm_amc),bNF_re6668474094832959905ar_o_o(bNF_eq_onp_list_char(aTP_Lamm_amc),fequal_o),lexordp_eq_char(aTP_Lamm_ama),lexordp_eq_char(aTP_Lamm_ama)) ).

% less_eq_literal.rsp
tff(fact_9579_less__eq__literal_Oabs__eq,axiom,
    ! [Xa: list_char,X: list_char] :
      ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_amc),Xa),Xa)
     => ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_amc),X),X)
       => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,aa_list_char_literal(abs_literal,Xa)),aa_list_char_literal(abs_literal,X))
        <=> aa_list_char_o(aa_lis7814493268959572977char_o(lexordp_eq_char(aTP_Lamm_ama),Xa),X) ) ) ) ).

% less_eq_literal.abs_eq
tff(fact_9580_less__literal_Oabs__eq,axiom,
    ! [Xa: list_char,X: list_char] :
      ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_amc),Xa),Xa)
     => ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_amc),X),X)
       => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_list_char_literal(abs_literal,Xa)),aa_list_char_literal(abs_literal,X))
        <=> aa_list_char_o(aa_lis7814493268959572977char_o(lexordp_char(aTP_Lamm_ama),Xa),X) ) ) ) ).

% less_literal.abs_eq
tff(fact_9581_literal_OAbs__literal__cases,axiom,
    ! [X: literal] :
      ~ ! [Y2: list_char] :
          ( ( X = aa_list_char_literal(abs_literal,Y2) )
         => ~ member_list_char(Y2,collect_list_char(aTP_Lamm_amc)) ) ).

% literal.Abs_literal_cases
tff(fact_9582_literal_OAbs__literal__induct,axiom,
    ! [P: fun_literal_o,X: literal] :
      ( ! [Y2: list_char] :
          ( member_list_char(Y2,collect_list_char(aTP_Lamm_amc))
         => aa_literal_o(P,aa_list_char_literal(abs_literal,Y2)) )
     => aa_literal_o(P,X) ) ).

% literal.Abs_literal_induct
tff(fact_9583_literal_OAbs__literal__inject,axiom,
    ! [X: list_char,Y: list_char] :
      ( member_list_char(X,collect_list_char(aTP_Lamm_amc))
     => ( member_list_char(Y,collect_list_char(aTP_Lamm_amc))
       => ( ( aa_list_char_literal(abs_literal,X) = aa_list_char_literal(abs_literal,Y) )
        <=> ( X = Y ) ) ) ) ).

% literal.Abs_literal_inject
tff(fact_9584_literal_Otype__definition__literal,axiom,
    type_d4752411451802217481t_char(explode,abs_literal,collect_list_char(aTP_Lamm_amc)) ).

% literal.type_definition_literal
tff(fact_9585_literal_OAbs__literal__inverse,axiom,
    ! [Y: list_char] :
      ( member_list_char(Y,collect_list_char(aTP_Lamm_amc))
     => ( aa_literal_list_char(explode,aa_list_char_literal(abs_literal,Y)) = Y ) ) ).

% literal.Abs_literal_inverse
tff(fact_9586_plus__literal_Oabs__eq,axiom,
    ! [Xa: list_char,X: list_char] :
      ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_amc),Xa),Xa)
     => ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_amc),X),X)
       => ( aa_literal_literal(aa_lit1193766896341886811iteral(plus_plus_literal,aa_list_char_literal(abs_literal,Xa)),aa_list_char_literal(abs_literal,X)) = aa_list_char_literal(abs_literal,aa_lis7614194356200183040t_char(aa_lis2542250943915203678t_char(append_char,Xa),X)) ) ) ) ).

% plus_literal.abs_eq
tff(fact_9587_String_OQuotient__literal,axiom,
    quotie6109894551476798619iteral(bNF_eq_onp_list_char(aTP_Lamm_amc),abs_literal,explode,cr_literal) ).

% String.Quotient_literal
tff(fact_9588_asciis__of__literal_Oabs__eq,axiom,
    ! [X: list_char] :
      ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_amc),X),X)
     => ( asciis_of_literal(aa_list_char_literal(abs_literal,X)) = aa_lis2940536026700532620nteger(map_ch5843194146513361140nteger(comm_s4049598766089966537nteger),X) ) ) ).

% asciis_of_literal.abs_eq
tff(fact_9589_size__literal_Oabs__eq,axiom,
    ! [X: list_char] :
      ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_amc),X),X)
     => ( aa_literal_nat(size_size_literal,aa_list_char_literal(abs_literal,X)) = aa_list_char_nat(size_size_list_char,X) ) ) ).

% size_literal.abs_eq
tff(fact_9590_literal_Odomain__par__left__total,axiom,
    ! [P4: fun_list_char_o] :
      ( left_t8440299596084406506t_char(list_all2_char_char(fequal_char))
     => ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(bNF_re6668474094832959905ar_o_o(list_all2_char_char(fequal_char),fequal_o),P4),aTP_Lamm_amc)
       => ( domain7888411570377768909iteral(pcr_literal) = P4 ) ) ) ).

% literal.domain_par_left_total
tff(fact_9591_literal_Odomain__par,axiom,
    ! [DR: fun_char_o,P22: fun_list_char_o] :
      ( ( domainp_char_char(fequal_char) = DR )
     => ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(bNF_re6668474094832959905ar_o_o(list_all2_char_char(fequal_char),fequal_o),P22),aTP_Lamm_amc)
       => ( domain7888411570377768909iteral(pcr_literal) = aa_fun3665998754152086460char_o(aa_fun7316373324528736215char_o(inf_in94156054473216830char_o,list_all_char(DR)),P22) ) ) ) ).

% literal.domain_par
tff(fact_9592_implode_Orsp,axiom,
    aa_fun8069410965070236123char_o(aa_fun8853676540986067637char_o(bNF_re1407854171455731685t_char(fequal_list_char,bNF_eq_onp_list_char(aTP_Lamm_amc)),map_char_char(ascii_of)),map_char_char(ascii_of)) ).

% implode.rsp
tff(fact_9593_literal__of__asciis_Orsp,axiom,
    bNF_re5828396963709814525t_char(fequal5224421480335545304nteger,bNF_eq_onp_list_char(aTP_Lamm_amc),map_Co843364623960961780r_char(comp_c4562247934908617188nteger(ascii_of,unique1708501630192214702nteger)),map_Co843364623960961780r_char(comp_c4562247934908617188nteger(ascii_of,unique1708501630192214702nteger))) ).

% literal_of_asciis.rsp
tff(fact_9594_Literal_Oabs__eq,axiom,
    ! [X: list_char,Xg: $o,Xf: $o,Xe: $o,Xd: $o,Xc: $o,Xb: $o,Xa: $o] :
      ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_amc),X),X)
     => ( aa_literal_literal(aa_o_f1412681823949618463iteral(aa_o_f837559538564922386iteral(aa_o_f8819669424537506293iteral(aa_o_f3445383346798733800iteral(aa_o_f2755856924670595275iteral(aa_o_f293512076901925310iteral(aa_o_f6593176798326225313iteral(literal2,(Xg)),(Xf)),(Xe)),(Xd)),(Xc)),(Xb)),(Xa)),aa_list_char_literal(abs_literal,X)) = aa_list_char_literal(abs_literal,aa_lis7614194356200183040t_char(cons_char(aa_o_char(char2((Xg),(Xf),(Xe),(Xd),(Xc),(Xb),(Xa)),$false)),X)) ) ) ).

% Literal.abs_eq
tff(fact_9595_String_OLiteral__def,axiom,
    literal2 = map_fu5639709166104062110iteral(id_o,map_fu3469583328629365786iteral(id_o,map_fu39305780192311794iteral(id_o,map_fu4012007721304947738iteral(id_o,map_fu7405565234982245702iteral(id_o,map_fu4664413887100446234iteral(id_o,map_fu8523206220227787418iteral(id_o,map_fu7398138089906338514iteral(explode,abs_literal))))))),aTP_Lamm_amd) ).

% String.Literal_def
tff(fact_9596_Literal_Otransfer,axiom,
    bNF_re7219000028618014437iteral(fequal_o,bNF_re2080241665920251617iteral(fequal_o,bNF_re2270469785393322297iteral(fequal_o,bNF_re7488407634572369889iteral(fequal_o,bNF_re6877862493807443341iteral(fequal_o,bNF_re7699176496855487201iteral(fequal_o,bNF_re4041988539332897761iteral(fequal_o,bNF_re5453467372598631581iteral(pcr_literal,pcr_literal))))))),aTP_Lamm_amd,literal2) ).

% Literal.transfer
tff(fact_9597_cr__integer__def,axiom,
    ! [X3: int,Xa3: code_integer] :
      ( aa_Code_integer_o(code_cr_integer(X3),Xa3)
    <=> ( X3 = aa_Code_integer_int(code_int_of_integer,Xa3) ) ) ).

% cr_integer_def
tff(fact_9598_int__encode__def,axiom,
    ! [I: int] :
      ( aa_int_nat(nat_int_encode,I) = aa_Sum8073061460641146765at_nat(nat_sum_encode,
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I),sum_Inl_nat_nat(aa_int_nat(nat2,I)),sum_Inr_nat_nat(aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,I)),one_one_int))))) ) ).

% int_encode_def
tff(fact_9599_surj__sum__encode,axiom,
    image_1320371278474632150at_nat(nat_sum_encode,top_to6661820994512907621at_nat) = top_top_set_nat ).

% surj_sum_encode
tff(fact_9600_bij__sum__encode,axiom,
    aa_set_nat_o(bij_be5432664580149595207at_nat(nat_sum_encode,top_to6661820994512907621at_nat),top_top_set_nat) ).

% bij_sum_encode
tff(fact_9601_sum__encode__def,axiom,
    ! [X: sum_sum_nat_nat] : ( aa_Sum8073061460641146765at_nat(nat_sum_encode,X) = sum_ca6763686470577984908at_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aTP_Lamm_ame,X) ) ).

% sum_encode_def
tff(fact_9602_surj__sum__decode,axiom,
    image_678696785212003926at_nat(nat_sum_decode,top_top_set_nat) = top_to6661820994512907621at_nat ).

% surj_sum_decode
tff(fact_9603_bij__sum__decode,axiom,
    bij_be4790990086886966983at_nat(nat_sum_decode,top_top_set_nat,top_to6661820994512907621at_nat) ).

% bij_sum_decode
tff(fact_9604_nth__item_Opinduct,axiom,
    ! [A0: nat,P: fun_nat_o] :
      ( aa_nat_o(accp_nat(nth_item_rel),A0)
     => ( ( aa_nat_o(accp_nat(nth_item_rel),zero_zero_nat)
         => aa_nat_o(P,zero_zero_nat) )
       => ( ! [N3: nat] :
              ( aa_nat_o(accp_nat(nth_item_rel),aa_nat_nat(suc,N3))
             => ( ! [A9: nat,Aa: nat] :
                    ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,N3) = sum_Inl_nat_nat(A9) )
                   => ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,A9) = sum_Inl_nat_nat(Aa) )
                     => aa_nat_o(P,Aa) ) )
               => ( ! [A9: nat,B9: nat] :
                      ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,N3) = sum_Inl_nat_nat(A9) )
                     => ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,A9) = sum_Inr_nat_nat(B9) )
                       => aa_nat_o(P,B9) ) )
                 => ( ! [B9: nat,Ba: nat,X3: nat,Y4: nat] :
                        ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,N3) = sum_Inr_nat_nat(B9) )
                       => ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,B9) = sum_Inr_nat_nat(Ba) )
                         => ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y4) = aa_nat7603221163087113075at_nat(nat_prod_decode,Ba) )
                           => aa_nat_o(P,X3) ) ) )
                   => ( ! [B9: nat,Ba: nat,X3: nat,Y4: nat] :
                          ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,N3) = sum_Inr_nat_nat(B9) )
                         => ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,B9) = sum_Inr_nat_nat(Ba) )
                           => ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y4) = aa_nat7603221163087113075at_nat(nat_prod_decode,Ba) )
                             => aa_nat_o(P,Y4) ) ) )
                     => aa_nat_o(P,aa_nat_nat(suc,N3)) ) ) ) ) )
         => aa_nat_o(P,A0) ) ) ) ).

% nth_item.pinduct
tff(fact_9605_int__decode__def,axiom,
    ! [N: nat] : ( aa_nat_int(nat_int_decode,N) = sum_ca7763040182479039464nt_nat(semiri1314217659103216013at_int,aTP_Lamm_amf,aa_nat7431386967378518541at_nat(nat_sum_decode,N)) ) ).

% int_decode_def
tff(fact_9606_sup__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Uu),Uv) = product_Unity ) ).

% sup_unit_def
tff(fact_9607_cone__def,axiom,
    bNF_Cardinal_cone = bNF_Ca7083678892733797993t_unit(insert_Product_unit(product_Unity,bot_bo3957492148770167129t_unit)) ).

% cone_def
tff(fact_9608_bot__unit__def,axiom,
    bot_bot_Product_unit = product_Unity ).

% bot_unit_def
tff(fact_9609_uminus__unit__def,axiom,
    ! [Uu: product_unit] : ( aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Uu) = product_Unity ) ).

% uminus_unit_def
tff(fact_9610_inf__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Uu),Uv) = product_Unity ) ).

% inf_unit_def
tff(fact_9611_top__unit__def,axiom,
    top_top_Product_unit = product_Unity ).

% top_unit_def
tff(fact_9612_UNIV__unit,axiom,
    top_to1996260823553986621t_unit = insert_Product_unit(product_Unity,bot_bo3957492148770167129t_unit) ).

% UNIV_unit
tff(fact_9613_natural__zero__minus__one,axiom,
    minus_7197305767214868737atural(zero_z2226904508553997617atural,one_one_Code_natural) = zero_z2226904508553997617atural ).

% natural_zero_minus_one
tff(fact_9614_log_Oelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y: code_natural] :
      ( ( log(X,Xa) = Y )
     => ( Y = $ite(
            ( aa_Code_natural_o(ord_le1926595141338095240atural(X),one_one_Code_natural)
            | aa_Code_natural_o(ord_le5570908160329646204atural(Xa),X) ),
            one_one_Code_natural,
            plus_p4538020629002901425atural(one_one_Code_natural,log(X,divide5121882707175180666atural(Xa,X))) ) ) ) ).

% log.elims
tff(fact_9615_log_Osimps,axiom,
    ! [B: code_natural,I: code_natural] :
      ( log(B,I) = $ite(
          ( aa_Code_natural_o(ord_le1926595141338095240atural(B),one_one_Code_natural)
          | aa_Code_natural_o(ord_le5570908160329646204atural(I),B) ),
          one_one_Code_natural,
          plus_p4538020629002901425atural(one_one_Code_natural,log(B,divide5121882707175180666atural(I,B))) ) ) ).

% log.simps
tff(fact_9616_Random_Orange__def,axiom,
    ! [K: code_natural] : ( range(K) = produc5538323210962509403atural(iterat8892046348760725948atural(log(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),K),aTP_Lamm_amh,one_one_Code_natural),aTP_Lamm_ami(K)) ) ).

% Random.range_def
tff(fact_9617_next_Osimps,axiom,
    ! [V: code_natural,W: code_natural] :
      ( aa_Pro7165575656943818355atural(next,produc3574140220909816553atural(V,W)) = $let(
          v: code_natural,
          v:= minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),times_2397367101498566445atural(modulo8411746178871703098atural(V,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),times_2397367101498566445atural(divide5121882707175180666atural(V,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))))))))))))),
          $let(
            w2: code_natural,
            w2:= minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),times_2397367101498566445atural(modulo8411746178871703098atural(W,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,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))))))))))))))))),times_2397367101498566445atural(divide5121882707175180666atural(W,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))),
            aa_Pro7165575656943818355atural(produc6639722614265839536atural(plus_p4538020629002901425atural(minus_shift(numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),v,plus_p4538020629002901425atural(w2,one_one_Code_natural)),one_one_Code_natural)),produc3574140220909816553atural(v,w2)) ) ) ) ).

% next.simps
tff(fact_9618_log_Ocases,axiom,
    ! [X: produc7822875418678951345atural] :
      ~ ! [B4: code_natural,I2: code_natural] : ( X != produc3574140220909816553atural(B4,I2) ) ).

% log.cases
tff(fact_9619_full__exhaustive__natural_H_Ocases,axiom,
    ! [X: produc9140211490168418807atural] :
      ~ ! [F2: fun_Pr285225112257423215e_term,D: code_natural,I2: code_natural] : ( X != produc3399137497101226409atural(F2,produc3574140220909816553atural(D,I2)) ) ).

% full_exhaustive_natural'.cases
tff(fact_9620_exhaustive__natural_H_Ocases,axiom,
    ! [X: produc7476732711998919211atural] :
      ~ ! [F2: fun_Co6386927884524458043e_term,D: code_natural,I2: code_natural] : ( X != produc3379161622255323613atural(F2,produc3574140220909816553atural(D,I2)) ) ).

% exhaustive_natural'.cases
tff(fact_9621_split__seed__def,axiom,
    ! [S2: produc7822875418678951345atural] : ( split_seed(S2) = produc8282080750456430313atural(aTP_Lamm_amk(S2),S2) ) ).

% split_seed_def
tff(fact_9622_inc__shift__def,axiom,
    ! [V: code_natural,K: code_natural] :
      ( inc_shift(V,K) = $ite(V = K,one_one_Code_natural,plus_p4538020629002901425atural(K,one_one_Code_natural)) ) ).

% inc_shift_def
tff(fact_9623_log_Opelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y: code_natural] :
      ( ( log(X,Xa) = Y )
     => ( accp_P8126237942716283194atural(log_rel,produc3574140220909816553atural(X,Xa))
       => ~ ( ( Y = $ite(
                  ( aa_Code_natural_o(ord_le1926595141338095240atural(X),one_one_Code_natural)
                  | aa_Code_natural_o(ord_le5570908160329646204atural(Xa),X) ),
                  one_one_Code_natural,
                  plus_p4538020629002901425atural(one_one_Code_natural,log(X,divide5121882707175180666atural(Xa,X))) ) )
           => ~ accp_P8126237942716283194atural(log_rel,produc3574140220909816553atural(X,Xa)) ) ) ) ).

% log.pelims
tff(fact_9624_one__natural__def,axiom,
    one_one_Code_natural = aa_nat_Code_natural(code_natural_of_nat,one_one_nat) ).

% one_natural_def
tff(fact_9625_natural__of__nat__cases,axiom,
    ! [X: code_natural] :
      ~ ! [Y2: nat] :
          ( ( X = aa_nat_Code_natural(code_natural_of_nat,Y2) )
         => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y2),top_top_set_nat) ) ).

% natural_of_nat_cases
tff(fact_9626_natural__of__nat__induct,axiom,
    ! [P: fun_Code_natural_o,X: code_natural] :
      ( ! [Y2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y2),top_top_set_nat)
         => aa_Code_natural_o(P,aa_nat_Code_natural(code_natural_of_nat,Y2)) )
     => aa_Code_natural_o(P,X) ) ).

% natural_of_nat_induct
tff(fact_9627_natural__of__nat__inject,axiom,
    ! [X: nat,Y: nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),top_top_set_nat)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y),top_top_set_nat)
       => ( ( aa_nat_Code_natural(code_natural_of_nat,X) = aa_nat_Code_natural(code_natural_of_nat,Y) )
        <=> ( X = Y ) ) ) ) ).

% natural_of_nat_inject
tff(fact_9628_one__natural_Orep__eq,axiom,
    aa_Code_natural_nat(code_nat_of_natural,one_one_Code_natural) = one_one_nat ).

% one_natural.rep_eq
tff(fact_9629_type__definition__natural,axiom,
    aa_set_nat_o(type_d4410041424927559462al_nat(code_nat_of_natural,code_natural_of_nat),top_top_set_nat) ).

% type_definition_natural
tff(fact_9630_natural__of__nat__inverse,axiom,
    ! [Y: nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y),top_top_set_nat)
     => ( aa_Code_natural_nat(code_nat_of_natural,aa_nat_Code_natural(code_natural_of_nat,Y)) = Y ) ) ).

% natural_of_nat_inverse
tff(fact_9631_nat__of__natural__induct,axiom,
    ! [Y: nat,P: fun_nat_o] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y),top_top_set_nat)
     => ( ! [X2: code_natural] : aa_nat_o(P,aa_Code_natural_nat(code_nat_of_natural,X2))
       => aa_nat_o(P,Y) ) ) ).

% nat_of_natural_induct
tff(fact_9632_nat__of__natural__cases,axiom,
    ! [Y: nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Y),top_top_set_nat)
     => ~ ! [X2: code_natural] : ( Y != aa_Code_natural_nat(code_nat_of_natural,X2) ) ) ).

% nat_of_natural_cases
tff(fact_9633_nat__of__natural,axiom,
    ! [X: code_natural] : aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_Code_natural_nat(code_nat_of_natural,X)),top_top_set_nat) ).

% nat_of_natural
tff(fact_9634_cr__natural__def,axiom,
    ! [X3: nat,Xa3: code_natural] :
      ( aa_Code_natural_o(code_cr_natural(X3),Xa3)
    <=> ( X3 = aa_Code_natural_nat(code_nat_of_natural,Xa3) ) ) ).

% cr_natural_def
tff(fact_9635_INFM__nat,axiom,
    ! [P: fun_nat_o] :
      ( frequently_nat(P,cofinite_nat)
    <=> ! [M2: nat] :
        ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N2)
          & aa_nat_o(P,N2) ) ) ).

% INFM_nat
tff(fact_9636_INFM__nat__le,axiom,
    ! [P: fun_nat_o] :
      ( frequently_nat(P,cofinite_nat)
    <=> ! [M2: nat] :
        ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N2)
          & aa_nat_o(P,N2) ) ) ).

% INFM_nat_le
tff(fact_9637_INFM__nat__inductI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [I2: nat] :
            ( aa_nat_o(P,I2)
           => ? [J3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J3)
                & aa_nat_o(P,J3)
                & aa_nat_o(Q,J3) ) )
       => frequently_nat(Q,cofinite_nat) ) ) ).

% INFM_nat_inductI
tff(fact_9638_integer__of__nat__1,axiom,
    aa_nat_Code_integer(code_integer_of_nat,one_one_nat) = one_one_Code_integer ).

% integer_of_nat_1
tff(fact_9639_Lcm__abs__eq,axiom,
    ! [K5: set_int] : ( gcd_Lcm_int(image_int_int(abs_abs_int,K5)) = gcd_Lcm_int(K5) ) ).

% Lcm_abs_eq
tff(fact_9640_Lcm__int__eq,axiom,
    ! [N4: set_nat] : ( gcd_Lcm_int(image_nat_int(semiri1314217659103216013at_int,N4)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Lcm_nat,N4)) ) ).

% Lcm_int_eq
tff(fact_9641_Lcm__nat__abs__eq,axiom,
    ! [K5: set_int] : ( aa_set_nat_nat(gcd_Lcm_nat,image_int_nat(aTP_Lamm_aki,K5)) = aa_int_nat(nat2,gcd_Lcm_int(K5)) ) ).

% Lcm_nat_abs_eq
tff(fact_9642_Lcm__nat__empty,axiom,
    aa_set_nat_nat(gcd_Lcm_nat,bot_bot_set_nat) = one_one_nat ).

% Lcm_nat_empty
tff(fact_9643_Gcd__nat__def,axiom,
    ! [M4: set_nat] : ( aa_set_nat_nat(gcd_Gcd_nat,M4) = aa_set_nat_nat(gcd_Lcm_nat,collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_aml,M4))) ) ).

% Gcd_nat_def
tff(fact_9644_Lcm__int__def,axiom,
    ! [K5: set_int] : ( gcd_Lcm_int(K5) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Lcm_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),K5))) ) ).

% Lcm_int_def
tff(fact_9645_Lcm__eq__Max__nat,axiom,
    ! [M4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M4)
     => ( ( M4 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),M4)
         => ( ! [M3: nat,N3: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,M3),M4)
               => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,N3),M4)
                 => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M3),N3)),M4) ) )
           => ( aa_set_nat_nat(gcd_Lcm_nat,M4) = aa_set_nat_nat(lattic8265883725875713057ax_nat,M4) ) ) ) ) ) ).

% Lcm_eq_Max_nat
tff(fact_9646_Lcm__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : ( aa_set_nat_nat(gcd_Lcm_nat,set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_lcm_nat,Xs),one_one_nat) ) ).

% Lcm_nat_set_eq_fold
tff(fact_9647_Lcm__in__lcm__closed__set__nat,axiom,
    ! [M4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M4)
     => ( ( M4 != bot_bot_set_nat )
       => ( ! [M3: nat,N3: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,M3),M4)
             => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,N3),M4)
               => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M3),N3)),M4) ) )
         => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_set_nat_nat(gcd_Lcm_nat,M4)),M4) ) ) ) ).

% Lcm_in_lcm_closed_set_nat
tff(fact_9648_Lcm__nat__def,axiom,
    ! [M4: set_nat] :
      ( aa_set_nat_nat(gcd_Lcm_nat,M4) = $ite(aa_set_nat_o(finite_finite_nat,M4),aa_set_nat_nat(aa_nat6910189468943142497at_nat(lattic7826324295020591184_F_nat(gcd_lcm_nat),one_one_nat),M4),zero_zero_nat) ) ).

% Lcm_nat_def
tff(fact_9649_lcm__1__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N) = one_one_int )
    <=> ( ( ( M = one_one_int )
          | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) )
        & ( ( N = one_one_int )
          | ( N = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% lcm_1_iff_int
tff(fact_9650_lcm__neg1__int,axiom,
    ! [X: int,Y: int] : ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_int_int(uminus_uminus_int,X)),Y) = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y) ) ).

% lcm_neg1_int
tff(fact_9651_lcm__neg2__int,axiom,
    ! [X: int,Y: int] : ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),aa_int_int(uminus_uminus_int,Y)) = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y) ) ).

% lcm_neg2_int
tff(fact_9652_lcm__cases__int,axiom,
    ! [X: int,Y: int,P: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),aa_int_int(uminus_uminus_int,Y))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_int_int(uminus_uminus_int,X)),Y)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_int_int(uminus_uminus_int,X)),aa_int_int(uminus_uminus_int,Y))) ) )
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y)) ) ) ) ) ).

% lcm_cases_int
tff(fact_9653_Lcm__int__set__eq__fold,axiom,
    ! [Xs: list_int] : ( gcd_Lcm_int(set_int2(Xs)) = aa_int_int(fold_int_int(gcd_lcm_int,Xs),one_one_int) ) ).

% Lcm_int_set_eq_fold
tff(fact_9654_unit__factor__simps_I2_J,axiom,
    ! [N: nat] : ( aa_nat_nat(unit_f2748546683901255202or_nat,aa_nat_nat(suc,N)) = one_one_nat ) ).

% unit_factor_simps(2)
tff(fact_9655_unit__factor__nat__def,axiom,
    ! [N: nat] :
      ( aa_nat_nat(unit_f2748546683901255202or_nat,N) = $ite(N = zero_zero_nat,zero_zero_nat,one_one_nat) ) ).

% unit_factor_nat_def
tff(fact_9656_unit__pred__cases,axiom,
    ! [P: fun_pr1768013098239751566unit_o,Q: pred_Product_unit] :
      ( aa_pre4150682718667570679unit_o(P,bot_bo2717538794563056311t_unit)
     => ( aa_pre4150682718667570679unit_o(P,single_Product_unit(product_Unity))
       => aa_pre4150682718667570679unit_o(P,Q) ) ) ).

% unit_pred_cases
tff(fact_9657_Random__Pred_Onot__randompred__def,axiom,
    ! [P: fun_Pr160617923510306311atural,X3: produc7822875418678951345atural] : ( aa_Pro6676232280479689400atural(random6974930770145893639ompred(P),X3) = produc6665183775751917029atural(aTP_Lamm_amm,aa_Pro6676232280479689400atural(P,X3)) ) ).

% Random_Pred.not_randompred_def
tff(fact_9658_not__pred__eq,axiom,
    ! [P: pred_Product_unit] :
      ( not_pred(P) = $ite(aa_Product_unit_o(eval_Product_unit(P),product_Unity),bot_bo2717538794563056311t_unit,single_Product_unit(product_Unity)) ) ).

% not_pred_eq
tff(fact_9659_if__pred__eq,axiom,
    ! [B: $o] :
      ( if_pred((B)) = $ite((B),single_Product_unit(product_Unity),bot_bo2717538794563056311t_unit) ) ).

% if_pred_eq
tff(fact_9660_not__predE,axiom,
    ! [P: $o,X: product_unit] :
      ( aa_Product_unit_o(eval_Product_unit(not_pred(pred_Product_unit2(aTP_Lamm_amn((P))))),X)
     => ~ (P) ) ).

% not_predE
tff(fact_9661_not__predI,axiom,
    ! [P: $o] :
      ( ~ (P)
     => aa_Product_unit_o(eval_Product_unit(not_pred(pred_Product_unit2(aTP_Lamm_amn((P))))),product_Unity) ) ).

% not_predI
tff(fact_9662_nat__of__num__code_I3_J,axiom,
    ! [N: num] :
      ( nat_of_num(aa_num_num(bit1,N)) = $let(
          m: nat,
          m:= nat_of_num(N),
          aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m)) ) ) ).

% nat_of_num_code(3)
tff(fact_9663_nat__of__num__code_I1_J,axiom,
    nat_of_num(one) = one_one_nat ).

% nat_of_num_code(1)
tff(fact_9664_nat__of__num__code_I2_J,axiom,
    ! [N: num] :
      ( nat_of_num(aa_num_num(bit0,N)) = $let(
          m: nat,
          m:= nat_of_num(N),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m) ) ) ).

% nat_of_num_code(2)
tff(fact_9665_Suc__natural__minus__one,axiom,
    ! [N: code_natural] : ( minus_7197305767214868737atural(code_Suc(N),one_one_Code_natural) = N ) ).

% Suc_natural_minus_one
tff(fact_9666_gcd__nat_Osemilattice__order__axioms,axiom,
    semila1248733672344298208er_nat(gcd_gcd_nat,dvd_dvd_nat,aTP_Lamm_ig) ).

% gcd_nat.semilattice_order_axioms
tff(fact_9667_random__aux__set_Ocases,axiom,
    ! [X: produc7822875418678951345atural] :
      ( ! [J2: code_natural] : ( X != produc3574140220909816553atural(zero_z2226904508553997617atural,J2) )
     => ~ ! [I2: code_natural,J2: code_natural] : ( X != produc3574140220909816553atural(code_Suc(I2),J2) ) ) ).

% random_aux_set.cases
tff(fact_9668_or__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( bit_un6697907153464112080or_num(X,Xa) = Y )
     => ( accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(X,Xa))
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = one )
               => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N3: num] :
                  ( ( Xa = aa_num_num(bit0,N3) )
                 => ( ( Y = aa_num_num(bit1,N3) )
                   => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(one,aa_num_num(bit0,N3))) ) ) )
           => ( ( ( X = one )
               => ! [N3: num] :
                    ( ( Xa = aa_num_num(bit1,N3) )
                   => ( ( Y = aa_num_num(bit1,N3) )
                     => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(one,aa_num_num(bit1,N3))) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa_num_num(bit0,M3) )
                   => ( ( Xa = one )
                     => ( ( Y = aa_num_num(bit1,M3) )
                       => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit0,M3),one)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa_num_num(bit0,M3) )
                     => ! [N3: num] :
                          ( ( Xa = aa_num_num(bit0,N3) )
                         => ( ( Y = aa_num_num(bit0,bit_un6697907153464112080or_num(M3,N3)) )
                           => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N3))) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa_num_num(bit0,M3) )
                       => ! [N3: num] :
                            ( ( Xa = aa_num_num(bit1,N3) )
                           => ( ( Y = aa_num_num(bit1,bit_un6697907153464112080or_num(M3,N3)) )
                             => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N3))) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa_num_num(bit1,M3) )
                         => ( ( Xa = one )
                           => ( ( Y = aa_num_num(bit1,M3) )
                             => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit1,M3),one)) ) ) )
                     => ( ! [M3: num] :
                            ( ( X = aa_num_num(bit1,M3) )
                           => ! [N3: num] :
                                ( ( Xa = aa_num_num(bit0,N3) )
                               => ( ( Y = aa_num_num(bit1,bit_un6697907153464112080or_num(M3,N3)) )
                                 => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N3))) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X = aa_num_num(bit1,M3) )
                             => ! [N3: num] :
                                  ( ( Xa = aa_num_num(bit1,N3) )
                                 => ( ( Y = aa_num_num(bit1,bit_un6697907153464112080or_num(M3,N3)) )
                                   => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_num.pelims
tff(fact_9669_less__nat__rel,axiom,
    ord_less_nat = transi2163837189807498211lp_nat(aTP_Lamm_akg) ).

% less_nat_rel
tff(fact_9670_sub_Otransfer,axiom,
    bNF_re4520166952630509772nteger(fequal_num,bNF_re6501075790457514782nteger(fequal_num,code_pcr_integer),aTP_Lamm_akw,code_sub) ).

% sub.transfer
tff(fact_9671_uminus__integer_Otransfer,axiom,
    bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer,uminus_uminus_int,uminus1351360451143612070nteger) ).

% uminus_integer.transfer
tff(fact_9672_integer_Oid__abs__transfer,axiom,
    bNF_re982302072995117890nteger(fequal_int,code_pcr_integer,aTP_Lamm_db,code_integer_of_int) ).

% integer.id_abs_transfer
tff(fact_9673_integer_Orep__transfer,axiom,
    bNF_re3804157879324367682nt_int(code_pcr_integer,fequal_int,aTP_Lamm_db,code_int_of_integer) ).

% integer.rep_transfer
tff(fact_9674_dup_Otransfer,axiom,
    bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer,aTP_Lamm_akv,code_dup) ).

% dup.transfer
tff(fact_9675_one__integer_Otransfer,axiom,
    aa_Code_integer_o(aa_int5993532643906289898eger_o(code_pcr_integer,one_one_int),one_one_Code_integer) ).

% one_integer.transfer
tff(fact_9676_gcd__nat_Oordering__axioms,axiom,
    ordering_nat(dvd_dvd_nat,aTP_Lamm_ig) ).

% gcd_nat.ordering_axioms
tff(fact_9677_Abs__int__cases,axiom,
    ! [X: int] :
      ~ ! [Y2: set_Pr1261947904930325089at_nat] :
          ( ( X = aa_set4746584241299694929at_int(abs_int,Y2) )
         => ~ member2643936169264416010at_nat(Y2,collec5514110066124741708at_nat(aTP_Lamm_amo)) ) ).

% Abs_int_cases
tff(fact_9678_Abs__int__induct,axiom,
    ! [P: fun_int_o,X: int] :
      ( ! [Y2: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(Y2,collec5514110066124741708at_nat(aTP_Lamm_amo))
         => aa_int_o(P,aa_set4746584241299694929at_int(abs_int,Y2)) )
     => aa_int_o(P,X) ) ).

% Abs_int_induct
tff(fact_9679_Abs__int__inject,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( member2643936169264416010at_nat(X,collec5514110066124741708at_nat(aTP_Lamm_amo))
     => ( member2643936169264416010at_nat(Y,collec5514110066124741708at_nat(aTP_Lamm_amo))
       => ( ( aa_set4746584241299694929at_int(abs_int,X) = aa_set4746584241299694929at_int(abs_int,Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_int_inject
tff(fact_9680_Abs__int__inverse,axiom,
    ! [Y: set_Pr1261947904930325089at_nat] :
      ( member2643936169264416010at_nat(Y,collec5514110066124741708at_nat(aTP_Lamm_amo))
     => ( aa_int8822742982810151799at_nat(rep_int,aa_set4746584241299694929at_int(abs_int,Y)) = Y ) ) ).

% Abs_int_inverse
tff(fact_9681_type__definition__int,axiom,
    type_d8752208705193531015at_nat(rep_int,abs_int,collec5514110066124741708at_nat(aTP_Lamm_amo)) ).

% type_definition_int
tff(fact_9682_Rep__int__induct,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( member2643936169264416010at_nat(Y,collec5514110066124741708at_nat(aTP_Lamm_amo))
     => ( ! [X2: int] : aa_set3702741264012733747_nat_o(P,aa_int8822742982810151799at_nat(rep_int,X2))
       => aa_set3702741264012733747_nat_o(P,Y) ) ) ).

% Rep_int_induct
tff(fact_9683_Rep__int__cases,axiom,
    ! [Y: set_Pr1261947904930325089at_nat] :
      ( member2643936169264416010at_nat(Y,collec5514110066124741708at_nat(aTP_Lamm_amo))
     => ~ ! [X2: int] : ( Y != aa_int8822742982810151799at_nat(rep_int,X2) ) ) ).

% Rep_int_cases
tff(fact_9684_Rep__int,axiom,
    ! [X: int] : member2643936169264416010at_nat(aa_int8822742982810151799at_nat(rep_int,X),collec5514110066124741708at_nat(aTP_Lamm_amo)) ).

% Rep_int
tff(fact_9685_Int_Osub__code_I9_J,axiom,
    ! [M: num,N: num] : ( sub(aa_num_num(bit0,M),aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(dup,sub(M,N))),one_one_int) ) ).

% Int.sub_code(9)
tff(fact_9686_Int_Osub__code_I8_J,axiom,
    ! [M: num,N: num] : ( sub(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(dup,sub(M,N))),one_one_int) ) ).

% Int.sub_code(8)
tff(fact_9687_gcd__nat_Opreordering__axioms,axiom,
    preordering_nat(dvd_dvd_nat,aTP_Lamm_ig) ).

% gcd_nat.preordering_axioms
tff(fact_9688_ATP_Olambda__1,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro6970504127699319548nt_int(aTP_Lamm_alf,Uu) = $ite(aa_Pro6799258434465214121nt_int(product_fst_int_int,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_Pro6799258434465214121nt_int(product_snd_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu))) ) ).

% ATP.lambda_1
tff(fact_9689_ATP_Olambda__2,axiom,
    ! [Uu: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aTP_Lamm_aa,Uu)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,y),Uu)
        & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,x),Uu) ) ) ).

% ATP.lambda_2
tff(fact_9690_ATP_Olambda__3,axiom,
    ! [Uu: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aTP_Lamm_a,Uu)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,x),Uu)
        & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,y),Uu) ) ) ).

% ATP.lambda_3
tff(fact_9691_ATP_Olambda__4,axiom,
    ! [Uu: product_prod_int_int] : ( aa_Pro6970504127699319548nt_int(aTP_Lamm_ald,Uu) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu))),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)) ) ).

% ATP.lambda_4
tff(fact_9692_ATP_Olambda__5,axiom,
    ! [Uu: nat] : ( aa_nat_int(aTP_Lamm_amf,Uu) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Uu))),one_one_int) ) ).

% ATP.lambda_5
tff(fact_9693_ATP_Olambda__6,axiom,
    ! [Uu: nat] : ( aa_nat_nat(aTP_Lamm_alm,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ) ).

% ATP.lambda_6
tff(fact_9694_ATP_Olambda__7,axiom,
    ! [Uu: int] : ( aa_int_int(aTP_Lamm_akv,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uu) ) ).

% ATP.lambda_7
tff(fact_9695_ATP_Olambda__8,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_alx,Uu)
    <=> aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Uu),Uu) ) ).

% ATP.lambda_8
tff(fact_9696_ATP_Olambda__9,axiom,
    ! [Uu: code_integer] : ( aa_Cod1783974652533396728nteger(aTP_Lamm_mz,Uu) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),one_one_Code_integer) ) ).

% ATP.lambda_9
tff(fact_9697_ATP_Olambda__10,axiom,
    ! [Uu: rat] : ( aa_rat_rat(aTP_Lamm_nc,Uu) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),one_one_rat) ) ).

% ATP.lambda_10
tff(fact_9698_ATP_Olambda__11,axiom,
    ! [Uu: nat] : ( aa_nat_nat(aTP_Lamm_nb,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ) ).

% ATP.lambda_11
tff(fact_9699_ATP_Olambda__12,axiom,
    ! [Uu: int] : ( aa_int_int(aTP_Lamm_na,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ) ).

% ATP.lambda_12
tff(fact_9700_ATP_Olambda__13,axiom,
    ! [Uu: nat] : ( aa_nat7603221163087113075at_nat(aTP_Lamm_alh,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ) ).

% ATP.lambda_13
tff(fact_9701_ATP_Olambda__14,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_alg,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_9702_ATP_Olambda__15,axiom,
    ! [Uu: code_natural] : ( aa_Cod6352003912080677724atural(aTP_Lamm_amh,Uu) = produc5538323210962509403atural(next,aTP_Lamm_amg(Uu)) ) ).

% ATP.lambda_15
tff(fact_9703_ATP_Olambda__16,axiom,
    ! [Uu: nat] : ( aa_nat_nat(aTP_Lamm_ame,Uu) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uu)) ) ).

% ATP.lambda_16
tff(fact_9704_ATP_Olambda__17,axiom,
    ! [Uu: nat] : ( aa_nat_set_nat(aTP_Lamm_akj,Uu) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_if,Uu)) ) ).

% ATP.lambda_17
tff(fact_9705_ATP_Olambda__18,axiom,
    ! [Uu: num] : ( aa_num_option_num(aTP_Lamm_akl,Uu) = some_num(aa_num_num(bit1,Uu)) ) ).

% ATP.lambda_18
tff(fact_9706_ATP_Olambda__19,axiom,
    ! [Uu: num] : ( aa_num_option_num(aTP_Lamm_akn,Uu) = some_num(aa_num_num(bit0,Uu)) ) ).

% ATP.lambda_19
tff(fact_9707_ATP_Olambda__20,axiom,
    ! [Uu: int] : ( aa_int2565445551602696444nt_int(aTP_Lamm_ajb,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,Uu)) ) ).

% ATP.lambda_20
tff(fact_9708_ATP_Olambda__21,axiom,
    ! [Uu: int] : ( aa_int2565445551602696444nt_int(aTP_Lamm_ajc,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(abs_abs_int,Uu)) ) ).

% ATP.lambda_21
tff(fact_9709_ATP_Olambda__22,axiom,
    ! [Uu: nat] : ( aa_nat8803788312118491388at_nat(aTP_Lamm_ia,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ) ).

% ATP.lambda_22
tff(fact_9710_ATP_Olambda__23,axiom,
    ! [Uu: int] : ( aa_int_nat(aTP_Lamm_aki,Uu) = aa_int_nat(nat2,aa_int_int(abs_abs_int,Uu)) ) ).

% ATP.lambda_23
tff(fact_9711_ATP_Olambda__24,axiom,
    ! [Uu: list_char] :
      ( aa_list_char_o(aTP_Lamm_amc,Uu)
    <=> ! [X4: char] :
          ( member_char(X4,set_char2(Uu))
         => ~ aa_char_o(digit7,X4) ) ) ).

% ATP.lambda_24
tff(fact_9712_ATP_Olambda__25,axiom,
    ! [Uu: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_aly,Uu)
    <=> ? [X4: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X4),X4)
          & ( Uu = collec213857154873943460nt_int(aa_Pro3952998155787061599_int_o(ratrel,X4)) ) ) ) ).

% ATP.lambda_25
tff(fact_9713_ATP_Olambda__26,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aTP_Lamm_amo,Uu)
    <=> ? [X4: product_prod_nat_nat] :
          ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,X4),X4)
          & ( Uu = collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(intrel,X4)) ) ) ) ).

% ATP.lambda_26
tff(fact_9714_ATP_Olambda__27,axiom,
    ! [Uu: num,Uua: nat] : ( aa_nat_option_num(aTP_Lamm_akq(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_ako(Uua),aTP_Lamm_akp(Uua),Uu) ) ).

% ATP.lambda_27
tff(fact_9715_ATP_Olambda__28,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_Code_integer(aTP_Lamm_ov(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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_28
tff(fact_9716_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_rat(aTP_Lamm_ox(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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_29
tff(fact_9717_ATP_Olambda__30,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ow,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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_30
tff(fact_9718_ATP_Olambda__31,axiom,
    ! [Uu: pred_Product_unit,Uua: produc7822875418678951345atural] :
      ( aa_Pro6676232280479689400atural(aa_pre2304647123621358054atural(aTP_Lamm_amm,Uu),Uua) = $ite(aa_Product_unit_o(eval_Product_unit(Uu),product_Unity),aa_Pro6676232280479689400atural(produc5069803637994805237atural(bot_bo2717538794563056311t_unit),Uua),aa_Pro6676232280479689400atural(produc5069803637994805237atural(single_Product_unit(product_Unity)),Uua)) ) ).

% ATP.lambda_31
tff(fact_9719_ATP_Olambda__32,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ajw,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_32
tff(fact_9720_ATP_Olambda__33,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_alc,Uu),Uua) = $ite(Uua = 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,Uu),Uua)) ) ).

% ATP.lambda_33
tff(fact_9721_ATP_Olambda__34,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_Code_integer(aTP_Lamm_os(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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_34
tff(fact_9722_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_rat(aTP_Lamm_ou(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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_35
tff(fact_9723_ATP_Olambda__36,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ot,Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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_36
tff(fact_9724_ATP_Olambda__37,axiom,
    ! [Uu: nat,Uua: num] : ( aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_akr,Uu),Uua) = aa_nat_option_num(case_nat_option_num(none_num,aTP_Lamm_akq(Uua)),Uu) ) ).

% ATP.lambda_37
tff(fact_9725_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: num] : ( aa_num_option_num(aTP_Lamm_ako(Uu),Uua) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_akn),bit_take_bit_num(Uu,Uua)) ) ).

% ATP.lambda_38
tff(fact_9726_ATP_Olambda__39,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : ( aa_nat_int(aTP_Lamm_aaa(Uu),Uua) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_zz(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ) ).

% ATP.lambda_39
tff(fact_9727_ATP_Olambda__40,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_zy(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oe(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ) ).

% ATP.lambda_40
tff(fact_9728_ATP_Olambda__41,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_of(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oe(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ) ).

% ATP.lambda_41
tff(fact_9729_ATP_Olambda__42,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : ( aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_ala,Uu),Uua) = 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_Pro6799258434465214121nt_int(product_fst_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua))) ) ).

% ATP.lambda_42
tff(fact_9730_ATP_Olambda__43,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_Code_integer(aTP_Lamm_sd(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_43
tff(fact_9731_ATP_Olambda__44,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_sf(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),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_44
tff(fact_9732_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_se,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(aa_int_fun_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_45
tff(fact_9733_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_oa(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Uua)) ) ).

% ATP.lambda_46
tff(fact_9734_ATP_Olambda__47,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_ka,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))),aa_Code_integer_int(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_47
tff(fact_9735_ATP_Olambda__48,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : ( aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_ale,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua))) ) ).

% ATP.lambda_48
tff(fact_9736_ATP_Olambda__49,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_akf(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_49
tff(fact_9737_ATP_Olambda__50,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_Code_integer(aTP_Lamm_oz(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_50
tff(fact_9738_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_pb(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ) ).

% ATP.lambda_51
tff(fact_9739_ATP_Olambda__52,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pa,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_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_52
tff(fact_9740_ATP_Olambda__53,axiom,
    ! [Uu: rat,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_alk(Uu),Uua)
    <=> ( ( Uu = aa_int_rat(aa_int_fun_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_53
tff(fact_9741_ATP_Olambda__54,axiom,
    ! [Uu: rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_nv(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))),Uua) ) ).

% ATP.lambda_54
tff(fact_9742_ATP_Olambda__55,axiom,
    ! [Uu: rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_nh(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Uu),numeral_numeral_rat(aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ) ).

% ATP.lambda_55
tff(fact_9743_ATP_Olambda__56,axiom,
    ! [Uu: rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_nw(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)) ) ).

% ATP.lambda_56
tff(fact_9744_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ig,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_57
tff(fact_9745_ATP_Olambda__58,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rw,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),numeral_numeral_nat(aa_num_num(bit0,one))) ) ).

% ATP.lambda_58
tff(fact_9746_ATP_Olambda__59,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_alq(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_59
tff(fact_9747_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aks,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_60
tff(fact_9748_ATP_Olambda__61,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ra,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_61
tff(fact_9749_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qz,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_62
tff(fact_9750_ATP_Olambda__63,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : ( aa_nat_Code_integer(aTP_Lamm_rk(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua))),aa_nat_Code_integer(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_63
tff(fact_9751_ATP_Olambda__64,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] : ( aa_nat_assn(aTP_Lamm_rl(Uu),Uua) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua))),aa_nat_assn(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_64
tff(fact_9752_ATP_Olambda__65,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_rm(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_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,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_65
tff(fact_9753_ATP_Olambda__66,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_rn(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(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,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_66
tff(fact_9754_ATP_Olambda__67,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : ( aa_nat_int(aTP_Lamm_ro(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,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,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_67
tff(fact_9755_ATP_Olambda__68,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_mn(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_68
tff(fact_9756_ATP_Olambda__69,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_mp(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,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,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_69
tff(fact_9757_ATP_Olambda__70,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : ( aa_nat_int(aTP_Lamm_mo(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,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,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_70
tff(fact_9758_ATP_Olambda__71,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_ly(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ) ).

% ATP.lambda_71
tff(fact_9759_ATP_Olambda__72,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : ( aa_nat_int(aTP_Lamm_lz(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_72
tff(fact_9760_ATP_Olambda__73,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_qq(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uua)) ) ).

% ATP.lambda_73
tff(fact_9761_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : ( aa_nat_int(aTP_Lamm_qs(Uu),Uua) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uua)) ) ).

% ATP.lambda_74
tff(fact_9762_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_lt(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uua)) ) ).

% ATP.lambda_75
tff(fact_9763_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_ur(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ) ).

% ATP.lambda_76
tff(fact_9764_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_ml(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ) ).

% ATP.lambda_77
tff(fact_9765_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : ( aa_nat_int(aTP_Lamm_mm(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_78
tff(fact_9766_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : ( aa_nat_Code_integer(aTP_Lamm_mi(Uu),Uua) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(Uu,Uua)),aa_nat_Code_integer(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ) ).

% ATP.lambda_79
tff(fact_9767_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_mj(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ) ).

% ATP.lambda_80
tff(fact_9768_ATP_Olambda__81,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(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ) ).

% ATP.lambda_81
tff(fact_9769_ATP_Olambda__82,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_ob(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_82
tff(fact_9770_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : ( aa_nat_int(aTP_Lamm_oc(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_83
tff(fact_9771_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_abt(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_84
tff(fact_9772_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : ( aa_nat_int(aTP_Lamm_abu(Uu),Uua) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ) ).

% ATP.lambda_85
tff(fact_9773_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_abs(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_86
tff(fact_9774_ATP_Olambda__87,axiom,
    ! [Uu: fun_se7586849273513198688nteger,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_aat(Uu),Uua)
    <=> ( aa_set871585252387937313nteger(Uu,Uua) = zero_z3403309356797280102nteger ) ) ).

% ATP.lambda_87
tff(fact_9775_ATP_Olambda__88,axiom,
    ! [Uu: fun_set_nat_rat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_aau(Uu),Uua)
    <=> ( aa_set_nat_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_88
tff(fact_9776_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aao(Uu),Uua)
    <=> ( aa_nat_Code_integer(Uu,Uua) = zero_z3403309356797280102nteger ) ) ).

% ATP.lambda_89
tff(fact_9777_ATP_Olambda__90,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aap(Uu),Uua)
    <=> ( aa_nat_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_90
tff(fact_9778_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aar(Uu),Uua)
    <=> ( aa_nat_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_91
tff(fact_9779_ATP_Olambda__92,axiom,
    ! [Uu: fun_nat_int,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aaq(Uu),Uua)
    <=> ( aa_nat_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_92
tff(fact_9780_ATP_Olambda__93,axiom,
    ! [Uu: fun_int_Code_integer,Uua: int] :
      ( aa_int_o(aTP_Lamm_aal(Uu),Uua)
    <=> ( aa_int_Code_integer(Uu,Uua) = zero_z3403309356797280102nteger ) ) ).

% ATP.lambda_93
tff(fact_9781_ATP_Olambda__94,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_aam(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_94
tff(fact_9782_ATP_Olambda__95,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_aan(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_95
tff(fact_9783_ATP_Olambda__96,axiom,
    ! [Uu: fun_int_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_aas(Uu),Uua)
    <=> ( aa_int_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_96
tff(fact_9784_ATP_Olambda__97,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_abj(Uu),Uua)
    <=> ( aa_nat_Code_integer(Uu,Uua) = one_one_Code_integer ) ) ).

% ATP.lambda_97
tff(fact_9785_ATP_Olambda__98,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_abk(Uu),Uua)
    <=> ( aa_nat_assn(Uu,Uua) = one_one_assn ) ) ).

% ATP.lambda_98
tff(fact_9786_ATP_Olambda__99,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_abl(Uu),Uua)
    <=> ( aa_nat_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_99
tff(fact_9787_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_abm(Uu),Uua)
    <=> ( aa_nat_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_100
tff(fact_9788_ATP_Olambda__101,axiom,
    ! [Uu: fun_nat_int,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_abn(Uu),Uua)
    <=> ( aa_nat_int(Uu,Uua) = one_one_int ) ) ).

% ATP.lambda_101
tff(fact_9789_ATP_Olambda__102,axiom,
    ! [Uu: fun_int_Code_integer,Uua: int] :
      ( aa_int_o(aTP_Lamm_abf(Uu),Uua)
    <=> ( aa_int_Code_integer(Uu,Uua) = one_one_Code_integer ) ) ).

% ATP.lambda_102
tff(fact_9790_ATP_Olambda__103,axiom,
    ! [Uu: fun_int_assn,Uua: int] :
      ( aa_int_o(aTP_Lamm_abg(Uu),Uua)
    <=> ( aa_int_assn(Uu,Uua) = one_one_assn ) ) ).

% ATP.lambda_103
tff(fact_9791_ATP_Olambda__104,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_abh(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_104
tff(fact_9792_ATP_Olambda__105,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_abi(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_105
tff(fact_9793_ATP_Olambda__106,axiom,
    ! [Uu: fun_int_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_abo(Uu),Uua)
    <=> ( aa_int_int(Uu,Uua) = one_one_int ) ) ).

% ATP.lambda_106
tff(fact_9794_ATP_Olambda__107,axiom,
    ! [Uu: code_integer,Uua: $o] : ( aa_o_char(aa_Cod7842134615204088097o_char(aTP_Lamm_ake,Uu),(Uua)) = produc4188289175737317920o_char(aTP_Lamm_akd((Uua)),code_bit_cut_integer(Uu)) ) ).

% ATP.lambda_107
tff(fact_9795_ATP_Olambda__108,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_alz,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_108
tff(fact_9796_ATP_Olambda__109,axiom,
    ! [Uu: num,Uua: num] : ( aa_num_int(aTP_Lamm_akh(Uu),Uua) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,numeral_numeral_nat(Uu)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),numeral_numeral_nat(Uu))),aa_num_int(numeral_numeral_int,Uua))) ) ).

% ATP.lambda_109
tff(fact_9797_ATP_Olambda__110,axiom,
    ! [Uu: assn,Uua: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_fh,Uu),Uua)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,Uua)
        & ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uua) ) ) ).

% ATP.lambda_110
tff(fact_9798_ATP_Olambda__111,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_em,Uu),Uua)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,Uua)
        & ~ aa_Pro653082453696054616_nat_o(Uu,Uua) ) ) ).

% ATP.lambda_111
tff(fact_9799_ATP_Olambda__112,axiom,
    ! [Uu: char,Uua: char] :
      ( aa_char_o(aa_char_fun_char_o(aTP_Lamm_ama,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,Uu)),aa_char_nat(comm_s629917340098488124ar_nat,Uua)) ) ).

% ATP.lambda_112
tff(fact_9800_ATP_Olambda__113,axiom,
    ! [Uu: num,Uua: num] : ( aa_num_int(aa_num_fun_num_int(aTP_Lamm_akw,Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Uu)),aa_num_int(numeral_numeral_int,Uua)) ) ).

% ATP.lambda_113
tff(fact_9801_ATP_Olambda__114,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_alr,Uu),Uua) = aa_int_rat(aa_int_fun_int_rat(fract,aa_nat_int(nat_int_decode,Uu)),aa_nat_int(nat_int_decode,Uua)) ) ).

% ATP.lambda_114
tff(fact_9802_ATP_Olambda__115,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_jx,Uu),Uua) = $let(
          l2: num,
          l2:= code_num_of_integer(Uu),
          $let(
            l3: num,
            l3:= plus_plus_num(l2,l2),
            $ite(Uua = zero_z3403309356797280102nteger,l3,plus_plus_num(l3,one)) ) ) ) ).

% ATP.lambda_115
tff(fact_9803_ATP_Olambda__116,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_jz,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_116
tff(fact_9804_ATP_Olambda__117,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_og(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ) ).

% ATP.lambda_117
tff(fact_9805_ATP_Olambda__118,axiom,
    ! [Uu: code_integer,Uua: code_integer] : ( aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_go,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)),one_one_Code_integer)) ) ).

% ATP.lambda_118
tff(fact_9806_ATP_Olambda__119,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gn,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,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)),one_one_nat)) ) ).

% ATP.lambda_119
tff(fact_9807_ATP_Olambda__120,axiom,
    ! [Uu: int,Uua: int] : ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gm,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_120
tff(fact_9808_ATP_Olambda__121,axiom,
    ! [Uu: code_integer,Uua: code_integer] : ( aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_gu,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)) ) ).

% ATP.lambda_121
tff(fact_9809_ATP_Olambda__122,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gt,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)) ) ).

% ATP.lambda_122
tff(fact_9810_ATP_Olambda__123,axiom,
    ! [Uu: int,Uua: int] : ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gs,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_123
tff(fact_9811_ATP_Olambda__124,axiom,
    ! [Uu: rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_or(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri681578069525770553at_rat,Uua)),numeral_numeral_rat(aa_num_num(bit0,one)))) ) ).

% ATP.lambda_124
tff(fact_9812_ATP_Olambda__125,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sg,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)) ) ).

% ATP.lambda_125
tff(fact_9813_ATP_Olambda__126,axiom,
    ! [Uu: int,Uua: int] : ( aa_int_int(aa_int_fun_int_int(aTP_Lamm_ih,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_126
tff(fact_9814_ATP_Olambda__127,axiom,
    ! [Uu: code_natural,Uua: nat] : ( aa_nat_Code_natural(aTP_Lamm_qi(Uu),Uua) = minus_7197305767214868737atural(Uu,aa_nat_Code_natural(semiri3763490453095760265atural,Uua)) ) ).

% ATP.lambda_127
tff(fact_9815_ATP_Olambda__128,axiom,
    ! [Uu: code_integer,Uua: nat] : ( aa_nat_Code_integer(aTP_Lamm_qh(Uu),Uua) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua)) ) ).

% ATP.lambda_128
tff(fact_9816_ATP_Olambda__129,axiom,
    ! [Uu: rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_qg(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ) ).

% ATP.lambda_129
tff(fact_9817_ATP_Olambda__130,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qj,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ) ).

% ATP.lambda_130
tff(fact_9818_ATP_Olambda__131,axiom,
    ! [Uu: int,Uua: nat] : ( aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_qk,Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ) ).

% ATP.lambda_131
tff(fact_9819_ATP_Olambda__132,axiom,
    ! [Uu: rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_rb(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ) ).

% ATP.lambda_132
tff(fact_9820_ATP_Olambda__133,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rc,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ) ).

% ATP.lambda_133
tff(fact_9821_ATP_Olambda__134,axiom,
    ! [Uu: int,Uua: nat] : ( aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_rd,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ) ).

% ATP.lambda_134
tff(fact_9822_ATP_Olambda__135,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_list_nat(aa_nat6157064778417537467st_nat(aTP_Lamm_als,Uu),Uua) = cons_nat(Uu,aa_nat_list_nat(nat_list_decode,Uua)) ) ).

% ATP.lambda_135
tff(fact_9823_ATP_Olambda__136,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_akg,Uu),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_136
tff(fact_9824_ATP_Olambda__137,axiom,
    ! [Uu: filter_nat,Uua: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(aTP_Lamm_gb,Uu),Uua)
    <=> aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Uua),Uu) ) ).

% ATP.lambda_137
tff(fact_9825_ATP_Olambda__138,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: set_Pr8551490117392284871at_nat] :
      ( aa_set192106157127879885_nat_o(aa_set8206400607124912991_nat_o(aTP_Lamm_fr,Uu),Uua)
    <=> ord_le8081472938463900775at_nat(Uua,Uu) ) ).

% ATP.lambda_138
tff(fact_9826_ATP_Olambda__139,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(aTP_Lamm_ft,Uu),Uua)
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Uua),Uu) ) ).

% ATP.lambda_139
tff(fact_9827_ATP_Olambda__140,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_fv,Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Uua),Uu) ) ).

% ATP.lambda_140
tff(fact_9828_ATP_Olambda__141,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(aTP_Lamm_nr,Uu),Uua)
    <=> aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(ord_le2843351958646193337nt_int,Uua),Uu) ) ).

% ATP.lambda_141
tff(fact_9829_ATP_Olambda__142,axiom,
    ! [Uu: set_o,Uua: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(aTP_Lamm_fx,Uu),Uua)
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Uua),Uu) ) ).

% ATP.lambda_142
tff(fact_9830_ATP_Olambda__143,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_gd,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Uua),Uu) ) ).

% ATP.lambda_143
tff(fact_9831_ATP_Olambda__144,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_fz,Uu),Uua)
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Uua),Uu) ) ).

% ATP.lambda_144
tff(fact_9832_ATP_Olambda__145,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_nt,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu) ) ).

% ATP.lambda_145
tff(fact_9833_ATP_Olambda__146,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gf,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_146
tff(fact_9834_ATP_Olambda__147,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_nu,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_147
tff(fact_9835_ATP_Olambda__148,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(aTP_Lamm_ns,Uu),Uua)
    <=> aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,Uua),Uu) ) ).

% ATP.lambda_148
tff(fact_9836_ATP_Olambda__149,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiu,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ) ).

% ATP.lambda_149
tff(fact_9837_ATP_Olambda__150,axiom,
    ! [Uu: literal,Uua: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(aTP_Lamm_xl,Uu),Uua)
    <=> aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Uua),Uu) ) ).

% ATP.lambda_150
tff(fact_9838_ATP_Olambda__151,axiom,
    ! [Uu: filter_nat,Uua: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(aTP_Lamm_gc,Uu),Uua)
    <=> aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,Uua),Uu) ) ).

% ATP.lambda_151
tff(fact_9839_ATP_Olambda__152,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: set_Pr8551490117392284871at_nat] :
      ( aa_set192106157127879885_nat_o(aa_set8206400607124912991_nat_o(aTP_Lamm_fs,Uu),Uua)
    <=> ord_le7642048601412989811at_nat(Uua,Uu) ) ).

% ATP.lambda_152
tff(fact_9840_ATP_Olambda__153,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(aTP_Lamm_fu,Uu),Uua)
    <=> aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(Uua),Uu) ) ).

% ATP.lambda_153
tff(fact_9841_ATP_Olambda__154,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_fw,Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,Uua),Uu) ) ).

% ATP.lambda_154
tff(fact_9842_ATP_Olambda__155,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(aTP_Lamm_xj,Uu),Uua)
    <=> aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(ord_le7563427860532173253nt_int,Uua),Uu) ) ).

% ATP.lambda_155
tff(fact_9843_ATP_Olambda__156,axiom,
    ! [Uu: set_o,Uua: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(aTP_Lamm_fy,Uu),Uua)
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,Uua),Uu) ) ).

% ATP.lambda_156
tff(fact_9844_ATP_Olambda__157,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_ge,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Uua),Uu) ) ).

% ATP.lambda_157
tff(fact_9845_ATP_Olambda__158,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_ga,Uu),Uua)
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,Uua),Uu) ) ).

% ATP.lambda_158
tff(fact_9846_ATP_Olambda__159,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_xm,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu) ) ).

% ATP.lambda_159
tff(fact_9847_ATP_Olambda__160,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gg,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_160
tff(fact_9848_ATP_Olambda__161,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_xn,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu) ) ).

% ATP.lambda_161
tff(fact_9849_ATP_Olambda__162,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(aTP_Lamm_xk,Uu),Uua)
    <=> aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le6684905062368609691_nat_o,Uua),Uu) ) ).

% ATP.lambda_162
tff(fact_9850_ATP_Olambda__163,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiv,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ) ).

% ATP.lambda_163
tff(fact_9851_ATP_Olambda__164,axiom,
    ! [Uu: code_integer,Uua: code_integer] : ( aa_Cod1783974652533396728nteger(aTP_Lamm_ait(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uua),Uu) ) ).

% ATP.lambda_164
tff(fact_9852_ATP_Olambda__165,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_all,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ) ).

% ATP.lambda_165
tff(fact_9853_ATP_Olambda__166,axiom,
    ! [Uu: int,Uua: int] : ( aa_int_int(aa_int_fun_int_int(aTP_Lamm_ais,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ) ).

% ATP.lambda_166
tff(fact_9854_ATP_Olambda__167,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_ie,Uu),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu) ) ).

% ATP.lambda_167
tff(fact_9855_ATP_Olambda__168,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_if,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_168
tff(fact_9856_ATP_Olambda__169,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_id,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_169
tff(fact_9857_ATP_Olambda__170,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ajh,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ) ).

% ATP.lambda_170
tff(fact_9858_ATP_Olambda__171,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qm,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ) ).

% ATP.lambda_171
tff(fact_9859_ATP_Olambda__172,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(aTP_Lamm_ch(Uu),Uua)
    <=> member6689249552917799696at_nat(Uua,Uu) ) ).

% ATP.lambda_172
tff(fact_9860_ATP_Olambda__173,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aTP_Lamm_ci(Uu),Uua)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uua),Uu) ) ).

% ATP.lambda_173
tff(fact_9861_ATP_Olambda__174,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_bt(Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uua),Uu) ) ).

% ATP.lambda_174
tff(fact_9862_ATP_Olambda__175,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_br(Uu),Uua)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(Uua),Uu) ) ).

% ATP.lambda_175
tff(fact_9863_ATP_Olambda__176,axiom,
    ! [Uu: set_list_char,Uua: list_char] :
      ( aa_list_char_o(aTP_Lamm_bs(Uu),Uua)
    <=> member_list_char(Uua,Uu) ) ).

% ATP.lambda_176
tff(fact_9864_ATP_Olambda__177,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_ag(Uu),Uua)
    <=> aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uua),Uu) ) ).

% ATP.lambda_177
tff(fact_9865_ATP_Olambda__178,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ah(Uu),Uua)
    <=> aa_set_set_nat_o(member_set_nat(Uua),Uu) ) ).

% ATP.lambda_178
tff(fact_9866_ATP_Olambda__179,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ai,Uu),Uua)
    <=> aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uua),Uu) ) ).

% ATP.lambda_179
tff(fact_9867_ATP_Olambda__180,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_aj(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_180
tff(fact_9868_ATP_Olambda__181,axiom,
    ! [Uu: set_fu3555266933746557007_nat_o,Uua: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aTP_Lamm_ak(Uu),Uua)
    <=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uua),Uu) ) ).

% ATP.lambda_181
tff(fact_9869_ATP_Olambda__182,axiom,
    ! [Uu: produc4166570645942440679at_nat,Uua: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(aTP_Lamm_jw(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_182
tff(fact_9870_ATP_Olambda__183,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_Pro1894297876208389727_nat_o(aTP_Lamm_ii,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_183
tff(fact_9871_ATP_Olambda__184,axiom,
    ! [Uu: product_prod_nat_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_im,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_184
tff(fact_9872_ATP_Olambda__185,axiom,
    ! [Uu: $o,Uua: $o] :
      ( aa_o_o(aa_o_fun_o_o(aTP_Lamm_in,(Uu)),(Uua))
    <=> ( (Uua)
      <=> (Uu) ) ) ).

% ATP.lambda_185
tff(fact_9873_ATP_Olambda__186,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(aTP_Lamm_ij,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_186
tff(fact_9874_ATP_Olambda__187,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_ik,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_187
tff(fact_9875_ATP_Olambda__188,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_io,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_188
tff(fact_9876_ATP_Olambda__189,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ip,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_189
tff(fact_9877_ATP_Olambda__190,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(aTP_Lamm_il,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_190
tff(fact_9878_ATP_Olambda__191,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_afj(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Pro4243173770044397973at_nat(Uu,Uua)) ) ).

% ATP.lambda_191
tff(fact_9879_ATP_Olambda__192,axiom,
    ! [Uu: fun_se4101287982975402108nt_nat,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_aez(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set4753215920113914285nt_nat(Uu,Uua)) ) ).

% ATP.lambda_192
tff(fact_9880_ATP_Olambda__193,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_afb(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_193
tff(fact_9881_ATP_Olambda__194,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aff(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_194
tff(fact_9882_ATP_Olambda__195,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_afh(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(Uu,Uua)) ) ).

% ATP.lambda_195
tff(fact_9883_ATP_Olambda__196,axiom,
    ! [Uu: fun_fu3504661237974713170_o_nat,Uua: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aTP_Lamm_afd(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_fun767828674014743683_o_nat(Uu,Uua)) ) ).

% ATP.lambda_196
tff(fact_9884_ATP_Olambda__197,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_set_nat(aTP_Lamm_alu(Uu),Uua) = aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Uu) ) ).

% ATP.lambda_197
tff(fact_9885_ATP_Olambda__198,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : ( aa_nat_Code_integer(aTP_Lamm_qn(Uu),Uua) = aa_nat_Code_integer(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_198
tff(fact_9886_ATP_Olambda__199,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] : ( aa_nat_assn(aTP_Lamm_qo(Uu),Uua) = aa_nat_assn(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_199
tff(fact_9887_ATP_Olambda__200,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aky(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_200
tff(fact_9888_ATP_Olambda__201,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_ma(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_201
tff(fact_9889_ATP_Olambda__202,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_lw(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_202
tff(fact_9890_ATP_Olambda__203,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : ( aa_nat_int(aTP_Lamm_mb(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_203
tff(fact_9891_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: num] : ( aa_num_option_num(aTP_Lamm_akp(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ) ).

% ATP.lambda_204
tff(fact_9892_ATP_Olambda__205,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_alj,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_ali(Uu),Uua)) ) ).

% ATP.lambda_205
tff(fact_9893_ATP_Olambda__206,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ajp,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ajo(Uu),Uua)) ) ).

% ATP.lambda_206
tff(fact_9894_ATP_Olambda__207,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ajn,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ajm(Uu),Uua)) ) ).

% ATP.lambda_207
tff(fact_9895_ATP_Olambda__208,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_ajl,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_ajk(Uu),Uua)) ) ).

% ATP.lambda_208
tff(fact_9896_ATP_Olambda__209,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_ajj,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_aji(Uu),Uua)) ) ).

% ATP.lambda_209
tff(fact_9897_ATP_Olambda__210,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_aiy,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aix(Uu),Uua)) ) ).

% ATP.lambda_210
tff(fact_9898_ATP_Olambda__211,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_akk,Uu),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_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,numeral_numeral_nat(aa_num_num(bit0,one))),Uua))) ) ).

% ATP.lambda_211
tff(fact_9899_ATP_Olambda__212,axiom,
    ! [Uu: code_natural,Uua: code_natural] : ( aa_Cod6352003912080677724atural(aTP_Lamm_amg(Uu),Uua) = produc6639722614265839536atural(plus_p4538020629002901425atural(Uua,times_2397367101498566445atural(Uu,numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))))))))))))))))))))) ) ).

% ATP.lambda_212
tff(fact_9900_ATP_Olambda__213,axiom,
    ! [Uu: rat,Uua: nat] : ( aa_nat_fun_rat_rat(aTP_Lamm_oy(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ) ).

% ATP.lambda_213
tff(fact_9901_ATP_Olambda__214,axiom,
    ! [Uu: code_integer,Uua: nat] : ( aa_nat6383147911832638789nteger(aTP_Lamm_nd(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))) ) ).

% ATP.lambda_214
tff(fact_9902_ATP_Olambda__215,axiom,
    ! [Uu: rat,Uua: nat] : ( aa_nat_fun_rat_rat(aTP_Lamm_nf(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ) ).

% ATP.lambda_215
tff(fact_9903_ATP_Olambda__216,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_fun_nat_nat(aTP_Lamm_ng(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_216
tff(fact_9904_ATP_Olambda__217,axiom,
    ! [Uu: int,Uua: nat] : ( aa_nat_fun_int_int(aTP_Lamm_ne(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_217
tff(fact_9905_ATP_Olambda__218,axiom,
    ! [Uu: code_natural,Uua: code_natural] : ( aa_Cod6352003912080677724atural(aTP_Lamm_ami(Uu),Uua) = produc6639722614265839536atural(modulo8411746178871703098atural(Uua,Uu)) ) ).

% ATP.lambda_218
tff(fact_9906_ATP_Olambda__219,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aku,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ) ).

% ATP.lambda_219
tff(fact_9907_ATP_Olambda__220,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_akt,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ) ).

% ATP.lambda_220
tff(fact_9908_ATP_Olambda__221,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_air,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ) ).

% ATP.lambda_221
tff(fact_9909_ATP_Olambda__222,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiq,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ) ).

% ATP.lambda_222
tff(fact_9910_ATP_Olambda__223,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_au(Uu),Uua)
    <=> ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uua),Uu) ) ).

% ATP.lambda_223
tff(fact_9911_ATP_Olambda__224,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_av(Uu),Uua)
    <=> ~ aa_set_set_nat_o(member_set_nat(Uua),Uu) ) ).

% ATP.lambda_224
tff(fact_9912_ATP_Olambda__225,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_aw,Uu),Uua)
    <=> ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uua),Uu) ) ).

% ATP.lambda_225
tff(fact_9913_ATP_Olambda__226,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ax(Uu),Uua)
    <=> ~ aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_226
tff(fact_9914_ATP_Olambda__227,axiom,
    ! [Uu: set_fu3555266933746557007_nat_o,Uua: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aTP_Lamm_ay(Uu),Uua)
    <=> ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uua),Uu) ) ).

% ATP.lambda_227
tff(fact_9915_ATP_Olambda__228,axiom,
    ! [Uu: fun_list_nat_o,Uua: list_nat] : ( aa_list_nat_rat(aTP_Lamm_ahc(Uu),Uua) = aa_o_rat(zero_n2052037380579107095ol_rat,aa_list_nat_o(Uu,Uua)) ) ).

% ATP.lambda_228
tff(fact_9916_ATP_Olambda__229,axiom,
    ! [Uu: fun_o_o,Uua: $o] : ( aa_o_rat(aTP_Lamm_agt(Uu),(Uua)) = aa_o_rat(zero_n2052037380579107095ol_rat,aa_o_o(Uu,(Uua))) ) ).

% ATP.lambda_229
tff(fact_9917_ATP_Olambda__230,axiom,
    ! [Uu: fun_o_o,Uua: $o] : ( aa_o_nat(aTP_Lamm_agy(Uu),(Uua)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_o_o(Uu,(Uua))) ) ).

% ATP.lambda_230
tff(fact_9918_ATP_Olambda__231,axiom,
    ! [Uu: fun_o_o,Uua: $o] : ( aa_o_int(aTP_Lamm_agw(Uu),(Uua)) = aa_o_int(zero_n2684676970156552555ol_int,aa_o_o(Uu,(Uua))) ) ).

% ATP.lambda_231
tff(fact_9919_ATP_Olambda__232,axiom,
    ! [Uu: fun_nat_o,Uua: nat] : ( aa_nat_rat(aTP_Lamm_agv(Uu),Uua) = aa_o_rat(zero_n2052037380579107095ol_rat,aa_nat_o(Uu,Uua)) ) ).

% ATP.lambda_232
tff(fact_9920_ATP_Olambda__233,axiom,
    ! [Uu: fun_nat_o,Uua: nat] : ( aa_nat_nat(aTP_Lamm_aha(Uu),Uua) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(Uu,Uua)) ) ).

% ATP.lambda_233
tff(fact_9921_ATP_Olambda__234,axiom,
    ! [Uu: fun_nat_o,Uua: nat] : ( aa_nat_int(aTP_Lamm_agx(Uu),Uua) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(Uu,Uua)) ) ).

% ATP.lambda_234
tff(fact_9922_ATP_Olambda__235,axiom,
    ! [Uu: fun_int_o,Uua: int] : ( aa_int_rat(aTP_Lamm_agu(Uu),Uua) = aa_o_rat(zero_n2052037380579107095ol_rat,aa_int_o(Uu,Uua)) ) ).

% ATP.lambda_235
tff(fact_9923_ATP_Olambda__236,axiom,
    ! [Uu: fun_int_o,Uua: int] : ( aa_int_nat(aTP_Lamm_agz(Uu),Uua) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_int_o(Uu,Uua)) ) ).

% ATP.lambda_236
tff(fact_9924_ATP_Olambda__237,axiom,
    ! [Uu: fun_int_o,Uua: int] : ( aa_int_int(aTP_Lamm_ahb(Uu),Uua) = aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(Uu,Uua)) ) ).

% ATP.lambda_237
tff(fact_9925_ATP_Olambda__238,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_mx(Uu),Uua) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_238
tff(fact_9926_ATP_Olambda__239,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_my(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_239
tff(fact_9927_ATP_Olambda__240,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : ( aa_nat_int(aTP_Lamm_mw(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_240
tff(fact_9928_ATP_Olambda__241,axiom,
    ! [Uu: fun_int_nat,Uua: int] : ( aa_int_int(aTP_Lamm_mv(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ) ).

% ATP.lambda_241
tff(fact_9929_ATP_Olambda__242,axiom,
    ! [Uu: fun_int_int,Uua: int] : ( aa_int_int(aTP_Lamm_lk(Uu),Uua) = aa_int_int(uminus_uminus_int,aa_int_int(Uu,Uua)) ) ).

% ATP.lambda_242
tff(fact_9930_ATP_Olambda__243,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : ( aa_nat6383147911832638789nteger(aTP_Lamm_qu(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uu,Uua)) ) ).

% ATP.lambda_243
tff(fact_9931_ATP_Olambda__244,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] : ( aa_nat_fun_assn_assn(aTP_Lamm_qv(Uu),Uua) = aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Uu,Uua)) ) ).

% ATP.lambda_244
tff(fact_9932_ATP_Olambda__245,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : ( aa_nat_fun_rat_rat(aTP_Lamm_qw(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uua)) ) ).

% ATP.lambda_245
tff(fact_9933_ATP_Olambda__246,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : ( aa_nat_fun_nat_nat(aTP_Lamm_qx(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_246
tff(fact_9934_ATP_Olambda__247,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : ( aa_nat_fun_int_int(aTP_Lamm_qy(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ) ).

% ATP.lambda_247
tff(fact_9935_ATP_Olambda__248,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : ( aa_nat_rat(aTP_Lamm_pf(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(Uu,Uua)) ) ).

% ATP.lambda_248
tff(fact_9936_ATP_Olambda__249,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : ( aa_nat_int(aTP_Lamm_pg(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ) ).

% ATP.lambda_249
tff(fact_9937_ATP_Olambda__250,axiom,
    ! [Uu: fun_int_int,Uua: int] : ( aa_int_rat(aTP_Lamm_kb(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ) ).

% ATP.lambda_250
tff(fact_9938_ATP_Olambda__251,axiom,
    ! [Uu: fun_int_int,Uua: int] : ( aa_int_int(aTP_Lamm_kc(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ) ).

% ATP.lambda_251
tff(fact_9939_ATP_Olambda__252,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : ( aa_nat6383147911832638789nteger(aTP_Lamm_md(Uu),Uua) = aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(Uu,Uua)) ) ).

% ATP.lambda_252
tff(fact_9940_ATP_Olambda__253,axiom,
    ! [Uu: fun_na2338080690716861136at_nat,Uua: nat] : ( aa_nat1862996181841379217at_nat(aTP_Lamm_me(Uu),Uua) = aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,aa_nat2240542744625857721at_nat(Uu,Uua)) ) ).

% ATP.lambda_253
tff(fact_9941_ATP_Olambda__254,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : ( aa_nat_fun_rat_rat(aTP_Lamm_mf(Uu),Uua) = aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uua)) ) ).

% ATP.lambda_254
tff(fact_9942_ATP_Olambda__255,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : ( aa_nat_fun_nat_nat(aTP_Lamm_mh(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_255
tff(fact_9943_ATP_Olambda__256,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : ( aa_nat_fun_int_int(aTP_Lamm_mg(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ) ).

% ATP.lambda_256
tff(fact_9944_ATP_Olambda__257,axiom,
    ! [Uu: fun_int_int,Uua: int] : ( aa_int_int(aTP_Lamm_ki(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ) ).

% ATP.lambda_257
tff(fact_9945_ATP_Olambda__258,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_aem,Uu),Uua)
    <=> ~ aa_Pro6150637604445964947_nat_o(Uu,Uua) ) ).

% ATP.lambda_258
tff(fact_9946_ATP_Olambda__259,axiom,
    ! [Uu: fun_list_nat_o,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_aen(Uu),Uua)
    <=> ~ aa_list_nat_o(Uu,Uua) ) ).

% ATP.lambda_259
tff(fact_9947_ATP_Olambda__260,axiom,
    ! [Uu: fun_o_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_aeo(Uu),(Uua))
    <=> ~ aa_o_o(Uu,(Uua)) ) ).

% ATP.lambda_260
tff(fact_9948_ATP_Olambda__261,axiom,
    ! [Uu: fun_se7249947736802449410_int_o,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_az(Uu),Uua)
    <=> ~ aa_set6170482117830564987_int_o(Uu,Uua) ) ).

% ATP.lambda_261
tff(fact_9949_ATP_Olambda__262,axiom,
    ! [Uu: fun_set_nat_o,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ba(Uu),Uua)
    <=> ~ aa_set_nat_o(Uu,Uua) ) ).

% ATP.lambda_262
tff(fact_9950_ATP_Olambda__263,axiom,
    ! [Uu: fun_rat_o,Uua: rat] :
      ( aa_rat_o(aTP_Lamm_aep(Uu),Uua)
    <=> ~ aa_rat_o(Uu,Uua) ) ).

% ATP.lambda_263
tff(fact_9951_ATP_Olambda__264,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_bb(Uu),Uua)
    <=> ~ aa_nat_o(Uu,Uua) ) ).

% ATP.lambda_264
tff(fact_9952_ATP_Olambda__265,axiom,
    ! [Uu: fun_int_o,Uua: int] :
      ( aa_int_o(aTP_Lamm_bc(Uu),Uua)
    <=> ~ aa_int_o(Uu,Uua) ) ).

% ATP.lambda_265
tff(fact_9953_ATP_Olambda__266,axiom,
    ! [Uu: fun_fu5135969500543783852at_o_o,Uua: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aTP_Lamm_bd(Uu),Uua)
    <=> ~ aa_fun1675921005893230629at_o_o(Uu,Uua) ) ).

% ATP.lambda_266
tff(fact_9954_ATP_Olambda__267,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_set_nat(aTP_Lamm_alv(Uu),Uua) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gg,Uu)) ) ).

% ATP.lambda_267
tff(fact_9955_ATP_Olambda__268,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_aml,Uu),Uua)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),Uu)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),X4) ) ) ).

% ATP.lambda_268
tff(fact_9956_ATP_Olambda__269,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      ( aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_gr(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu)),Uub),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uub),aa_num_Code_integer(numera6620942414471956472nteger,Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),Uub)) ) ).

% ATP.lambda_269
tff(fact_9957_ATP_Olambda__270,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gq(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),numeral_numeral_nat(Uu))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)),Uub)) ) ).

% ATP.lambda_270
tff(fact_9958_ATP_Olambda__271,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gp(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_271
tff(fact_9959_ATP_Olambda__272,axiom,
    ! [Uu: fun_se7586849273513198688nteger,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set871585252387937313nteger(aa_set4557450419886607578nteger(aTP_Lamm_aaj(Uu),Uua),Uub) = $ite(aa_set_set_nat_o(member_set_nat(Uub),Uua),aa_set871585252387937313nteger(Uu,Uub),zero_z3403309356797280102nteger) ) ).

% ATP.lambda_272
tff(fact_9960_ATP_Olambda__273,axiom,
    ! [Uu: fun_set_nat_rat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_rat(aa_set8854944969235456213at_rat(aTP_Lamm_aak(Uu),Uua),Uub) = $ite(aa_set_set_nat_o(member_set_nat(Uub),Uua),aa_set_nat_rat(Uu,Uub),zero_zero_rat) ) ).

% ATP.lambda_273
tff(fact_9961_ATP_Olambda__274,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: set_nat,Uub: nat] :
      ( aa_nat_Code_integer(aa_set753585224467312750nteger(aTP_Lamm_aae(Uu),Uua),Uub) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua),aa_nat_Code_integer(Uu,Uub),zero_z3403309356797280102nteger) ) ).

% ATP.lambda_274
tff(fact_9962_ATP_Olambda__275,axiom,
    ! [Uu: fun_nat_rat,Uua: set_nat,Uub: nat] :
      ( aa_nat_rat(aa_set3028864993726649449at_rat(aTP_Lamm_aaf(Uu),Uua),Uub) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua),aa_nat_rat(Uu,Uub),zero_zero_rat) ) ).

% ATP.lambda_275
tff(fact_9963_ATP_Olambda__276,axiom,
    ! [Uu: fun_nat_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_aah(Uu),Uua),Uub) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua),aa_nat_nat(Uu,Uub),zero_zero_nat) ) ).

% ATP.lambda_276
tff(fact_9964_ATP_Olambda__277,axiom,
    ! [Uu: fun_nat_int,Uua: set_nat,Uub: nat] :
      ( aa_nat_int(aa_set2359242532610849469at_int(aTP_Lamm_aag(Uu),Uua),Uub) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua),aa_nat_int(Uu,Uub),zero_zero_int) ) ).

% ATP.lambda_277
tff(fact_9965_ATP_Olambda__278,axiom,
    ! [Uu: fun_int_Code_integer,Uua: set_int,Uub: int] :
      ( aa_int_Code_integer(aa_set5517701783735052070nteger(aTP_Lamm_aab(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_Code_integer(Uu,Uub),zero_z3403309356797280102nteger) ) ).

% ATP.lambda_278
tff(fact_9966_ATP_Olambda__279,axiom,
    ! [Uu: fun_int_rat,Uua: set_int,Uub: int] :
      ( aa_int_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_aac(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_rat(Uu,Uub),zero_zero_rat) ) ).

% ATP.lambda_279
tff(fact_9967_ATP_Olambda__280,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: int] :
      ( aa_int_nat(aa_set2264519855582034201nt_nat(aTP_Lamm_aad(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_nat(Uu,Uub),zero_zero_nat) ) ).

% ATP.lambda_280
tff(fact_9968_ATP_Olambda__281,axiom,
    ! [Uu: fun_int_int,Uua: set_int,Uub: int] :
      ( aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_aai(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_int(Uu,Uub),zero_zero_int) ) ).

% ATP.lambda_281
tff(fact_9969_ATP_Olambda__282,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: set_nat,Uub: nat] :
      ( aa_nat_Code_integer(aa_set753585224467312750nteger(aTP_Lamm_aaz(Uu),Uua),Uub) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua),aa_nat_Code_integer(Uu,Uub),one_one_Code_integer) ) ).

% ATP.lambda_282
tff(fact_9970_ATP_Olambda__283,axiom,
    ! [Uu: fun_nat_assn,Uua: set_nat,Uub: nat] :
      ( aa_nat_assn(aa_set7559964055741201337t_assn(aTP_Lamm_aba(Uu),Uua),Uub) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua),aa_nat_assn(Uu,Uub),one_one_assn) ) ).

% ATP.lambda_283
tff(fact_9971_ATP_Olambda__284,axiom,
    ! [Uu: fun_nat_rat,Uua: set_nat,Uub: nat] :
      ( aa_nat_rat(aa_set3028864993726649449at_rat(aTP_Lamm_abb(Uu),Uua),Uub) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua),aa_nat_rat(Uu,Uub),one_one_rat) ) ).

% ATP.lambda_284
tff(fact_9972_ATP_Olambda__285,axiom,
    ! [Uu: fun_nat_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_abc(Uu),Uua),Uub) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua),aa_nat_nat(Uu,Uub),one_one_nat) ) ).

% ATP.lambda_285
tff(fact_9973_ATP_Olambda__286,axiom,
    ! [Uu: fun_nat_int,Uua: set_nat,Uub: nat] :
      ( aa_nat_int(aa_set2359242532610849469at_int(aTP_Lamm_abd(Uu),Uua),Uub) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua),aa_nat_int(Uu,Uub),one_one_int) ) ).

% ATP.lambda_286
tff(fact_9974_ATP_Olambda__287,axiom,
    ! [Uu: fun_int_Code_integer,Uua: set_int,Uub: int] :
      ( aa_int_Code_integer(aa_set5517701783735052070nteger(aTP_Lamm_aav(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_Code_integer(Uu,Uub),one_one_Code_integer) ) ).

% ATP.lambda_287
tff(fact_9975_ATP_Olambda__288,axiom,
    ! [Uu: fun_int_assn,Uua: set_int,Uub: int] :
      ( aa_int_assn(aa_set5310648174735401585t_assn(aTP_Lamm_aaw(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_assn(Uu,Uub),one_one_assn) ) ).

% ATP.lambda_288
tff(fact_9976_ATP_Olambda__289,axiom,
    ! [Uu: fun_int_rat,Uua: set_int,Uub: int] :
      ( aa_int_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_aax(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_rat(Uu,Uub),one_one_rat) ) ).

% ATP.lambda_289
tff(fact_9977_ATP_Olambda__290,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: int] :
      ( aa_int_nat(aa_set2264519855582034201nt_nat(aTP_Lamm_aay(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_nat(Uu,Uub),one_one_nat) ) ).

% ATP.lambda_290
tff(fact_9978_ATP_Olambda__291,axiom,
    ! [Uu: fun_int_int,Uua: set_int,Uub: int] :
      ( aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_abe(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_int(Uu,Uub),one_one_int) ) ).

% ATP.lambda_291
tff(fact_9979_ATP_Olambda__292,axiom,
    ! [Uu: set_nat,Uua: fun_se7586849273513198688nteger,Uub: set_nat] :
      ( aa_set871585252387937313nteger(aa_fun4587287018503165694nteger(aTP_Lamm_sr(Uu),Uua),Uub) = $ite(Uu = Uub,aa_set871585252387937313nteger(Uua,Uub),zero_z3403309356797280102nteger) ) ).

% ATP.lambda_292
tff(fact_9980_ATP_Olambda__293,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_rat,Uub: set_nat] :
      ( aa_set_nat_rat(aa_fun8478617152544252148at_rat(aTP_Lamm_ss(Uu),Uua),Uub) = $ite(Uu = Uub,aa_set_nat_rat(Uua,Uub),zero_zero_rat) ) ).

% ATP.lambda_293
tff(fact_9981_ATP_Olambda__294,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_sj(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ) ).

% ATP.lambda_294
tff(fact_9982_ATP_Olambda__295,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_sl(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),zero_zero_rat) ) ).

% ATP.lambda_295
tff(fact_9983_ATP_Olambda__296,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_sp(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_nat(Uua,Uub),zero_zero_nat) ) ).

% ATP.lambda_296
tff(fact_9984_ATP_Olambda__297,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      ( aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_so(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_int(Uua,Uub),zero_zero_int) ) ).

% ATP.lambda_297
tff(fact_9985_ATP_Olambda__298,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_sk(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ) ).

% ATP.lambda_298
tff(fact_9986_ATP_Olambda__299,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_sm(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_rat(Uua,Uub),zero_zero_rat) ) ).

% ATP.lambda_299
tff(fact_9987_ATP_Olambda__300,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      ( aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_sn(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_nat(Uua,Uub),zero_zero_nat) ) ).

% ATP.lambda_300
tff(fact_9988_ATP_Olambda__301,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] :
      ( aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_sq(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_int(Uua,Uub),zero_zero_int) ) ).

% ATP.lambda_301
tff(fact_9989_ATP_Olambda__302,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_tn(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_Code_integer(Uua,Uub),one_one_Code_integer) ) ).

% ATP.lambda_302
tff(fact_9990_ATP_Olambda__303,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: nat] :
      ( aa_nat_assn(aa_fun785744391571715112t_assn(aTP_Lamm_tp(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_assn(Uua,Uub),one_one_assn) ) ).

% ATP.lambda_303
tff(fact_9991_ATP_Olambda__304,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_tr(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),one_one_rat) ) ).

% ATP.lambda_304
tff(fact_9992_ATP_Olambda__305,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_tu(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_nat(Uua,Uub),one_one_nat) ) ).

% ATP.lambda_305
tff(fact_9993_ATP_Olambda__306,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      ( aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_tv(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_int(Uua,Uub),one_one_int) ) ).

% ATP.lambda_306
tff(fact_9994_ATP_Olambda__307,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_to(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_Code_integer(Uua,Uub),one_one_Code_integer) ) ).

% ATP.lambda_307
tff(fact_9995_ATP_Olambda__308,axiom,
    ! [Uu: int,Uua: fun_int_assn,Uub: int] :
      ( aa_int_assn(aa_fun919007281442365664t_assn(aTP_Lamm_tq(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_assn(Uua,Uub),one_one_assn) ) ).

% ATP.lambda_308
tff(fact_9996_ATP_Olambda__309,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ts(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_rat(Uua,Uub),one_one_rat) ) ).

% ATP.lambda_309
tff(fact_9997_ATP_Olambda__310,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      ( aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_tt(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_nat(Uua,Uub),one_one_nat) ) ).

% ATP.lambda_310
tff(fact_9998_ATP_Olambda__311,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] :
      ( aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_tw(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_int(Uua,Uub),one_one_int) ) ).

% ATP.lambda_311
tff(fact_9999_ATP_Olambda__312,axiom,
    ! [Uu: set_nat,Uua: fun_se7586849273513198688nteger,Uub: set_nat] :
      ( aa_set871585252387937313nteger(aa_fun4587287018503165694nteger(aTP_Lamm_tb(Uu),Uua),Uub) = $ite(Uub = Uu,aa_set871585252387937313nteger(Uua,Uub),zero_z3403309356797280102nteger) ) ).

% ATP.lambda_312
tff(fact_10000_ATP_Olambda__313,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_rat,Uub: set_nat] :
      ( aa_set_nat_rat(aa_fun8478617152544252148at_rat(aTP_Lamm_tc(Uu),Uua),Uub) = $ite(Uub = Uu,aa_set_nat_rat(Uua,Uub),zero_zero_rat) ) ).

% ATP.lambda_313
tff(fact_10001_ATP_Olambda__314,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_st(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ) ).

% ATP.lambda_314
tff(fact_10002_ATP_Olambda__315,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_sv(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),zero_zero_rat) ) ).

% ATP.lambda_315
tff(fact_10003_ATP_Olambda__316,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_sz(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ) ).

% ATP.lambda_316
tff(fact_10004_ATP_Olambda__317,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      ( aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_sy(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ) ).

% ATP.lambda_317
tff(fact_10005_ATP_Olambda__318,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_su(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ) ).

% ATP.lambda_318
tff(fact_10006_ATP_Olambda__319,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_sw(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_rat(Uua,Uub),zero_zero_rat) ) ).

% ATP.lambda_319
tff(fact_10007_ATP_Olambda__320,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      ( aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_sx(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_nat(Uua,Uub),zero_zero_nat) ) ).

% ATP.lambda_320
tff(fact_10008_ATP_Olambda__321,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] :
      ( aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ta(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_int(Uua,Uub),zero_zero_int) ) ).

% ATP.lambda_321
tff(fact_10009_ATP_Olambda__322,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_td(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_Code_integer(Uua,Uub),one_one_Code_integer) ) ).

% ATP.lambda_322
tff(fact_10010_ATP_Olambda__323,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: nat] :
      ( aa_nat_assn(aa_fun785744391571715112t_assn(aTP_Lamm_tf(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_assn(Uua,Uub),one_one_assn) ) ).

% ATP.lambda_323
tff(fact_10011_ATP_Olambda__324,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_th(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),one_one_rat) ) ).

% ATP.lambda_324
tff(fact_10012_ATP_Olambda__325,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_tk(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),one_one_nat) ) ).

% ATP.lambda_325
tff(fact_10013_ATP_Olambda__326,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      ( aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_tl(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),one_one_int) ) ).

% ATP.lambda_326
tff(fact_10014_ATP_Olambda__327,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_te(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_Code_integer(Uua,Uub),one_one_Code_integer) ) ).

% ATP.lambda_327
tff(fact_10015_ATP_Olambda__328,axiom,
    ! [Uu: int,Uua: fun_int_assn,Uub: int] :
      ( aa_int_assn(aa_fun919007281442365664t_assn(aTP_Lamm_tg(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_assn(Uua,Uub),one_one_assn) ) ).

% ATP.lambda_328
tff(fact_10016_ATP_Olambda__329,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ti(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_rat(Uua,Uub),one_one_rat) ) ).

% ATP.lambda_329
tff(fact_10017_ATP_Olambda__330,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      ( aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_tj(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_nat(Uua,Uub),one_one_nat) ) ).

% ATP.lambda_330
tff(fact_10018_ATP_Olambda__331,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] :
      ( aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_tm(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_int(Uua,Uub),one_one_int) ) ).

% ATP.lambda_331
tff(fact_10019_ATP_Olambda__332,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      ( aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_lr(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu)),Uub))) ) ).

% ATP.lambda_332
tff(fact_10020_ATP_Olambda__333,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      ( aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aip(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Uu)),Uub))) ) ).

% ATP.lambda_333
tff(fact_10021_ATP_Olambda__334,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      ( aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_lq(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uu),Uub))) ) ).

% ATP.lambda_334
tff(fact_10022_ATP_Olambda__335,axiom,
    ! [Uu: fun_se7586849273513198688nteger,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set871585252387937313nteger(aa_fun7102796863437819535nteger(aTP_Lamm_zk(Uu),Uua),Uub) = $ite(aa_set_nat_o(Uua,Uub),aa_set871585252387937313nteger(Uu,Uub),zero_z3403309356797280102nteger) ) ).

% ATP.lambda_335
tff(fact_10023_ATP_Olambda__336,axiom,
    ! [Uu: fun_set_nat_rat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_rat(aa_fun7116522257906611850at_rat(aTP_Lamm_zl(Uu),Uua),Uub) = $ite(aa_set_nat_o(Uua,Uub),aa_set_nat_rat(Uu,Uub),zero_zero_rat) ) ).

% ATP.lambda_336
tff(fact_10024_ATP_Olambda__337,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_Code_integer(aa_fun3969837373441324047nteger(aTP_Lamm_zc(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_Code_integer(Uu,Uub),zero_z3403309356797280102nteger) ) ).

% ATP.lambda_337
tff(fact_10025_ATP_Olambda__338,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_ze(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),zero_zero_rat) ) ).

% ATP.lambda_338
tff(fact_10026_ATP_Olambda__339,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_zi(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_nat(Uu,Uub),zero_zero_nat) ) ).

% ATP.lambda_339
tff(fact_10027_ATP_Olambda__340,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_int(aa_fun27009930614080094at_int(aTP_Lamm_zh(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_int(Uu,Uub),zero_zero_int) ) ).

% ATP.lambda_340
tff(fact_10028_ATP_Olambda__341,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_o,Uub: int] :
      ( aa_int_Code_integer(aa_fun6019004165459918095nteger(aTP_Lamm_zd(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_Code_integer(Uu,Uub),zero_z3403309356797280102nteger) ) ).

% ATP.lambda_341
tff(fact_10029_ATP_Olambda__342,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] :
      ( aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_zf(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu,Uub),zero_zero_rat) ) ).

% ATP.lambda_342
tff(fact_10030_ATP_Olambda__343,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      ( aa_int_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_zg(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu,Uub),zero_zero_nat) ) ).

% ATP.lambda_343
tff(fact_10031_ATP_Olambda__344,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_int(aa_fun8822586851003501406nt_int(aTP_Lamm_zj(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_int(Uu,Uub),zero_zero_int) ) ).

% ATP.lambda_344
tff(fact_10032_ATP_Olambda__345,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_Code_integer(aa_fun3969837373441324047nteger(aTP_Lamm_zm(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_Code_integer(Uu,Uub),one_one_Code_integer) ) ).

% ATP.lambda_345
tff(fact_10033_ATP_Olambda__346,axiom,
    ! [Uu: fun_nat_assn,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_assn(aa_fun3990007403578212698t_assn(aTP_Lamm_zo(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_assn(Uu,Uub),one_one_assn) ) ).

% ATP.lambda_346
tff(fact_10034_ATP_Olambda__347,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_zq(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),one_one_rat) ) ).

% ATP.lambda_347
tff(fact_10035_ATP_Olambda__348,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_zt(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_nat(Uu,Uub),one_one_nat) ) ).

% ATP.lambda_348
tff(fact_10036_ATP_Olambda__349,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_int(aa_fun27009930614080094at_int(aTP_Lamm_zu(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_int(Uu,Uub),one_one_int) ) ).

% ATP.lambda_349
tff(fact_10037_ATP_Olambda__350,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_o,Uub: int] :
      ( aa_int_Code_integer(aa_fun6019004165459918095nteger(aTP_Lamm_zn(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_Code_integer(Uu,Uub),one_one_Code_integer) ) ).

% ATP.lambda_350
tff(fact_10038_ATP_Olambda__351,axiom,
    ! [Uu: fun_int_assn,Uua: fun_int_o,Uub: int] :
      ( aa_int_assn(aa_fun9028805181350005338t_assn(aTP_Lamm_zp(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_assn(Uu,Uub),one_one_assn) ) ).

% ATP.lambda_351
tff(fact_10039_ATP_Olambda__352,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] :
      ( aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_zr(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu,Uub),one_one_rat) ) ).

% ATP.lambda_352
tff(fact_10040_ATP_Olambda__353,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      ( aa_int_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_zs(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu,Uub),one_one_nat) ) ).

% ATP.lambda_353
tff(fact_10041_ATP_Olambda__354,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_int(aa_fun8822586851003501406nt_int(aTP_Lamm_zv(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_int(Uu,Uub),one_one_int) ) ).

% ATP.lambda_354
tff(fact_10042_ATP_Olambda__355,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oe(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_355
tff(fact_10043_ATP_Olambda__356,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_zz(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_356
tff(fact_10044_ATP_Olambda__357,axiom,
    ! [Uu: fun_se1179353343636554400at_nat,Uua: nat,Uub: set_nat] : ( aa_set_nat_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_vw(Uu),Uua),Uub) = aa_nat_nat(aa_set6537093552120046177at_nat(Uu,Uub),Uua) ) ).

% ATP.lambda_357
tff(fact_10045_ATP_Olambda__358,axiom,
    ! [Uu: fun_se7912642235559603452at_int,Uua: nat,Uub: set_nat] : ( aa_set_nat_int(aa_nat2732338449433945789at_int(aTP_Lamm_xf(Uu),Uua),Uub) = aa_nat_int(aa_set2359242532610849469at_int(Uu,Uub),Uua) ) ).

% ATP.lambda_358
tff(fact_10046_ATP_Olambda__359,axiom,
    ! [Uu: fun_se876191773397849432nt_int,Uua: int,Uub: set_nat] : ( aa_set_nat_int(aa_int2448287375063835929at_int(aTP_Lamm_wb(Uu),Uua),Uub) = aa_int_int(aa_set3358596244511904025nt_int(Uu,Uub),Uua) ) ).

% ATP.lambda_359
tff(fact_10047_ATP_Olambda__360,axiom,
    ! [Uu: fun_na106483476154090400at_nat,Uua: set_nat,Uub: nat] : ( aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_wg(Uu),Uua),Uub) = aa_set_nat_nat(aa_nat6910189468943142497at_nat(Uu,Uub),Uua) ) ).

% ATP.lambda_360
tff(fact_10048_ATP_Olambda__361,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ku(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ) ).

% ATP.lambda_361
tff(fact_10049_ATP_Olambda__362,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pk(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ) ).

% ATP.lambda_362
tff(fact_10050_ATP_Olambda__363,axiom,
    ! [Uu: fun_nat_fun_int_nat,Uua: int,Uub: nat] : ( aa_nat_nat(aa_int_fun_nat_nat(aTP_Lamm_vi(Uu),Uua),Uub) = aa_int_nat(aa_nat_fun_int_nat(Uu,Uub),Uua) ) ).

% ATP.lambda_363
tff(fact_10051_ATP_Olambda__364,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: int,Uub: nat] : ( aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_pn(Uu),Uua),Uub) = aa_int_int(aa_nat_fun_int_int(Uu,Uub),Uua) ) ).

% ATP.lambda_364
tff(fact_10052_ATP_Olambda__365,axiom,
    ! [Uu: fun_in6555721293707029592at_int,Uua: set_nat,Uub: int] : ( aa_int_int(aa_set3358596244511904025nt_int(aTP_Lamm_wl(Uu),Uua),Uub) = aa_set_nat_int(aa_int2448287375063835929at_int(Uu,Uub),Uua) ) ).

% ATP.lambda_365
tff(fact_10053_ATP_Olambda__366,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : ( aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_va(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ) ).

% ATP.lambda_366
tff(fact_10054_ATP_Olambda__367,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: nat,Uub: int] : ( aa_int_int(aa_nat_fun_int_int(aTP_Lamm_pq(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(Uu,Uub),Uua) ) ).

% ATP.lambda_367
tff(fact_10055_ATP_Olambda__368,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: int,Uub: int] : ( aa_int_int(aa_int_fun_int_int(aTP_Lamm_kx(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(Uu,Uub),Uua) ) ).

% ATP.lambda_368
tff(fact_10056_ATP_Olambda__369,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      ( aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_jy(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),Uub)),Uub = one_one_Code_integer) ) ).

% ATP.lambda_369
tff(fact_10057_ATP_Olambda__370,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qt(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_pk(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ) ).

% ATP.lambda_370
tff(fact_10058_ATP_Olambda__371,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qr(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ku(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ) ).

% ATP.lambda_371
tff(fact_10059_ATP_Olambda__372,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lu(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ku(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ) ).

% ATP.lambda_372
tff(fact_10060_ATP_Olambda__373,axiom,
    ! [Uu: produc7822875418678951345atural,Uua: code_natural,Uub: code_natural] : ( aa_Cod6808739660676082561atural(aa_Cod4292056611156568242atural(aTP_Lamm_amk(Uu),Uua),Uub) = produc8282080750456430313atural(aa_Cod8811915439870959971atural(aTP_Lamm_amj(Uua),Uub),produc6591912806276919810atural(aa_Pro7165575656943818355atural(next,Uu))) ) ).

% ATP.lambda_373
tff(fact_10061_ATP_Olambda__374,axiom,
    ! [Uu: $o,Uua: code_integer,Uub: $o] : ( aa_o_char(aa_Cod7842134615204088097o_char(aTP_Lamm_akd((Uu)),Uua),(Uub)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aTP_Lamm_akc((Uu)),(Uub)),code_bit_cut_integer(Uua)) ) ).

% ATP.lambda_374
tff(fact_10062_ATP_Olambda__375,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ajv(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aju(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_375
tff(fact_10063_ATP_Olambda__376,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ajt(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ajs(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_376
tff(fact_10064_ATP_Olambda__377,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ajr(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ajq(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_377
tff(fact_10065_ATP_Olambda__378,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ajg(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_ajf(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_378
tff(fact_10066_ATP_Olambda__379,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aje(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_ajd(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_379
tff(fact_10067_ATP_Olambda__380,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aja(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aiz(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_380
tff(fact_10068_ATP_Olambda__381,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: set_int,Uub: nat] : ( aa_nat_int(aa_set6310687161026558745at_int(aTP_Lamm_pr(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_pq(Uu),Uub),Uua) ) ).

% ATP.lambda_381
tff(fact_10069_ATP_Olambda__382,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: set_nat,Uub: int] : ( aa_int_int(aa_set3358596244511904025nt_int(aTP_Lamm_po(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_pn(Uu),Uub)),Uua) ) ).

% ATP.lambda_382
tff(fact_10070_ATP_Olambda__383,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: set_nat,Uub: nat] : ( aa_nat_int(aa_set2359242532610849469at_int(aTP_Lamm_pl(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_pk(Uu),Uub)),Uua) ) ).

% ATP.lambda_383
tff(fact_10071_ATP_Olambda__384,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : ( aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_pt(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_kx(Uu),Uub),Uua) ) ).

% ATP.lambda_384
tff(fact_10072_ATP_Olambda__385,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : ( aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_pi(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ku(Uu),Uub)),Uua) ) ).

% ATP.lambda_385
tff(fact_10073_ATP_Olambda__386,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : ( aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_ky(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_kx(Uu),Uub),Uua) ) ).

% ATP.lambda_386
tff(fact_10074_ATP_Olambda__387,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : ( aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_kv(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ku(Uu),Uub)),Uua) ) ).

% ATP.lambda_387
tff(fact_10075_ATP_Olambda__388,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : ( aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_si(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub))),aa_nat_rat(Uua,Uub)) ) ).

% ATP.lambda_388
tff(fact_10076_ATP_Olambda__389,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_alo(Uu),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( aa_list_nat_nat(groups4561878855575611511st_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_389
tff(fact_10077_ATP_Olambda__390,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_alp(Uu),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_nat_nat(groups4561878855575611511st_nat,Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_390
tff(fact_10078_ATP_Olambda__391,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aln(Uu),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_list_nat_nat(groups4561878855575611511st_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_391
tff(fact_10079_ATP_Olambda__392,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aio(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_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_392
tff(fact_10080_ATP_Olambda__393,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_aiw(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),pow_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uu)))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_393
tff(fact_10081_ATP_Olambda__394,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : ( aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_rp(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uub))),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ) ).

% ATP.lambda_394
tff(fact_10082_ATP_Olambda__395,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_akx(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_395
tff(fact_10083_ATP_Olambda__396,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_nat1727191234135941576_nat_o(aTP_Lamm_ahi(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Uub),Uu)
        & ( finite711546835091564841at_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_396
tff(fact_10084_ATP_Olambda__397,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: nat,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_nat4194932087953772816_int_o(aTP_Lamm_ahd(Uu),Uua),Uub)
    <=> ( aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(ord_le2843351958646193337nt_int,Uub),Uu)
        & ( aa_set4753215920113914285nt_nat(finite6756421564338198497nt_int,Uub) = Uua ) ) ) ).

% ATP.lambda_397
tff(fact_10085_ATP_Olambda__398,axiom,
    ! [Uu: set_list_nat,Uua: nat,Uub: set_list_nat] :
      ( aa_set_list_nat_o(aa_nat7721107121239158477_nat_o(aTP_Lamm_ahe(Uu),Uua),Uub)
    <=> ( aa_set_list_nat_o(aa_set8027019173009287559_nat_o(ord_le6045566169113846134st_nat,Uub),Uu)
        & ( finite_card_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_398
tff(fact_10086_ATP_Olambda__399,axiom,
    ! [Uu: set_o,Uua: nat,Uub: set_o] :
      ( aa_set_o_o(aa_nat_fun_set_o_o(aTP_Lamm_ahf(Uu),Uua),Uub)
    <=> ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Uub),Uu)
        & ( finite_card_o(Uub) = Uua ) ) ) ).

% ATP.lambda_399
tff(fact_10087_ATP_Olambda__400,axiom,
    ! [Uu: set_set_nat,Uua: nat,Uub: set_set_nat] :
      ( aa_set_set_nat_o(aa_nat5160323483289836711_nat_o(aTP_Lamm_ahg(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(Uub),Uu)
        & ( finite_card_set_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_400
tff(fact_10088_ATP_Olambda__401,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_ahj(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Uub),Uu)
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_401
tff(fact_10089_ATP_Olambda__402,axiom,
    ! [Uu: set_int,Uua: nat,Uub: set_int] :
      ( aa_set_int_o(aa_nat_fun_set_int_o(aTP_Lamm_ahh(Uu),Uua),Uub)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Uub),Uu)
        & ( finite_card_int(Uub) = Uua ) ) ) ).

% ATP.lambda_402
tff(fact_10090_ATP_Olambda__403,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ain(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_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_403
tff(fact_10091_ATP_Olambda__404,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_re(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_404
tff(fact_10092_ATP_Olambda__405,axiom,
    ! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_ac(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uub)
        | aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uua),Uub) ) ) ).

% ATP.lambda_405
tff(fact_10093_ATP_Olambda__406,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: set_Pr8551490117392284871at_nat,Uub: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(aa_set1219845518824332607_nat_o(aTP_Lamm_ce(Uu),Uua),Uub)
    <=> ( member6689249552917799696at_nat(Uub,Uu)
        | member6689249552917799696at_nat(Uub,Uua) ) ) ).

% ATP.lambda_406
tff(fact_10094_ATP_Olambda__407,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_set6353399069419443519_nat_o(aTP_Lamm_cf(Uu),Uua),Uub)
    <=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uub),Uu)
        | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uub),Uua) ) ) ).

% ATP.lambda_407
tff(fact_10095_ATP_Olambda__408,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: set_se6260736226359567993nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4062574822058255039_int_o(aTP_Lamm_ca(Uu),Uua),Uub)
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uu)
        | aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uua) ) ) ).

% ATP.lambda_408
tff(fact_10096_ATP_Olambda__409,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_cb(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        | aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_409
tff(fact_10097_ATP_Olambda__410,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_cg(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        | aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua) ) ) ).

% ATP.lambda_410
tff(fact_10098_ATP_Olambda__411,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_cc(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_411
tff(fact_10099_ATP_Olambda__412,axiom,
    ! [Uu: set_fu3555266933746557007_nat_o,Uua: set_fu3555266933746557007_nat_o,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_set7516337885093543743at_o_o(aTP_Lamm_cd(Uu),Uua),Uub)
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uub),Uu)
        | aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uub),Uua) ) ) ).

% ATP.lambda_412
tff(fact_10100_ATP_Olambda__413,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_set6353399069419443519_nat_o(aTP_Lamm_iy(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uub),Uua) ) ) ).

% ATP.lambda_413
tff(fact_10101_ATP_Olambda__414,axiom,
    ! [Uu: product_prod_nat_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_iz(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_414
tff(fact_10102_ATP_Olambda__415,axiom,
    ! [Uu: $o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_ja((Uu)),Uua),(Uub))
    <=> ( ( (Uub)
        <=> (Uu) )
        | aa_set_o_o(member_o((Uub)),Uua) ) ) ).

% ATP.lambda_415
tff(fact_10103_ATP_Olambda__416,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_se6260736226359567993nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4062574822058255039_int_o(aTP_Lamm_jb(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uua) ) ) ).

% ATP.lambda_416
tff(fact_10104_ATP_Olambda__417,axiom,
    ! [Uu: set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_jc(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_417
tff(fact_10105_ATP_Olambda__418,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_jd(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua) ) ) ).

% ATP.lambda_418
tff(fact_10106_ATP_Olambda__419,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_je(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_419
tff(fact_10107_ATP_Olambda__420,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: set_fu3555266933746557007_nat_o,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_set7516337885093543743at_o_o(aTP_Lamm_jf(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uub),Uua) ) ) ).

% ATP.lambda_420
tff(fact_10108_ATP_Olambda__421,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ic(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_421
tff(fact_10109_ATP_Olambda__422,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aer(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_422
tff(fact_10110_ATP_Olambda__423,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ib(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_423
tff(fact_10111_ATP_Olambda__424,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aet(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_424
tff(fact_10112_ATP_Olambda__425,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aeu(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_425
tff(fact_10113_ATP_Olambda__426,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aes(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_426
tff(fact_10114_ATP_Olambda__427,axiom,
    ! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_ab(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uub)
        & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uua),Uub) ) ) ).

% ATP.lambda_427
tff(fact_10115_ATP_Olambda__428,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_alb(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_428
tff(fact_10116_ATP_Olambda__429,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_akz(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_429
tff(fact_10117_ATP_Olambda__430,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_bp(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uu)
        & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_430
tff(fact_10118_ATP_Olambda__431,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: set_se6260736226359567993nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4062574822058255039_int_o(aTP_Lamm_bl(Uu),Uua),Uub)
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uu)
        & aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uua) ) ) ).

% ATP.lambda_431
tff(fact_10119_ATP_Olambda__432,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_bm(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_432
tff(fact_10120_ATP_Olambda__433,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_bq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua) ) ) ).

% ATP.lambda_433
tff(fact_10121_ATP_Olambda__434,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_bn(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_434
tff(fact_10122_ATP_Olambda__435,axiom,
    ! [Uu: set_fu3555266933746557007_nat_o,Uua: set_fu3555266933746557007_nat_o,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_set7516337885093543743at_o_o(aTP_Lamm_bo(Uu),Uua),Uub)
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uub),Uu)
        & aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uub),Uua) ) ) ).

% ATP.lambda_435
tff(fact_10123_ATP_Olambda__436,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_uu(Uu),Uua),Uub)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,Uub),Uu) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,Uua),Uu) ) ) ).

% ATP.lambda_436
tff(fact_10124_ATP_Olambda__437,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(aTP_Lamm_us(Uu),Uua),Uub)
    <=> ( minus_1052850069191792384nt_int(Uub,Uu) = minus_1052850069191792384nt_int(Uua,Uu) ) ) ).

% ATP.lambda_437
tff(fact_10125_ATP_Olambda__438,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_uv(Uu),Uua),Uub)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Uub),Uu) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Uua),Uu) ) ) ).

% ATP.lambda_438
tff(fact_10126_ATP_Olambda__439,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_ut(Uu),Uua),Uub)
    <=> ( minus_minus_set_int(Uub,Uu) = minus_minus_set_int(Uua,Uu) ) ) ).

% ATP.lambda_439
tff(fact_10127_ATP_Olambda__440,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_gh(Uu),Uua),Uub)
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uu)
        & aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_440
tff(fact_10128_ATP_Olambda__441,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_gi(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_441
tff(fact_10129_ATP_Olambda__442,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_gl(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_442
tff(fact_10130_ATP_Olambda__443,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_gj(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_443
tff(fact_10131_ATP_Olambda__444,axiom,
    ! [Uu: set_fu3555266933746557007_nat_o,Uua: fun_fu5135969500543783852at_o_o,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_gk(Uu),Uua),Uub)
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uub),Uu)
        & aa_fun1675921005893230629at_o_o(Uua,Uub) ) ) ).

% ATP.lambda_444
tff(fact_10132_ATP_Olambda__445,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_jg(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_445
tff(fact_10133_ATP_Olambda__446,axiom,
    ! [Uu: product_prod_nat_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_jk(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_446
tff(fact_10134_ATP_Olambda__447,axiom,
    ! [Uu: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_jl((Uu)),Uua),(Uub))
    <=> ( ( (Uu)
        <=> (Uub) )
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_447
tff(fact_10135_ATP_Olambda__448,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_jh(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_448
tff(fact_10136_ATP_Olambda__449,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ji(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_449
tff(fact_10137_ATP_Olambda__450,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_jm(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_450
tff(fact_10138_ATP_Olambda__451,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_jn(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_451
tff(fact_10139_ATP_Olambda__452,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_fu5135969500543783852at_o_o,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_jj(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_fun1675921005893230629at_o_o(Uua,Uub) ) ) ).

% ATP.lambda_452
tff(fact_10140_ATP_Olambda__453,axiom,
    ! [Uu: fun_se7249947736802449410_int_o,Uua: set_Pr958786334691620121nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(aTP_Lamm_ho(Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa_set6170482117830564987_int_o(Uu,Uua) ) ) ).

% ATP.lambda_453
tff(fact_10141_ATP_Olambda__454,axiom,
    ! [Uu: fun_set_nat_o,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_hp(Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa_set_nat_o(Uu,Uua) ) ) ).

% ATP.lambda_454
tff(fact_10142_ATP_Olambda__455,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_hq(Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa_nat_o(Uu,Uua) ) ) ).

% ATP.lambda_455
tff(fact_10143_ATP_Olambda__456,axiom,
    ! [Uu: fun_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hs(Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa_int_o(Uu,Uua) ) ) ).

% ATP.lambda_456
tff(fact_10144_ATP_Olambda__457,axiom,
    ! [Uu: fun_fu5135969500543783852at_o_o,Uua: fun_Pr6460944699146269935_nat_o,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(aTP_Lamm_hr(Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa_fun1675921005893230629at_o_o(Uu,Uua) ) ) ).

% ATP.lambda_457
tff(fact_10145_ATP_Olambda__458,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_jo(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_458
tff(fact_10146_ATP_Olambda__459,axiom,
    ! [Uu: product_prod_nat_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_js(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_459
tff(fact_10147_ATP_Olambda__460,axiom,
    ! [Uu: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_jt((Uu)),Uua),(Uub))
    <=> ( ( (Uub)
        <=> (Uu) )
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_460
tff(fact_10148_ATP_Olambda__461,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_jp(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_461
tff(fact_10149_ATP_Olambda__462,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_jq(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_462
tff(fact_10150_ATP_Olambda__463,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ju(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_463
tff(fact_10151_ATP_Olambda__464,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_jv(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_464
tff(fact_10152_ATP_Olambda__465,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_fu5135969500543783852at_o_o,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_jr(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_fun1675921005893230629at_o_o(Uua,Uub) ) ) ).

% ATP.lambda_465
tff(fact_10153_ATP_Olambda__466,axiom,
    ! [Uu: set_o,Uua: fun_o_Code_integer,Uub: $o] :
      ( aa_o_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_afx(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ( aa_o_Code_integer(Uua,(Uub)) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_466
tff(fact_10154_ATP_Olambda__467,axiom,
    ! [Uu: set_o,Uua: fun_o_rat,Uub: $o] :
      ( aa_o_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_afy(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ( aa_o_rat(Uua,(Uub)) != zero_zero_rat ) ) ) ).

% ATP.lambda_467
tff(fact_10155_ATP_Olambda__468,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: $o] :
      ( aa_o_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_ags(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ( aa_o_nat(Uua,(Uub)) != zero_zero_nat ) ) ) ).

% ATP.lambda_468
tff(fact_10156_ATP_Olambda__469,axiom,
    ! [Uu: set_o,Uua: fun_o_int,Uub: $o] :
      ( aa_o_o(aa_fun_o_int_fun_o_o(aTP_Lamm_afz(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ( aa_o_int(Uua,(Uub)) != zero_zero_int ) ) ) ).

% ATP.lambda_469
tff(fact_10157_ATP_Olambda__470,axiom,
    ! [Uu: set_set_nat,Uua: fun_se7586849273513198688nteger,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun5627232717963835469_nat_o(aTP_Lamm_ye(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & ( aa_set871585252387937313nteger(Uua,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_470
tff(fact_10158_ATP_Olambda__471,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_rat,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun3672760304936660370_nat_o(aTP_Lamm_yg(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & ( aa_set_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_471
tff(fact_10159_ATP_Olambda__472,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aTP_Lamm_xo(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_Code_integer(Uua,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_472
tff(fact_10160_ATP_Olambda__473,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_xs(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_473
tff(fact_10161_ATP_Olambda__474,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_xw(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_474
tff(fact_10162_ATP_Olambda__475,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      ( aa_nat_o(aa_fun2180201864171524266_nat_o(aTP_Lamm_ya(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_int(Uua,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_475
tff(fact_10163_ATP_Olambda__476,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aTP_Lamm_xq(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_Code_integer(Uua,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_476
tff(fact_10164_ATP_Olambda__477,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aTP_Lamm_xu(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_477
tff(fact_10165_ATP_Olambda__478,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_xy(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_478
tff(fact_10166_ATP_Olambda__479,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: int] :
      ( aa_int_o(aa_fun8155382727227203058_int_o(aTP_Lamm_yc(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_int(Uua,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_479
tff(fact_10167_ATP_Olambda__480,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aTP_Lamm_yi(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_Code_integer(Uua,Uub) != one_one_Code_integer ) ) ) ).

% ATP.lambda_480
tff(fact_10168_ATP_Olambda__481,axiom,
    ! [Uu: set_nat,Uua: fun_nat_assn,Uub: nat] :
      ( aa_nat_o(aa_fun7183233715851567790_nat_o(aTP_Lamm_ym(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_481
tff(fact_10169_ATP_Olambda__482,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_yq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_482
tff(fact_10170_ATP_Olambda__483,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_yu(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_nat(Uua,Uub) != one_one_nat ) ) ) ).

% ATP.lambda_483
tff(fact_10171_ATP_Olambda__484,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      ( aa_nat_o(aa_fun2180201864171524266_nat_o(aTP_Lamm_yy(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_int(Uua,Uub) != one_one_int ) ) ) ).

% ATP.lambda_484
tff(fact_10172_ATP_Olambda__485,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aTP_Lamm_yk(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_Code_integer(Uua,Uub) != one_one_Code_integer ) ) ) ).

% ATP.lambda_485
tff(fact_10173_ATP_Olambda__486,axiom,
    ! [Uu: set_int,Uua: fun_int_assn,Uub: int] :
      ( aa_int_o(aa_fun202081646129719286_int_o(aTP_Lamm_yo(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_486
tff(fact_10174_ATP_Olambda__487,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aTP_Lamm_ys(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_487
tff(fact_10175_ATP_Olambda__488,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_yw(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_nat(Uua,Uub) != one_one_nat ) ) ) ).

% ATP.lambda_488
tff(fact_10176_ATP_Olambda__489,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: int] :
      ( aa_int_o(aa_fun8155382727227203058_int_o(aTP_Lamm_za(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_int(Uua,Uub) != one_one_int ) ) ) ).

% ATP.lambda_489
tff(fact_10177_ATP_Olambda__490,axiom,
    ! [Uu: fun_se7586849273513198688nteger,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_ahs(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uua)
        & ( aa_set871585252387937313nteger(Uu,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_490
tff(fact_10178_ATP_Olambda__491,axiom,
    ! [Uu: fun_set_nat_rat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_aht(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uua)
        & ( aa_set_nat_rat(Uu,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_491
tff(fact_10179_ATP_Olambda__492,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ahk(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua)
        & ( aa_nat_Code_integer(Uu,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_492
tff(fact_10180_ATP_Olambda__493,axiom,
    ! [Uu: fun_nat_rat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ahm(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua)
        & ( aa_nat_rat(Uu,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_493
tff(fact_10181_ATP_Olambda__494,axiom,
    ! [Uu: fun_nat_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_aho(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua)
        & ( aa_nat_nat(Uu,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_494
tff(fact_10182_ATP_Olambda__495,axiom,
    ! [Uu: fun_nat_int,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ahq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua)
        & ( aa_nat_int(Uu,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_495
tff(fact_10183_ATP_Olambda__496,axiom,
    ! [Uu: fun_int_Code_integer,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_ahl(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uua)
        & ( aa_int_Code_integer(Uu,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_496
tff(fact_10184_ATP_Olambda__497,axiom,
    ! [Uu: fun_int_rat,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_ahn(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uua)
        & ( aa_int_rat(Uu,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_497
tff(fact_10185_ATP_Olambda__498,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_ahp(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uua)
        & ( aa_int_nat(Uu,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_498
tff(fact_10186_ATP_Olambda__499,axiom,
    ! [Uu: fun_int_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_ahr(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uua)
        & ( aa_int_int(Uu,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_499
tff(fact_10187_ATP_Olambda__500,axiom,
    ! [Uu: set_o,Uua: fun_o_Code_natural,Uub: $o] :
      ( aa_o_o(aa_fun2872501519293664346un_o_o(aTP_Lamm_agh(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_o_Code_natural(Uua,(Uub))) ) ) ).

% ATP.lambda_500
tff(fact_10188_ATP_Olambda__501,axiom,
    ! [Uu: set_o,Uua: fun_o_Code_integer,Uub: $o] :
      ( aa_o_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_age(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_o_Code_integer(Uua,(Uub))) ) ) ).

% ATP.lambda_501
tff(fact_10189_ATP_Olambda__502,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: $o] :
      ( aa_o_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_aga(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_o_nat(Uua,(Uub))) ) ) ).

% ATP.lambda_502
tff(fact_10190_ATP_Olambda__503,axiom,
    ! [Uu: set_o,Uua: fun_o_int,Uub: $o] :
      ( aa_o_o(aa_fun_o_int_fun_o_o(aTP_Lamm_agc(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_o_int(Uua,(Uub))) ) ) ).

% ATP.lambda_503
tff(fact_10191_ATP_Olambda__504,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_natural,Uub: nat] :
      ( aa_nat_o(aa_fun440078198722963374_nat_o(aTP_Lamm_agi(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_nat_Code_natural(Uua,Uub)) ) ) ).

% ATP.lambda_504
tff(fact_10192_ATP_Olambda__505,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aTP_Lamm_agf(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(Uua,Uub)) ) ) ).

% ATP.lambda_505
tff(fact_10193_ATP_Olambda__506,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      ( aa_nat_o(aa_fun2180201864171524266_nat_o(aTP_Lamm_agd(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ~ 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(Uua,Uub)) ) ) ).

% ATP.lambda_506
tff(fact_10194_ATP_Olambda__507,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_natural,Uub: int] :
      ( aa_int_o(aa_fun296549085072024822_int_o(aTP_Lamm_agj(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_int_Code_natural(Uua,Uub)) ) ) ).

% ATP.lambda_507
tff(fact_10195_ATP_Olambda__508,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aTP_Lamm_agg(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(Uua,Uub)) ) ) ).

% ATP.lambda_508
tff(fact_10196_ATP_Olambda__509,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_agb(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_int_nat(Uua,Uub)) ) ) ).

% ATP.lambda_509
tff(fact_10197_ATP_Olambda__510,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: set_se6260736226359567993nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4062574822058255039_int_o(aTP_Lamm_fm(Uu),Uua),Uub)
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uu)
        & ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uua) ) ) ).

% ATP.lambda_510
tff(fact_10198_ATP_Olambda__511,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_fn(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & ~ aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_511
tff(fact_10199_ATP_Olambda__512,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_fq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua) ) ) ).

% ATP.lambda_512
tff(fact_10200_ATP_Olambda__513,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_fo(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_513
tff(fact_10201_ATP_Olambda__514,axiom,
    ! [Uu: set_fu3555266933746557007_nat_o,Uua: set_fu3555266933746557007_nat_o,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_set7516337885093543743at_o_o(aTP_Lamm_fp(Uu),Uua),Uub)
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uub),Uu)
        & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uub),Uua) ) ) ).

% ATP.lambda_514
tff(fact_10202_ATP_Olambda__515,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_akm(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_515
tff(fact_10203_ATP_Olambda__516,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_od(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_516
tff(fact_10204_ATP_Olambda__517,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_sh(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_517
tff(fact_10205_ATP_Olambda__518,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: multis2468970476368604999at_nat,Uub: multis2468970476368604999at_nat] :
      ( aa_mul869302825175964237_nat_o(aa_mul3916465897682844511_nat_o(aTP_Lamm_dp(Uu),Uua),Uub)
    <=> member6689249552917799696at_nat(produc4348348721325984599at_nat(Uua,Uub),Uu) ) ).

% ATP.lambda_518
tff(fact_10206_ATP_Olambda__519,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_dq(Uu),Uua),Uub)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(Uua,Uub)),Uu) ) ).

% ATP.lambda_519
tff(fact_10207_ATP_Olambda__520,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_dr(Uu),Uua),Uub)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)),Uu) ) ).

% ATP.lambda_520
tff(fact_10208_ATP_Olambda__521,axiom,
    ! [Uu: set_Pr3486312109254157132nteger,Uua: fun_Pr5098669352780829690e_term,Uub: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(aTP_Lamm_dk(Uu),Uua),Uub)
    <=> member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Uua),Uub),Uu) ) ).

% ATP.lambda_521
tff(fact_10209_ATP_Olambda__522,axiom,
    ! [Uu: set_Pr1312406793078187350et_nat,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_dl(Uu),Uua),Uub)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_522
tff(fact_10210_ATP_Olambda__523,axiom,
    ! [Uu: set_Pr4040394263813967220et_nat,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(aTP_Lamm_dm(Uu),Uua),Uub)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_523
tff(fact_10211_ATP_Olambda__524,axiom,
    ! [Uu: set_Pr6700895561065273913nt_int,Uua: fun_Pr7763680713840462443e_term,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(aTP_Lamm_dn(Uu),Uua),Uub)
    <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Uua),Uub),Uu) ) ).

% ATP.lambda_524
tff(fact_10212_ATP_Olambda__525,axiom,
    ! [Uu: set_Pr4166680992137330533nt_int,Uua: fun_in5730379987311621695e_term,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(aTP_Lamm_do(Uu),Uua),Uub)
    <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Uua),Uub),Uu) ) ).

% ATP.lambda_525
tff(fact_10213_ATP_Olambda__526,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_uw(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_526
tff(fact_10214_ATP_Olambda__527,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : ( aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pu(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_527
tff(fact_10215_ATP_Olambda__528,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : ( aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_pv(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ) ).

% ATP.lambda_528
tff(fact_10216_ATP_Olambda__529,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : ( aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_pw(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ) ).

% ATP.lambda_529
tff(fact_10217_ATP_Olambda__530,axiom,
    ! [Uu: fun_se1399197888071115401nteger,Uua: fun_se1399197888071115401nteger,Uub: set_Pr958786334691620121nt_int] : ( aa_set4224706125354629690nteger(aa_fun1473822802349397756nteger(aTP_Lamm_ks(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set4224706125354629690nteger(Uua,Uub)),aa_set4224706125354629690nteger(Uu,Uub)) ) ).

% ATP.lambda_530
tff(fact_10218_ATP_Olambda__531,axiom,
    ! [Uu: fun_se7586849273513198688nteger,Uua: fun_se7586849273513198688nteger,Uub: set_nat] : ( aa_set871585252387937313nteger(aa_fun4587287018503165694nteger(aTP_Lamm_kp(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(Uua,Uub)),aa_set871585252387937313nteger(Uu,Uub)) ) ).

% ATP.lambda_531
tff(fact_10219_ATP_Olambda__532,axiom,
    ! [Uu: fun_set_nat_rat,Uua: fun_set_nat_rat,Uub: set_nat] : ( aa_set_nat_rat(aa_fun8478617152544252148at_rat(aTP_Lamm_kq(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(Uua,Uub)),aa_set_nat_rat(Uu,Uub)) ) ).

% ATP.lambda_532
tff(fact_10220_ATP_Olambda__533,axiom,
    ! [Uu: fun_set_nat_int,Uua: fun_set_nat_int,Uub: set_nat] : ( aa_set_nat_int(aa_fun3600621996379952540at_int(aTP_Lamm_kr(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(Uua,Uub)),aa_set_nat_int(Uu,Uub)) ) ).

% ATP.lambda_533
tff(fact_10221_ATP_Olambda__534,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : ( aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_kj(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_534
tff(fact_10222_ATP_Olambda__535,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : ( aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kl(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ) ).

% ATP.lambda_535
tff(fact_10223_ATP_Olambda__536,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : ( aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_kn(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uua,Uub)),aa_nat_int(Uu,Uub)) ) ).

% ATP.lambda_536
tff(fact_10224_ATP_Olambda__537,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : ( aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_kk(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_537
tff(fact_10225_ATP_Olambda__538,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_rat,Uub: int] : ( aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_km(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uua,Uub)),aa_int_rat(Uu,Uub)) ) ).

% ATP.lambda_538
tff(fact_10226_ATP_Olambda__539,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : ( aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ko(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uua,Uub)),aa_int_int(Uu,Uub)) ) ).

% ATP.lambda_539
tff(fact_10227_ATP_Olambda__540,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : ( aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_lj(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ) ).

% ATP.lambda_540
tff(fact_10228_ATP_Olambda__541,axiom,
    ! [Uu: fun_se4101287982975402108nt_nat,Uua: fun_se4101287982975402108nt_nat,Uub: set_Pr958786334691620121nt_int] : ( aa_set4753215920113914285nt_nat(aa_fun209864160807355708nt_nat(aTP_Lamm_ll(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4753215920113914285nt_nat(Uua,Uub)),aa_set4753215920113914285nt_nat(Uu,Uub)) ) ).

% ATP.lambda_541
tff(fact_10229_ATP_Olambda__542,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : ( aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_lm(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_542
tff(fact_10230_ATP_Olambda__543,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : ( aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lp(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_543
tff(fact_10231_ATP_Olambda__544,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : ( aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ln(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_544
tff(fact_10232_ATP_Olambda__545,axiom,
    ! [Uu: fun_fu3504661237974713170_o_nat,Uua: fun_fu3504661237974713170_o_nat,Uub: fun_Pr6460944699146269935_nat_o] : ( aa_fun767828674014743683_o_nat(aa_fun6312600589078465724_o_nat(aTP_Lamm_lo(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_fun767828674014743683_o_nat(Uua,Uub)),aa_fun767828674014743683_o_nat(Uu,Uub)) ) ).

% ATP.lambda_545
tff(fact_10233_ATP_Olambda__546,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : ( aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kz(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_546
tff(fact_10234_ATP_Olambda__547,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : ( aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_la(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_547
tff(fact_10235_ATP_Olambda__548,axiom,
    ! [Uu: fun_Pr1099658227000185396_nat_o,Uua: fun_Pr1099658227000185396_nat_o,Uub: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(aa_fun9125434449681309094_nat_o(aTP_Lamm_cn(Uu),Uua),Uub)
    <=> ( aa_Pro8143006071014924973_nat_o(Uu,Uub)
       => aa_Pro8143006071014924973_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_548
tff(fact_10236_ATP_Olambda__549,axiom,
    ! [Uu: fun_Pr3059359313727237556_nat_o,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_co(Uu),Uua),Uub)
    <=> ( aa_Pro5947980084102686253_nat_o(Uu,Uub)
       => aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_549
tff(fact_10237_ATP_Olambda__550,axiom,
    ! [Uu: fun_se7249947736802449410_int_o,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_cj(Uu),Uua),Uub)
    <=> ( aa_set6170482117830564987_int_o(Uu,Uub)
       => aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_550
tff(fact_10238_ATP_Olambda__551,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ck(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
       => aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_551
tff(fact_10239_ATP_Olambda__552,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cp(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_552
tff(fact_10240_ATP_Olambda__553,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_cl(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_553
tff(fact_10241_ATP_Olambda__554,axiom,
    ! [Uu: fun_fu5135969500543783852at_o_o,Uua: fun_fu5135969500543783852at_o_o,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_cm(Uu),Uua),Uub)
    <=> ( aa_fun1675921005893230629at_o_o(Uu,Uub)
       => aa_fun1675921005893230629at_o_o(Uua,Uub) ) ) ).

% ATP.lambda_554
tff(fact_10242_ATP_Olambda__555,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_ae(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(Uu,Uub)
        | aa_Pro653082453696054616_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_555
tff(fact_10243_ATP_Olambda__556,axiom,
    ! [Uu: fun_Pr1099658227000185396_nat_o,Uua: fun_Pr1099658227000185396_nat_o,Uub: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(aa_fun9125434449681309094_nat_o(aTP_Lamm_cu(Uu),Uua),Uub)
    <=> ( aa_Pro8143006071014924973_nat_o(Uu,Uub)
        | aa_Pro8143006071014924973_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_556
tff(fact_10244_ATP_Olambda__557,axiom,
    ! [Uu: fun_Pr3059359313727237556_nat_o,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_cv(Uu),Uua),Uub)
    <=> ( aa_Pro5947980084102686253_nat_o(Uu,Uub)
        | aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_557
tff(fact_10245_ATP_Olambda__558,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_aeq(Uu),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        | aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_558
tff(fact_10246_ATP_Olambda__559,axiom,
    ! [Uu: fun_se7249947736802449410_int_o,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_cq(Uu),Uua),Uub)
    <=> ( aa_set6170482117830564987_int_o(Uu,Uub)
        | aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_559
tff(fact_10247_ATP_Olambda__560,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_cr(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        | aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_560
tff(fact_10248_ATP_Olambda__561,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cw(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_561
tff(fact_10249_ATP_Olambda__562,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_cs(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_562
tff(fact_10250_ATP_Olambda__563,axiom,
    ! [Uu: fun_fu5135969500543783852at_o_o,Uua: fun_fu5135969500543783852at_o_o,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_ct(Uu),Uua),Uub)
    <=> ( aa_fun1675921005893230629at_o_o(Uu,Uub)
        | aa_fun1675921005893230629at_o_o(Uua,Uub) ) ) ).

% ATP.lambda_563
tff(fact_10251_ATP_Olambda__564,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_af(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(Uu,Uub)
        & aa_Pro653082453696054616_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_564
tff(fact_10252_ATP_Olambda__565,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_by(Uu),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_565
tff(fact_10253_ATP_Olambda__566,axiom,
    ! [Uu: fun_se7249947736802449410_int_o,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_bu(Uu),Uua),Uub)
    <=> ( aa_set6170482117830564987_int_o(Uu,Uub)
        & aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_566
tff(fact_10254_ATP_Olambda__567,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_bv(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_567
tff(fact_10255_ATP_Olambda__568,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bz(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_568
tff(fact_10256_ATP_Olambda__569,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bw(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_569
tff(fact_10257_ATP_Olambda__570,axiom,
    ! [Uu: fun_fu5135969500543783852at_o_o,Uua: fun_fu5135969500543783852at_o_o,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_bx(Uu),Uua),Uub)
    <=> ( aa_fun1675921005893230629at_o_o(Uu,Uub)
        & aa_fun1675921005893230629at_o_o(Uua,Uub) ) ) ).

% ATP.lambda_570
tff(fact_10258_ATP_Olambda__571,axiom,
    ! [Uu: fun_se7586849273513198688nteger,Uua: fun_set_nat_o,Uub: set_nat] : ( aa_set871585252387937313nteger(aa_fun7102796863437819535nteger(aTP_Lamm_uf(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(Uu,Uub)),aa_o_Code_integer(zero_n356916108424825756nteger,aa_set_nat_o(Uua,Uub))) ) ).

% ATP.lambda_571
tff(fact_10259_ATP_Olambda__572,axiom,
    ! [Uu: fun_set_nat_rat,Uua: fun_set_nat_o,Uub: set_nat] : ( aa_set_nat_rat(aa_fun7116522257906611850at_rat(aTP_Lamm_ug(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(Uu,Uub)),aa_o_rat(zero_n2052037380579107095ol_rat,aa_set_nat_o(Uua,Uub))) ) ).

% ATP.lambda_572
tff(fact_10260_ATP_Olambda__573,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_o,Uub: nat] : ( aa_nat_Code_integer(aa_fun3969837373441324047nteger(aTP_Lamm_ty(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uu,Uub)),aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(Uua,Uub))) ) ).

% ATP.lambda_573
tff(fact_10261_ATP_Olambda__574,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] : ( aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_ua(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uub)),aa_o_rat(zero_n2052037380579107095ol_rat,aa_nat_o(Uua,Uub))) ) ).

% ATP.lambda_574
tff(fact_10262_ATP_Olambda__575,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_o,Uub: nat] : ( aa_nat_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_ud(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(Uua,Uub))) ) ).

% ATP.lambda_575
tff(fact_10263_ATP_Olambda__576,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_o,Uub: nat] : ( aa_nat_int(aa_fun27009930614080094at_int(aTP_Lamm_ub(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(Uua,Uub))) ) ).

% ATP.lambda_576
tff(fact_10264_ATP_Olambda__577,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_o,Uub: int] : ( aa_int_Code_integer(aa_fun6019004165459918095nteger(aTP_Lamm_tx(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uu,Uub)),aa_o_Code_integer(zero_n356916108424825756nteger,aa_int_o(Uua,Uub))) ) ).

% ATP.lambda_577
tff(fact_10265_ATP_Olambda__578,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] : ( aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_tz(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uu,Uub)),aa_o_rat(zero_n2052037380579107095ol_rat,aa_int_o(Uua,Uub))) ) ).

% ATP.lambda_578
tff(fact_10266_ATP_Olambda__579,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] : ( aa_int_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_uc(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(Uu,Uub)),aa_o_nat(zero_n2687167440665602831ol_nat,aa_int_o(Uua,Uub))) ) ).

% ATP.lambda_579
tff(fact_10267_ATP_Olambda__580,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_o,Uub: int] : ( aa_int_int(aa_fun8822586851003501406nt_int(aTP_Lamm_ue(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(Uua,Uub))) ) ).

% ATP.lambda_580
tff(fact_10268_ATP_Olambda__581,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : ( aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_ph(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_581
tff(fact_10269_ATP_Olambda__582,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: set_nat,Uub: nat] : ( aa_nat_int(aa_set2359242532610849469at_int(aTP_Lamm_pj(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uub)),Uua) ) ).

% ATP.lambda_582
tff(fact_10270_ATP_Olambda__583,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: set_int,Uub: nat] : ( aa_nat_int(aa_set6310687161026558745at_int(aTP_Lamm_pm(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_nat_fun_int_int(Uu,Uub),Uua) ) ).

% ATP.lambda_583
tff(fact_10271_ATP_Olambda__584,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: set_nat,Uub: int] : ( aa_int_int(aa_set3358596244511904025nt_int(aTP_Lamm_pp(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(Uu,Uub)),Uua) ) ).

% ATP.lambda_584
tff(fact_10272_ATP_Olambda__585,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : ( aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_ps(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ) ).

% ATP.lambda_585
tff(fact_10273_ATP_Olambda__586,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : ( aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_kt(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_586
tff(fact_10274_ATP_Olambda__587,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : ( aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_kw(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ) ).

% ATP.lambda_587
tff(fact_10275_ATP_Olambda__588,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ux(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_588
tff(fact_10276_ATP_Olambda__589,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kd(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_589
tff(fact_10277_ATP_Olambda__590,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : ( aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_qa(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(Uu,Uub)),Uua) ) ).

% ATP.lambda_590
tff(fact_10278_ATP_Olambda__591,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : ( aa_int_int(aa_int_fun_int_int(aTP_Lamm_ke(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(Uu,Uub)),Uua) ) ).

% ATP.lambda_591
tff(fact_10279_ATP_Olambda__592,axiom,
    ! [Uu: fun_set_nat_nat,Uua: nat,Uub: set_nat] : ( aa_set_nat_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_adr(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_592
tff(fact_10280_ATP_Olambda__593,axiom,
    ! [Uu: fun_set_nat_int,Uua: int,Uub: set_nat] : ( aa_set_nat_int(aa_int2448287375063835929at_int(aTP_Lamm_adu(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_set_nat_int(Uu,Uub)),Uua) ) ).

% ATP.lambda_593
tff(fact_10281_ATP_Olambda__594,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: code_natural,Uub: nat] : ( aa_nat_Code_natural(aa_Cod5320533616963873627atural(aTP_Lamm_adi(Uu),Uua),Uub) = divide5121882707175180666atural(aa_nat_Code_natural(Uu,Uub),Uua) ) ).

% ATP.lambda_594
tff(fact_10282_ATP_Olambda__595,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat] : ( aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_adc(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(Uu,Uub)),Uua) ) ).

% ATP.lambda_595
tff(fact_10283_ATP_Olambda__596,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_adl(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_596
tff(fact_10284_ATP_Olambda__597,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : ( aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_acw(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(Uu,Uub)),Uua) ) ).

% ATP.lambda_597
tff(fact_10285_ATP_Olambda__598,axiom,
    ! [Uu: fun_int_Code_natural,Uua: code_natural,Uub: int] : ( aa_int_Code_natural(aa_Cod5692187177200718263atural(aTP_Lamm_adf(Uu),Uua),Uub) = divide5121882707175180666atural(aa_int_Code_natural(Uu,Uub),Uua) ) ).

% ATP.lambda_598
tff(fact_10286_ATP_Olambda__599,axiom,
    ! [Uu: fun_int_Code_integer,Uua: code_integer,Uub: int] : ( aa_int_Code_integer(aa_Cod1949805079358391713nteger(aTP_Lamm_acz(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_int_Code_integer(Uu,Uub)),Uua) ) ).

% ATP.lambda_599
tff(fact_10287_ATP_Olambda__600,axiom,
    ! [Uu: fun_int_nat,Uua: nat,Uub: int] : ( aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_act(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_600
tff(fact_10288_ATP_Olambda__601,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : ( aa_int_int(aa_int_fun_int_int(aTP_Lamm_ado(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(Uu,Uub)),Uua) ) ).

% ATP.lambda_601
tff(fact_10289_ATP_Olambda__602,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lf(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_602
tff(fact_10290_ATP_Olambda__603,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : ( aa_int_int(aa_int_fun_int_int(aTP_Lamm_lg(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),Uua) ) ).

% ATP.lambda_603
tff(fact_10291_ATP_Olambda__604,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_px(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_604
tff(fact_10292_ATP_Olambda__605,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_py(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(Uu,Uub)),Uua) ) ).

% ATP.lambda_605
tff(fact_10293_ATP_Olambda__606,axiom,
    ! [Uu: fun_int_int,Uua: nat,Uub: int] : ( aa_int_int(aa_nat_fun_int_int(aTP_Lamm_pz(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(Uu,Uub)),Uua) ) ).

% ATP.lambda_606
tff(fact_10294_ATP_Olambda__607,axiom,
    ! [Uu: fun_se4101287982975402108nt_nat,Uua: set_nat,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set5909024289875323334_int_o(aTP_Lamm_fg(Uu),Uua),Uub)
    <=> aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_set4753215920113914285nt_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_607
tff(fact_10295_ATP_Olambda__608,axiom,
    ! [Uu: fun_set_nat_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_ff(Uu),Uua),Uub)
    <=> aa_set_set_nat_o(member_set_nat(aa_set_nat_set_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_608
tff(fact_10296_ATP_Olambda__609,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_fb(Uu),Uua),Uub)
    <=> aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_set_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_609
tff(fact_10297_ATP_Olambda__610,axiom,
    ! [Uu: fun_set_nat_int,Uua: set_int,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7670721151442836043_nat_o(aTP_Lamm_fc(Uu),Uua),Uub)
    <=> aa_set_int_o(member_int(aa_set_nat_int(Uu,Uub)),Uua) ) ).

% ATP.lambda_610
tff(fact_10298_ATP_Olambda__611,axiom,
    ! [Uu: fun_nat_set_nat,Uua: set_set_nat,Uub: nat] :
      ( aa_nat_o(aa_set1169776424217490599_nat_o(aTP_Lamm_fd(Uu),Uua),Uub)
    <=> aa_set_set_nat_o(member_set_nat(aa_nat_set_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_611
tff(fact_10299_ATP_Olambda__612,axiom,
    ! [Uu: fun_nat_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ex(Uu),Uua),Uub)
    <=> aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_612
tff(fact_10300_ATP_Olambda__613,axiom,
    ! [Uu: fun_nat_int,Uua: set_int,Uub: nat] :
      ( aa_nat_o(aa_set_int_fun_nat_o(aTP_Lamm_ey(Uu),Uua),Uub)
    <=> aa_set_int_o(member_int(aa_nat_int(Uu,Uub)),Uua) ) ).

% ATP.lambda_613
tff(fact_10301_ATP_Olambda__614,axiom,
    ! [Uu: fun_int_set_nat,Uua: set_set_nat,Uub: int] :
      ( aa_int_o(aa_set8572106115639578443_int_o(aTP_Lamm_fe(Uu),Uua),Uub)
    <=> aa_set_set_nat_o(member_set_nat(aa_int_set_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_614
tff(fact_10302_ATP_Olambda__615,axiom,
    ! [Uu: fun_int_nat,Uua: set_nat,Uub: int] :
      ( aa_int_o(aa_set_nat_fun_int_o(aTP_Lamm_ez(Uu),Uua),Uub)
    <=> aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_615
tff(fact_10303_ATP_Olambda__616,axiom,
    ! [Uu: fun_int_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_fa(Uu),Uua),Uub)
    <=> aa_set_int_o(member_int(aa_int_int(Uu,Uub)),Uua) ) ).

% ATP.lambda_616
tff(fact_10304_ATP_Olambda__617,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_iq(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_617
tff(fact_10305_ATP_Olambda__618,axiom,
    ! [Uu: product_prod_nat_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_ir(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_618
tff(fact_10306_ATP_Olambda__619,axiom,
    ! [Uu: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_is((Uu)),Uua),(Uub))
    <=> ( ~ ( (Uub)
          <=> (Uu) )
       => aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_619
tff(fact_10307_ATP_Olambda__620,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_it(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_620
tff(fact_10308_ATP_Olambda__621,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_iu(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_621
tff(fact_10309_ATP_Olambda__622,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_iv(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_622
tff(fact_10310_ATP_Olambda__623,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_iw(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_623
tff(fact_10311_ATP_Olambda__624,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_fu5135969500543783852at_o_o,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_ix(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_fun1675921005893230629at_o_o(Uua,Uub) ) ) ).

% ATP.lambda_624
tff(fact_10312_ATP_Olambda__625,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : ( aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_qf(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub))) ) ).

% ATP.lambda_625
tff(fact_10313_ATP_Olambda__626,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_se7586849273513198688nteger,Uub: set_nat] : ( aa_set871585252387937313nteger(aa_fun4587287018503165694nteger(aTP_Lamm_up(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_set_nat_o(Uu,Uub))),aa_set871585252387937313nteger(Uua,Uub)) ) ).

% ATP.lambda_626
tff(fact_10314_ATP_Olambda__627,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_rat,Uub: set_nat] : ( aa_set_nat_rat(aa_fun8478617152544252148at_rat(aTP_Lamm_uq(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(zero_n2052037380579107095ol_rat,aa_set_nat_o(Uu,Uub))),aa_set_nat_rat(Uua,Uub)) ) ).

% ATP.lambda_627
tff(fact_10315_ATP_Olambda__628,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_Code_integer,Uub: nat] : ( aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_ui(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(Uu,Uub))),aa_nat_Code_integer(Uua,Uub)) ) ).

% ATP.lambda_628
tff(fact_10316_ATP_Olambda__629,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_rat,Uub: nat] : ( aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_uk(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(zero_n2052037380579107095ol_rat,aa_nat_o(Uu,Uub))),aa_nat_rat(Uua,Uub)) ) ).

% ATP.lambda_629
tff(fact_10317_ATP_Olambda__630,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] : ( aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_un(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(Uu,Uub))),aa_nat_nat(Uua,Uub)) ) ).

% ATP.lambda_630
tff(fact_10318_ATP_Olambda__631,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: nat] : ( aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ul(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(Uu,Uub))),aa_nat_int(Uua,Uub)) ) ).

% ATP.lambda_631
tff(fact_10319_ATP_Olambda__632,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_Code_integer,Uub: int] : ( aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_uh(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_int_o(Uu,Uub))),aa_int_Code_integer(Uua,Uub)) ) ).

% ATP.lambda_632
tff(fact_10320_ATP_Olambda__633,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_rat,Uub: int] : ( aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_uj(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(zero_n2052037380579107095ol_rat,aa_int_o(Uu,Uub))),aa_int_rat(Uua,Uub)) ) ).

% ATP.lambda_633
tff(fact_10321_ATP_Olambda__634,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_nat,Uub: int] : ( aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_um(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_int_o(Uu,Uub))),aa_int_nat(Uua,Uub)) ) ).

% ATP.lambda_634
tff(fact_10322_ATP_Olambda__635,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_int,Uub: int] : ( aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_uo(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(Uu,Uub))),aa_int_int(Uua,Uub)) ) ).

% ATP.lambda_635
tff(fact_10323_ATP_Olambda__636,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abq(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(Uu),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ) ).

% ATP.lambda_636
tff(fact_10324_ATP_Olambda__637,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_abr(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(Uu),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ) ).

% ATP.lambda_637
tff(fact_10325_ATP_Olambda__638,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abp(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(Uu),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ) ).

% ATP.lambda_638
tff(fact_10326_ATP_Olambda__639,axiom,
    ! [Uu: code_natural,Uua: code_natural,Uub: nat] : ( aa_nat_Code_natural(aa_Cod5320533616963873627atural(aTP_Lamm_mr(Uu),Uua),Uub) = plus_p4538020629002901425atural(Uu,times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,Uub),Uua)) ) ).

% ATP.lambda_639
tff(fact_10327_ATP_Olambda__640,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : ( aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_mq(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Uub)),Uua)) ) ).

% ATP.lambda_640
tff(fact_10328_ATP_Olambda__641,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : ( aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_mu(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ) ).

% ATP.lambda_641
tff(fact_10329_ATP_Olambda__642,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mt(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_642
tff(fact_10330_ATP_Olambda__643,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : ( aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ms(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_643
tff(fact_10331_ATP_Olambda__644,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : ( aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_aej(Uu),Uua),Uub) = aa_nat_Code_integer(power_8256067586552552935nteger(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ) ).

% ATP.lambda_644
tff(fact_10332_ATP_Olambda__645,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : ( aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_aek(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_645
tff(fact_10333_ATP_Olambda__646,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ael(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_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_646
tff(fact_10334_ATP_Olambda__647,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : ( aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_kf(Uu),Uua),Uub) = aa_nat_Code_integer(power_8256067586552552935nteger(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ) ).

% ATP.lambda_647
tff(fact_10335_ATP_Olambda__648,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : ( aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_kg(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_648
tff(fact_10336_ATP_Olambda__649,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_kh(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ) ).

% ATP.lambda_649
tff(fact_10337_ATP_Olambda__650,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ls(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_650
tff(fact_10338_ATP_Olambda__651,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : ( aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lh(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ) ).

% ATP.lambda_651
tff(fact_10339_ATP_Olambda__652,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] : ( aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_li(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,Uu),aa_int_int(Uua,Uub)) ) ).

% ATP.lambda_652
tff(fact_10340_ATP_Olambda__653,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : ( aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qb(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),aa_nat_nat(Uua,Uub)) ) ).

% ATP.lambda_653
tff(fact_10341_ATP_Olambda__654,axiom,
    ! [Uu: int,Uua: fun_nat_nat,Uub: nat] : ( aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_qc(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(Uua,Uub)) ) ).

% ATP.lambda_654
tff(fact_10342_ATP_Olambda__655,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : ( aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_qd(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_int_nat(Uua,Uub)) ) ).

% ATP.lambda_655
tff(fact_10343_ATP_Olambda__656,axiom,
    ! [Uu: fun_set_nat_nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_ads(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_set_nat_nat(Uu,Uub)) ) ).

% ATP.lambda_656
tff(fact_10344_ATP_Olambda__657,axiom,
    ! [Uu: fun_set_nat_int,Uua: int,Uub: set_nat] :
      ( aa_set_nat_o(aa_int_fun_set_nat_o(aTP_Lamm_adv(Uu),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_set_nat_int(Uu,Uub)) ) ).

% ATP.lambda_657
tff(fact_10345_ATP_Olambda__658,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: code_natural,Uub: nat] :
      ( aa_nat_o(aa_Cod9120328075896114107_nat_o(aTP_Lamm_adj(Uu),Uua),Uub)
    <=> aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_nat_Code_natural(Uu,Uub)) ) ).

% ATP.lambda_658
tff(fact_10346_ATP_Olambda__659,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aTP_Lamm_add(Uu),Uua),Uub)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_nat_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_659
tff(fact_10347_ATP_Olambda__660,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_adm(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_nat_nat(Uu,Uub)) ) ).

% ATP.lambda_660
tff(fact_10348_ATP_Olambda__661,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aTP_Lamm_acx(Uu),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_nat_int(Uu,Uub)) ) ).

% ATP.lambda_661
tff(fact_10349_ATP_Olambda__662,axiom,
    ! [Uu: fun_int_Code_natural,Uua: code_natural,Uub: int] :
      ( aa_int_o(aa_Cod7299285730463426143_int_o(aTP_Lamm_adg(Uu),Uua),Uub)
    <=> aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_int_Code_natural(Uu,Uub)) ) ).

% ATP.lambda_662
tff(fact_10350_ATP_Olambda__663,axiom,
    ! [Uu: fun_int_Code_integer,Uua: code_integer,Uub: int] :
      ( aa_int_o(aa_Cod8477617969808203498_int_o(aTP_Lamm_ada(Uu),Uua),Uub)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_int_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_663
tff(fact_10351_ATP_Olambda__664,axiom,
    ! [Uu: fun_int_nat,Uua: nat,Uub: int] :
      ( aa_int_o(aa_nat_fun_int_o(aTP_Lamm_acu(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_int_nat(Uu,Uub)) ) ).

% ATP.lambda_664
tff(fact_10352_ATP_Olambda__665,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_adp(Uu),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_int_int(Uu,Uub)) ) ).

% ATP.lambda_665
tff(fact_10353_ATP_Olambda__666,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : ( aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_rg(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_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_666
tff(fact_10354_ATP_Olambda__667,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rh(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_667
tff(fact_10355_ATP_Olambda__668,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ri(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_668
tff(fact_10356_ATP_Olambda__669,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_afq(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(Uu,Uub)),aa_Pro4243173770044397973at_nat(Uu,Uub))) ) ).

% ATP.lambda_669
tff(fact_10357_ATP_Olambda__670,axiom,
    ! [Uu: fun_se4101287982975402108nt_nat,Uua: set_Pr958786334691620121nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(aTP_Lamm_afl(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_set4753215920113914285nt_nat(Uu,Uub)),aa_set4753215920113914285nt_nat(Uu,Uub))) ) ).

% ATP.lambda_670
tff(fact_10358_ATP_Olambda__671,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_afm(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_set_nat_nat(Uu,Uub)),aa_set_nat_nat(Uu,Uub))) ) ).

% ATP.lambda_671
tff(fact_10359_ATP_Olambda__672,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_afo(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uu,Uub))) ) ).

% ATP.lambda_672
tff(fact_10360_ATP_Olambda__673,axiom,
    ! [Uu: fun_int_nat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_afp(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_int_nat(Uu,Uub)),aa_int_nat(Uu,Uub))) ) ).

% ATP.lambda_673
tff(fact_10361_ATP_Olambda__674,axiom,
    ! [Uu: fun_fu3504661237974713170_o_nat,Uua: fun_Pr6460944699146269935_nat_o,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(aTP_Lamm_afn(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_fun767828674014743683_o_nat(Uu,Uub)),aa_fun767828674014743683_o_nat(Uu,Uub))) ) ).

% ATP.lambda_674
tff(fact_10362_ATP_Olambda__675,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_afw(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_Pro6150637604445964947_nat_o(Uua,Uub),aa_Pro4243173770044397973at_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_675
tff(fact_10363_ATP_Olambda__676,axiom,
    ! [Uu: fun_se4101287982975402108nt_nat,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_afr(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_set6170482117830564987_int_o(Uua,Uub),aa_set4753215920113914285nt_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_676
tff(fact_10364_ATP_Olambda__677,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_afs(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_set_nat_o(Uua,Uub),aa_set_nat_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_677
tff(fact_10365_ATP_Olambda__678,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_afu(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_nat_o(Uua,Uub),aa_nat_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_678
tff(fact_10366_ATP_Olambda__679,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_afv(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_679
tff(fact_10367_ATP_Olambda__680,axiom,
    ! [Uu: fun_fu3504661237974713170_o_nat,Uua: fun_fu5135969500543783852at_o_o,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_aft(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_fun1675921005893230629at_o_o(Uua,Uub),aa_fun767828674014743683_o_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_680
tff(fact_10368_ATP_Olambda__681,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: fun_Pr2602117396471556196at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun741873263805325634_nat_o(aTP_Lamm_afk(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro4243173770044397973at_nat(Uu,Uub)),aa_Pro4243173770044397973at_nat(Uua,Uub))) ) ).

% ATP.lambda_681
tff(fact_10369_ATP_Olambda__682,axiom,
    ! [Uu: fun_se4101287982975402108nt_nat,Uua: fun_se4101287982975402108nt_nat,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun3785083049450726210_int_o(aTP_Lamm_afa(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4753215920113914285nt_nat(Uu,Uub)),aa_set4753215920113914285nt_nat(Uua,Uub))) ) ).

% ATP.lambda_682
tff(fact_10370_ATP_Olambda__683,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun1332161875654541722_nat_o(aTP_Lamm_afc(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(Uu,Uub)),aa_set_nat_nat(Uua,Uub))) ) ).

% ATP.lambda_683
tff(fact_10371_ATP_Olambda__684,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_afg(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub))) ) ).

% ATP.lambda_684
tff(fact_10372_ATP_Olambda__685,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_afi(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uua,Uub))) ) ).

% ATP.lambda_685
tff(fact_10373_ATP_Olambda__686,axiom,
    ! [Uu: fun_fu3504661237974713170_o_nat,Uua: fun_fu3504661237974713170_o_nat,Uub: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun112519265078508482at_o_o(aTP_Lamm_afe(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_fun767828674014743683_o_nat(Uu,Uub)),aa_fun767828674014743683_o_nat(Uua,Uub))) ) ).

% ATP.lambda_686
tff(fact_10374_ATP_Olambda__687,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zw(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_687
tff(fact_10375_ATP_Olambda__688,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_zx(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_688
tff(fact_10376_ATP_Olambda__689,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_alt(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_689
tff(fact_10377_ATP_Olambda__690,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lx(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_690
tff(fact_10378_ATP_Olambda__691,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qe(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_691
tff(fact_10379_ATP_Olambda__692,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_692
tff(fact_10380_ATP_Olambda__693,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_693
tff(fact_10381_ATP_Olambda__694,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_694
tff(fact_10382_ATP_Olambda__695,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_695
tff(fact_10383_ATP_Olambda__696,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_696
tff(fact_10384_ATP_Olambda__697,axiom,
    ! [Uu: fun_op8989173174992673105on_num,Uua: fun_nat_option_num,Uub: nat] : ( aa_nat_option_num(aa_fun7716400287237121964on_num(aTP_Lamm_nq(Uu),Uua),Uub) = aa_opt5369159329878751634on_num(Uu,aa_nat_option_num(Uua,Uub)) ) ).

% ATP.lambda_697
tff(fact_10385_ATP_Olambda__698,axiom,
    ! [Uu: fun_option_num_o,Uua: fun_nat_option_num,Uub: nat] :
      ( aa_nat_o(aa_fun1675964276245824108_nat_o(aTP_Lamm_no(Uu),Uua),Uub)
    <=> aa_option_num_o(Uu,aa_nat_option_num(Uua,Uub)) ) ).

% ATP.lambda_698
tff(fact_10386_ATP_Olambda__699,axiom,
    ! [Uu: fun_option_num_nat,Uua: fun_nat_option_num,Uub: nat] : ( aa_nat_nat(aa_fun7387201323300553874at_nat(aTP_Lamm_np(Uu),Uua),Uub) = aa_option_num_nat(Uu,aa_nat_option_num(Uua,Uub)) ) ).

% ATP.lambda_699
tff(fact_10387_ATP_Olambda__700,axiom,
    ! [Uu: fun_o_option_num,Uua: fun_nat_o,Uub: nat] : ( aa_nat_option_num(aa_fun6990899178391151900on_num(aTP_Lamm_nk(Uu),Uua),Uub) = aa_o_option_num(Uu,aa_nat_o(Uua,Uub)) ) ).

% ATP.lambda_700
tff(fact_10388_ATP_Olambda__701,axiom,
    ! [Uu: fun_o_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ni(Uu),Uua),Uub)
    <=> aa_o_o(Uu,aa_nat_o(Uua,Uub)) ) ).

% ATP.lambda_701
tff(fact_10389_ATP_Olambda__702,axiom,
    ! [Uu: fun_o_nat,Uua: fun_nat_o,Uub: nat] : ( aa_nat_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_nj(Uu),Uua),Uub) = aa_o_nat(Uu,aa_nat_o(Uua,Uub)) ) ).

% ATP.lambda_702
tff(fact_10390_ATP_Olambda__703,axiom,
    ! [Uu: fun_nat_option_num,Uua: fun_nat_nat,Uub: nat] : ( aa_nat_option_num(aa_fun6517009837249254290on_num(aTP_Lamm_nn(Uu),Uua),Uub) = aa_nat_option_num(Uu,aa_nat_nat(Uua,Uub)) ) ).

% ATP.lambda_703
tff(fact_10391_ATP_Olambda__704,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_nl(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(Uua,Uub)) ) ).

% ATP.lambda_704
tff(fact_10392_ATP_Olambda__705,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : ( aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nm(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(Uua,Uub)) ) ).

% ATP.lambda_705
tff(fact_10393_ATP_Olambda__706,axiom,
    ! [Uu: fun_set_nat_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ek(Uu),Uua),Uub)
    <=> aa_set_nat_o(Uua,aa_set_nat_set_nat(Uu,Uub)) ) ).

% ATP.lambda_706
tff(fact_10394_ATP_Olambda__707,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun5113018963486751686_nat_o(aTP_Lamm_ei(Uu),Uua),Uub)
    <=> aa_nat_o(Uua,aa_set_nat_nat(Uu,Uub)) ) ).

% ATP.lambda_707
tff(fact_10395_ATP_Olambda__708,axiom,
    ! [Uu: fun_set_nat_int,Uua: fun_int_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun6321136096415131682_nat_o(aTP_Lamm_ej(Uu),Uua),Uub)
    <=> aa_int_o(Uua,aa_set_nat_int(Uu,Uub)) ) ).

% ATP.lambda_708
tff(fact_10396_ATP_Olambda__709,axiom,
    ! [Uu: fun_na5449828587677346466nt_int,Uua: fun_se7249947736802449410_int_o,Uub: nat] :
      ( aa_nat_o(aa_fun7485545044466630625_nat_o(aTP_Lamm_el(Uu),Uua),Uub)
    <=> aa_set6170482117830564987_int_o(Uua,aa_nat6759544748818335627nt_int(Uu,Uub)) ) ).

% ATP.lambda_709
tff(fact_10397_ATP_Olambda__710,axiom,
    ! [Uu: fun_nat_set_nat,Uua: fun_set_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun7902669250853151666_nat_o(aTP_Lamm_eg(Uu),Uua),Uub)
    <=> aa_set_nat_o(Uua,aa_nat_set_nat(Uu,Uub)) ) ).

% ATP.lambda_710
tff(fact_10398_ATP_Olambda__711,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ec(Uu),Uua),Uub)
    <=> aa_nat_o(Uua,aa_nat_nat(Uu,Uub)) ) ).

% ATP.lambda_711
tff(fact_10399_ATP_Olambda__712,axiom,
    ! [Uu: fun_nat_int,Uua: fun_int_o,Uub: nat] :
      ( aa_nat_o(aa_fun8064560132910005592_nat_o(aTP_Lamm_ee(Uu),Uua),Uub)
    <=> aa_int_o(Uua,aa_nat_int(Uu,Uub)) ) ).

% ATP.lambda_712
tff(fact_10400_ATP_Olambda__713,axiom,
    ! [Uu: fun_int_set_nat,Uua: fun_set_nat_o,Uub: int] :
      ( aa_int_o(aa_fun6081626905420463702_int_o(aTP_Lamm_eh(Uu),Uua),Uub)
    <=> aa_set_nat_o(Uua,aa_int_set_nat(Uu,Uub)) ) ).

% ATP.lambda_713
tff(fact_10401_ATP_Olambda__714,axiom,
    ! [Uu: fun_int_nat,Uua: fun_nat_o,Uub: int] :
      ( aa_int_o(aa_fun4116151057475255200_int_o(aTP_Lamm_ed(Uu),Uua),Uub)
    <=> aa_nat_o(Uua,aa_int_nat(Uu,Uub)) ) ).

% ATP.lambda_714
tff(fact_10402_ATP_Olambda__715,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ef(Uu),Uua),Uub)
    <=> aa_int_o(Uua,aa_int_int(Uu,Uub)) ) ).

% ATP.lambda_715
tff(fact_10403_ATP_Olambda__716,axiom,
    ! [Uu: fun_set_nat_nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_adt(Uu),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_set_nat_nat(Uu,Uub)) ) ).

% ATP.lambda_716
tff(fact_10404_ATP_Olambda__717,axiom,
    ! [Uu: fun_set_nat_int,Uua: int,Uub: set_nat] :
      ( aa_set_nat_o(aa_int_fun_set_nat_o(aTP_Lamm_adw(Uu),Uua),Uub)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_set_nat_int(Uu,Uub)) ) ).

% ATP.lambda_717
tff(fact_10405_ATP_Olambda__718,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: code_natural,Uub: nat] :
      ( aa_nat_o(aa_Cod9120328075896114107_nat_o(aTP_Lamm_adk(Uu),Uua),Uub)
    <=> ~ aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_nat_Code_natural(Uu,Uub)) ) ).

% ATP.lambda_718
tff(fact_10406_ATP_Olambda__719,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aTP_Lamm_ade(Uu),Uua),Uub)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_nat_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_719
tff(fact_10407_ATP_Olambda__720,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_adn(Uu),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_nat_nat(Uu,Uub)) ) ).

% ATP.lambda_720
tff(fact_10408_ATP_Olambda__721,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aTP_Lamm_acy(Uu),Uua),Uub)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_nat_int(Uu,Uub)) ) ).

% ATP.lambda_721
tff(fact_10409_ATP_Olambda__722,axiom,
    ! [Uu: fun_int_Code_natural,Uua: code_natural,Uub: int] :
      ( aa_int_o(aa_Cod7299285730463426143_int_o(aTP_Lamm_adh(Uu),Uua),Uub)
    <=> ~ aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_int_Code_natural(Uu,Uub)) ) ).

% ATP.lambda_722
tff(fact_10410_ATP_Olambda__723,axiom,
    ! [Uu: fun_int_Code_integer,Uua: code_integer,Uub: int] :
      ( aa_int_o(aa_Cod8477617969808203498_int_o(aTP_Lamm_adb(Uu),Uua),Uub)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_int_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_723
tff(fact_10411_ATP_Olambda__724,axiom,
    ! [Uu: fun_int_nat,Uua: nat,Uub: int] :
      ( aa_int_o(aa_nat_fun_int_o(aTP_Lamm_acv(Uu),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_int_nat(Uu,Uub)) ) ).

% ATP.lambda_724
tff(fact_10412_ATP_Olambda__725,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_adq(Uu),Uua),Uub)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_int_int(Uu,Uub)) ) ).

% ATP.lambda_725
tff(fact_10413_ATP_Olambda__726,axiom,
    ! [Uu: set_nat,Uua: fun_Pr6984107991879650211_nat_o,Uub: product_prod_nat_nat] : ( aa_Pro4243173770044397973at_nat(aa_fun2860604400581938115at_nat(aTP_Lamm_aim(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_Pro4733806262894120092_nat_o(aa_fun5503210535175347332_nat_o(aTP_Lamm_ail(Uu),Uua),Uub))) ) ).

% ATP.lambda_726
tff(fact_10414_ATP_Olambda__727,axiom,
    ! [Uu: set_list_nat,Uua: fun_na7234455742697417626_nat_o,Uub: nat] : ( aa_nat_nat(aa_fun5158605793559895163at_nat(aTP_Lamm_aij(Uu),Uua),Uub) = finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aa_fun6745918892298368316_nat_o(aTP_Lamm_aii(Uu),Uua),Uub))) ) ).

% ATP.lambda_727
tff(fact_10415_ATP_Olambda__728,axiom,
    ! [Uu: set_nat,Uua: fun_li2421049240157457050_nat_o,Uub: list_nat] : ( aa_list_nat_nat(aa_fun8348502103087546635at_nat(aTP_Lamm_aif(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_lis1686035672126679939_nat_o(aa_fun6275404141584987452_nat_o(aTP_Lamm_aie(Uu),Uua),Uub))) ) ).

% ATP.lambda_728
tff(fact_10416_ATP_Olambda__729,axiom,
    ! [Uu: set_o,Uua: fun_nat_fun_o_o,Uub: nat] : ( aa_nat_nat(aa_fun4178193321845537899at_nat(aTP_Lamm_aia(Uu),Uua),Uub) = finite_card_o(collect_o(aa_nat_fun_o_o(aa_fun3555063939605388254un_o_o(aTP_Lamm_ahz(Uu),Uua),Uub))) ) ).

% ATP.lambda_729
tff(fact_10417_ATP_Olambda__730,axiom,
    ! [Uu: set_nat,Uua: fun_o_fun_nat_o,Uub: $o] : ( aa_o_nat(aa_fun6229755271245457521_o_nat(aTP_Lamm_ahw(Uu),Uua),(Uub)) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_o_fun_nat_o(aa_fun9138341750119017558_nat_o(aTP_Lamm_ahv(Uu),Uua),(Uub)))) ) ).

% ATP.lambda_730
tff(fact_10418_ATP_Olambda__731,axiom,
    ! [Uu: set_set_nat,Uua: fun_na8563630686737515764_nat_o,Uub: nat] : ( aa_nat_nat(aa_fun7187098123162309461at_nat(aTP_Lamm_aik(Uu),Uua),Uub) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aa_fun1792963693083279932_nat_o(aTP_Lamm_we(Uu),Uua),Uub))) ) ).

% ATP.lambda_731
tff(fact_10419_ATP_Olambda__732,axiom,
    ! [Uu: set_nat,Uua: fun_se382663106265794036_nat_o,Uub: set_nat] : ( aa_set_nat_nat(aa_fun6238879321244136459at_nat(aTP_Lamm_aig(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_set_nat_fun_nat_o(aa_fun924172432987161148_nat_o(aTP_Lamm_vu(Uu),Uua),Uub))) ) ).

% ATP.lambda_732
tff(fact_10420_ATP_Olambda__733,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_o,Uub: nat] : ( aa_nat_nat(aa_fun8857679445107423627at_nat(aTP_Lamm_aic(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_vk(Uu),Uua),Uub))) ) ).

% ATP.lambda_733
tff(fact_10421_ATP_Olambda__734,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_o,Uub: nat] : ( aa_nat_nat(aa_fun8666973529875159599at_nat(aTP_Lamm_aib(Uu),Uua),Uub) = finite_card_int(collect_int(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_vd(Uu),Uua),Uub))) ) ).

% ATP.lambda_734
tff(fact_10422_ATP_Olambda__735,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_o,Uub: int] : ( aa_int_nat(aa_fun4650797359919590027nt_nat(aTP_Lamm_ahx(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_uy(Uu),Uua),Uub))) ) ).

% ATP.lambda_735
tff(fact_10423_ATP_Olambda__736,axiom,
    ! [Uu: fun_na1668100129766719005at_nat,Uua: product_prod_nat_nat,Uub: nat,Uuc: nat] : ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_Pro6895555690143131534at_nat(aTP_Lamm_hf(Uu),Uua),Uub),Uuc) = aa_Pro262675715343573244at_nat(aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Uu,Uub),Uuc),Uua) ) ).

% ATP.lambda_736
tff(fact_10424_ATP_Olambda__737,axiom,
    ! [Uu: fun_na5914473681949881652_nat_o,Uua: product_prod_nat_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_Pro1957341097241355813_nat_o(aTP_Lamm_hg(Uu),Uua),Uub),Uuc)
    <=> aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uu,Uub),Uuc),Uua) ) ).

% ATP.lambda_737
tff(fact_10425_ATP_Olambda__738,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] :
      ( aa_nat_Code_integer(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_oh(Uu),Uua),Uub),Uuc) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
          aa_nat_Code_integer(Uua,Uuc),
          $ite(Uuc = Uu,zero_z3403309356797280102nteger,aa_nat_Code_integer(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ) ).

% ATP.lambda_738
tff(fact_10426_ATP_Olambda__739,axiom,
    ! [Uu: nat,Uua: fun_na2338080690716861136at_nat,Uub: fun_na2338080690716861136at_nat,Uuc: nat] :
      ( aa_nat2240542744625857721at_nat(aa_fun4141531684813184444at_nat(aa_fun8230936523094545239at_nat(aTP_Lamm_oj(Uu),Uua),Uub),Uuc) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
          aa_nat2240542744625857721at_nat(Uua,Uuc),
          $ite(Uuc = Uu,zero_z1048942125864253310at_nat,aa_nat2240542744625857721at_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ) ).

% ATP.lambda_739
tff(fact_10427_ATP_Olambda__740,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ol(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_740
tff(fact_10428_ATP_Olambda__741,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      ( aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_op(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_741
tff(fact_10429_ATP_Olambda__742,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      ( aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_on(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_742
tff(fact_10430_ATP_Olambda__743,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] :
      ( aa_nat_Code_integer(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_rx(Uu),Uua),Uub),Uuc) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
          aa_nat_Code_integer(Uua,Uuc),
          $ite(Uuc = Uu,one_one_Code_integer,aa_nat_Code_integer(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ) ).

% ATP.lambda_743
tff(fact_10431_ATP_Olambda__744,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      ( aa_nat_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_ry(Uu),Uua),Uub),Uuc) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
          aa_nat_assn(Uua,Uuc),
          $ite(Uuc = Uu,one_one_assn,aa_nat_assn(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ) ).

% ATP.lambda_744
tff(fact_10432_ATP_Olambda__745,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_sa(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_745
tff(fact_10433_ATP_Olambda__746,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      ( aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sb(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_746
tff(fact_10434_ATP_Olambda__747,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      ( aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_sc(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_747
tff(fact_10435_ATP_Olambda__748,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] :
      ( aa_nat_Code_integer(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_oi(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_Code_integer(Uua,Uuc),aa_nat_Code_integer(Uub,Uuc)) ) ).

% ATP.lambda_748
tff(fact_10436_ATP_Olambda__749,axiom,
    ! [Uu: nat,Uua: fun_na2338080690716861136at_nat,Uub: fun_na2338080690716861136at_nat,Uuc: nat] :
      ( aa_nat2240542744625857721at_nat(aa_fun4141531684813184444at_nat(aa_fun8230936523094545239at_nat(aTP_Lamm_ok(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat2240542744625857721at_nat(Uua,Uuc),aa_nat2240542744625857721at_nat(Uub,Uuc)) ) ).

% ATP.lambda_749
tff(fact_10437_ATP_Olambda__750,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      ( aa_nat_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_rz(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_assn(Uua,Uuc),aa_nat_assn(Uub,Uuc)) ) ).

% ATP.lambda_750
tff(fact_10438_ATP_Olambda__751,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_om(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_751
tff(fact_10439_ATP_Olambda__752,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      ( aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_oq(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_752
tff(fact_10440_ATP_Olambda__753,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      ( aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_oo(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_753
tff(fact_10441_ATP_Olambda__754,axiom,
    ! [Uu: $o,Uua: fun_o_Code_integer,Uub: fun_o_Code_integer,Uuc: $o] :
      ( aa_o_Code_integer(aa_fun8520278569782659900nteger(aa_fun5804704703283808911nteger(aTP_Lamm_adx((Uu)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu),aa_o_Code_integer(Uua,(Uuc)),aa_o_Code_integer(Uub,(Uuc))) ) ).

% ATP.lambda_754
tff(fact_10442_ATP_Olambda__755,axiom,
    ! [Uu: $o,Uua: fun_o_assn,Uub: fun_o_assn,Uuc: $o] :
      ( aa_o_assn(aa_fun4536517337817438332o_assn(aa_fun3110273112593859354o_assn(aTP_Lamm_aea((Uu)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu),aa_o_assn(Uua,(Uuc)),aa_o_assn(Uub,(Uuc))) ) ).

% ATP.lambda_755
tff(fact_10443_ATP_Olambda__756,axiom,
    ! [Uu: $o,Uua: fun_o_rat,Uub: fun_o_rat,Uuc: $o] :
      ( aa_o_rat(aa_fun4336163974892599420_o_rat(aa_fun2919298112881465802_o_rat(aTP_Lamm_acj((Uu)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu),aa_o_rat(Uua,(Uuc)),aa_o_rat(Uub,(Uuc))) ) ).

% ATP.lambda_756
tff(fact_10444_ATP_Olambda__757,axiom,
    ! [Uu: $o,Uua: fun_o_nat,Uub: fun_o_nat,Uuc: $o] :
      ( aa_o_nat(aa_fun6176012476734143100_o_nat(aa_fun6251207968994965954_o_nat(aTP_Lamm_ack((Uu)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu),aa_o_nat(Uua,(Uuc)),aa_o_nat(Uub,(Uuc))) ) ).

% ATP.lambda_757
tff(fact_10445_ATP_Olambda__758,axiom,
    ! [Uu: $o,Uua: fun_o_int,Uub: fun_o_int,Uuc: $o] :
      ( aa_o_int(aa_fun4727766026823762812_o_int(aa_fun5648456818326528798_o_int(aTP_Lamm_acl((Uu)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu),aa_o_int(Uua,(Uuc)),aa_o_int(Uub,(Uuc))) ) ).

% ATP.lambda_758
tff(fact_10446_ATP_Olambda__759,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_rat,Uub: fun_set_nat_rat,Uuc: set_nat] :
      ( aa_set_nat_rat(aa_fun8478617152544252148at_rat(aa_fun2377924995230810828at_rat(aTP_Lamm_acs(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_set_nat_rat(Uua,Uuc),aa_set_nat_rat(Uub,Uuc)) ) ).

% ATP.lambda_759
tff(fact_10447_ATP_Olambda__760,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] :
      ( aa_nat_Code_integer(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_adz(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_nat_Code_integer(Uua,Uuc),aa_nat_Code_integer(Uub,Uuc)) ) ).

% ATP.lambda_760
tff(fact_10448_ATP_Olambda__761,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      ( aa_nat_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_aec(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_nat_assn(Uua,Uuc),aa_nat_assn(Uub,Uuc)) ) ).

% ATP.lambda_761
tff(fact_10449_ATP_Olambda__762,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_aco(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ) ).

% ATP.lambda_762
tff(fact_10450_ATP_Olambda__763,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      ( aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_acq(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ) ).

% ATP.lambda_763
tff(fact_10451_ATP_Olambda__764,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      ( aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_acp(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ) ).

% ATP.lambda_764
tff(fact_10452_ATP_Olambda__765,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: fun_int_Code_integer,Uuc: int] :
      ( aa_int_Code_integer(aa_fun5304498221708291146nteger(aa_fun3938006231294128461nteger(aTP_Lamm_ady(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_Code_integer(Uua,Uuc),aa_int_Code_integer(Uub,Uuc)) ) ).

% ATP.lambda_765
tff(fact_10453_ATP_Olambda__766,axiom,
    ! [Uu: int,Uua: fun_int_assn,Uub: fun_int_assn,Uuc: int] :
      ( aa_int_assn(aa_fun919007281442365664t_assn(aa_fun7795160494388424622t_assn(aTP_Lamm_aeb(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_assn(Uua,Uuc),aa_int_assn(Uub,Uuc)) ) ).

% ATP.lambda_766
tff(fact_10454_ATP_Olambda__767,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_acm(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_rat(Uua,Uuc),aa_int_rat(Uub,Uuc)) ) ).

% ATP.lambda_767
tff(fact_10455_ATP_Olambda__768,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] :
      ( aa_int_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_acn(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_nat(Uua,Uuc),aa_int_nat(Uub,Uuc)) ) ).

% ATP.lambda_768
tff(fact_10456_ATP_Olambda__769,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: fun_int_int,Uuc: int] :
      ( aa_int_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_acr(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_int(Uua,Uuc),aa_int_int(Uub,Uuc)) ) ).

% ATP.lambda_769
tff(fact_10457_ATP_Olambda__770,axiom,
    ! [Uu: set_o,Uua: nat,Uub: nat,Uuc: $o] :
      ( aa_o_nat(aa_nat_fun_o_nat(aa_nat7267195842670305526_o_nat(aTP_Lamm_ds(Uu),Uua),Uub),(Uuc)) = $ite(aa_set_o_o(member_o((Uuc)),Uu),Uua,Uub) ) ).

% ATP.lambda_770
tff(fact_10458_ATP_Olambda__771,axiom,
    ! [Uu: set_o,Uua: int,Uub: int,Uuc: $o] :
      ( aa_o_int(aa_int_fun_o_int(aa_int1171789168926488730_o_int(aTP_Lamm_dt(Uu),Uua),Uub),(Uuc)) = $ite(aa_set_o_o(member_o((Uuc)),Uu),Uua,Uub) ) ).

% ATP.lambda_771
tff(fact_10459_ATP_Olambda__772,axiom,
    ! [Uu: set_set_nat,Uua: nat,Uub: nat,Uuc: set_nat] :
      ( aa_set_nat_nat(aa_nat6910189468943142497at_nat(aa_nat8373145576387461038at_nat(aTP_Lamm_ea(Uu),Uua),Uub),Uuc) = $ite(aa_set_set_nat_o(member_set_nat(Uuc),Uu),Uua,Uub) ) ).

% ATP.lambda_772
tff(fact_10460_ATP_Olambda__773,axiom,
    ! [Uu: set_set_nat,Uua: int,Uub: int,Uuc: set_nat] :
      ( aa_set_nat_int(aa_int2448287375063835929at_int(aa_int7007698265767296706at_int(aTP_Lamm_eb(Uu),Uua),Uub),Uuc) = $ite(aa_set_set_nat_o(member_set_nat(Uuc),Uu),Uua,Uub) ) ).

% ATP.lambda_773
tff(fact_10461_ATP_Olambda__774,axiom,
    ! [Uu: set_rat,Uua: nat,Uub: nat,Uuc: rat] :
      ( aa_rat_nat(aa_nat_fun_rat_nat(aa_nat8605836375616589056at_nat(aTP_Lamm_du(Uu),Uua),Uub),Uuc) = $ite(aa_set_rat_o(member_rat(Uuc),Uu),Uua,Uub) ) ).

% ATP.lambda_774
tff(fact_10462_ATP_Olambda__775,axiom,
    ! [Uu: set_rat,Uua: int,Uub: int,Uuc: rat] :
      ( aa_rat_int(aa_int_fun_rat_int(aa_int1442183312183061524at_int(aTP_Lamm_dv(Uu),Uua),Uub),Uuc) = $ite(aa_set_rat_o(member_rat(Uuc),Uu),Uua,Uub) ) ).

% ATP.lambda_775
tff(fact_10463_ATP_Olambda__776,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_dw(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu),Uua,Uub) ) ).

% ATP.lambda_776
tff(fact_10464_ATP_Olambda__777,axiom,
    ! [Uu: set_nat,Uua: int,Uub: int,Uuc: nat] :
      ( aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_dx(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu),Uua,Uub) ) ).

% ATP.lambda_777
tff(fact_10465_ATP_Olambda__778,axiom,
    ! [Uu: set_int,Uua: nat,Uub: nat,Uuc: int] :
      ( aa_int_nat(aa_nat_fun_int_nat(aa_nat5970070103963837268nt_nat(aTP_Lamm_dy(Uu),Uua),Uub),Uuc) = $ite(aa_set_int_o(member_int(Uuc),Uu),Uua,Uub) ) ).

% ATP.lambda_778
tff(fact_10466_ATP_Olambda__779,axiom,
    ! [Uu: set_int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_int(aa_int_fun_int_int(aa_int8029789077385085544nt_int(aTP_Lamm_dz(Uu),Uua),Uub),Uuc) = $ite(aa_set_int_o(member_int(Uuc),Uu),Uua,Uub) ) ).

% ATP.lambda_779
tff(fact_10467_ATP_Olambda__780,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr8317260874932935276at_rat,Uub: fun_Pr8317260874932935276at_rat,Uuc: product_prod_nat_nat] :
      ( aa_Pro3608043709957902237at_rat(aa_fun1488288443938188604at_rat(aa_fun126448503906362355at_rat(aTP_Lamm_ace(Uu),Uua),Uub),Uuc) = $ite(aa_Pro6150637604445964947_nat_o(Uu,Uuc),aa_Pro3608043709957902237at_rat(Uua,Uuc),aa_Pro3608043709957902237at_rat(Uub,Uuc)) ) ).

% ATP.lambda_780
tff(fact_10468_ATP_Olambda__781,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_rat,Uub: fun_set_nat_rat,Uuc: set_nat] :
      ( aa_set_nat_rat(aa_fun8478617152544252148at_rat(aa_fun2377924995230810828at_rat(aTP_Lamm_acb(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(Uu,Uuc),aa_set_nat_rat(Uua,Uuc),aa_set_nat_rat(Uub,Uuc)) ) ).

% ATP.lambda_781
tff(fact_10469_ATP_Olambda__782,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_nat,Uub: fun_set_nat_nat,Uuc: set_nat] :
      ( aa_set_nat_nat(aa_fun3386010016858254564at_nat(aa_fun8118849421014274228at_nat(aTP_Lamm_acc(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(Uu,Uuc),aa_set_nat_nat(Uua,Uuc),aa_set_nat_nat(Uub,Uuc)) ) ).

% ATP.lambda_782
tff(fact_10470_ATP_Olambda__783,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_int,Uub: fun_set_nat_int,Uuc: set_nat] :
      ( aa_set_nat_int(aa_fun3600621996379952540at_int(aa_fun4894436714634582984at_int(aTP_Lamm_acd(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(Uu,Uuc),aa_set_nat_int(Uua,Uuc),aa_set_nat_int(Uub,Uuc)) ) ).

% ATP.lambda_783
tff(fact_10471_ATP_Olambda__784,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] :
      ( aa_nat_Code_integer(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_acg(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_Code_integer(Uua,Uuc),aa_nat_Code_integer(Uub,Uuc)) ) ).

% ATP.lambda_784
tff(fact_10472_ATP_Olambda__785,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      ( aa_nat_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_aci(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_assn(Uua,Uuc),aa_nat_assn(Uub,Uuc)) ) ).

% ATP.lambda_785
tff(fact_10473_ATP_Olambda__786,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_abx(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ) ).

% ATP.lambda_786
tff(fact_10474_ATP_Olambda__787,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      ( aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_abz(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ) ).

% ATP.lambda_787
tff(fact_10475_ATP_Olambda__788,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      ( aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_aby(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ) ).

% ATP.lambda_788
tff(fact_10476_ATP_Olambda__789,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_Code_integer,Uub: fun_int_Code_integer,Uuc: int] :
      ( aa_int_Code_integer(aa_fun5304498221708291146nteger(aa_fun3938006231294128461nteger(aTP_Lamm_acf(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_Code_integer(Uua,Uuc),aa_int_Code_integer(Uub,Uuc)) ) ).

% ATP.lambda_789
tff(fact_10477_ATP_Olambda__790,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_assn,Uub: fun_int_assn,Uuc: int] :
      ( aa_int_assn(aa_fun919007281442365664t_assn(aa_fun7795160494388424622t_assn(aTP_Lamm_ach(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_assn(Uua,Uuc),aa_int_assn(Uub,Uuc)) ) ).

% ATP.lambda_790
tff(fact_10478_ATP_Olambda__791,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_abv(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_rat(Uua,Uuc),aa_int_rat(Uub,Uuc)) ) ).

% ATP.lambda_791
tff(fact_10479_ATP_Olambda__792,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] :
      ( aa_int_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_abw(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_nat(Uua,Uuc),aa_int_nat(Uub,Uuc)) ) ).

% ATP.lambda_792
tff(fact_10480_ATP_Olambda__793,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_int,Uub: fun_int_int,Uuc: int] :
      ( aa_int_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_aca(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_int(Uua,Uuc),aa_int_int(Uub,Uuc)) ) ).

% ATP.lambda_793
tff(fact_10481_ATP_Olambda__794,axiom,
    ! [Uu: $o,Uua: $o,Uub: code_integer,Uuc: $o] : ( aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aTP_Lamm_akc((Uu)),(Uua)),Uub),(Uuc)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aTP_Lamm_akb((Uu)),(Uua)),(Uuc)),code_bit_cut_integer(Uub)) ) ).

% ATP.lambda_794
tff(fact_10482_ATP_Olambda__795,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: set_int,Uuc: int] : ( aa_int_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_le(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_ld(Uu),Uua),Uuc),Uub) ) ).

% ATP.lambda_795
tff(fact_10483_ATP_Olambda__796,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: set_nat,Uuc: nat] : ( aa_nat_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_lc(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_lb(Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_796
tff(fact_10484_ATP_Olambda__797,axiom,
    ! [Uu: set_set_nat,Uua: fun_se7912642235559603452at_int,Uub: fun_se382663106265794036_nat_o,Uuc: nat] : ( aa_nat_int(aa_fun6199171195344186545at_int(aa_fun2727794920753907526at_int(aTP_Lamm_xg(Uu),Uua),Uub),Uuc) = groups4246057289670975065at_int(aa_nat2732338449433945789at_int(aTP_Lamm_xf(Uua),Uuc),collect_set_nat(aa_nat_fun_set_nat_o(aa_fun9105140013458882876_nat_o(aTP_Lamm_vx(Uu),Uub),Uuc))) ) ).

% ATP.lambda_797
tff(fact_10485_ATP_Olambda__798,axiom,
    ! [Uu: set_set_nat,Uua: fun_se876191773397849432nt_int,Uub: fun_se7360890175450123928_int_o,Uuc: int] : ( aa_int_int(aa_fun7007818992012977073nt_int(aa_fun7130641361752856810nt_int(aTP_Lamm_xi(Uu),Uua),Uub),Uuc) = groups4246057289670975065at_int(aa_int2448287375063835929at_int(aTP_Lamm_wb(Uua),Uuc),collect_set_nat(aa_int_fun_set_nat_o(aa_fun9086678093774146108_nat_o(aTP_Lamm_wc(Uu),Uub),Uuc))) ) ).

% ATP.lambda_798
tff(fact_10486_ATP_Olambda__799,axiom,
    ! [Uu: set_set_nat,Uua: fun_se1179353343636554400at_nat,Uub: fun_se382663106265794036_nat_o,Uuc: nat] : ( aa_nat_nat(aa_fun1153650177998607445at_nat(aa_fun5868473082678171206at_nat(aTP_Lamm_xd(Uu),Uua),Uub),Uuc) = groups4248547760180025341at_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_vw(Uua),Uuc),collect_set_nat(aa_nat_fun_set_nat_o(aa_fun9105140013458882876_nat_o(aTP_Lamm_vx(Uu),Uub),Uuc))) ) ).

% ATP.lambda_799
tff(fact_10487_ATP_Olambda__800,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_wr(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_int_fun_nat_nat(aTP_Lamm_vi(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_vf(Uu),Uub),Uuc))) ) ).

% ATP.lambda_800
tff(fact_10488_ATP_Olambda__801,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_wp(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(aTP_Lamm_va(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_vb(Uu),Uub),Uuc))) ) ).

% ATP.lambda_801
tff(fact_10489_ATP_Olambda__802,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_int,Uub: fun_int_fun_nat_o,Uuc: nat] : ( aa_nat_int(aa_fun8696964665364114571at_int(aa_fun3875905286069234426at_int(aTP_Lamm_wz(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_pq(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_vb(Uu),Uub),Uuc))) ) ).

% ATP.lambda_802
tff(fact_10490_ATP_Olambda__803,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_wx(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_pn(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_vf(Uu),Uub),Uuc))) ) ).

% ATP.lambda_803
tff(fact_10491_ATP_Olambda__804,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_int,Uub: fun_nat_fun_nat_o,Uuc: nat] : ( aa_nat_int(aa_fun4679828425598226919at_int(aa_fun5734951013428723122at_int(aTP_Lamm_wv(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_pk(Uua),Uuc)),collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_vm(Uu),Uub),Uuc))) ) ).

% ATP.lambda_804
tff(fact_10492_ATP_Olambda__805,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_int,Uub: fun_int_fun_int_o,Uuc: int] : ( aa_int_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_xb(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_kx(Uua),Uuc),collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_vs(Uu),Uub),Uuc))) ) ).

% ATP.lambda_805
tff(fact_10493_ATP_Olambda__806,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: fun_nat_fun_nat_o,Uuc: nat] : ( aa_nat_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_wt(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ku(Uua),Uuc)),collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_vm(Uu),Uub),Uuc))) ) ).

% ATP.lambda_806
tff(fact_10494_ATP_Olambda__807,axiom,
    ! [Uu: set_int,Uua: fun_in6555721293707029592at_int,Uub: fun_in7136481858371106712_nat_o,Uuc: set_nat] : ( aa_set_nat_int(aa_fun5805668902077597195at_int(aa_fun5576367408028688196at_int(aTP_Lamm_wn(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_set3358596244511904025nt_int(aTP_Lamm_wl(Uua),Uuc),collect_int(aa_set_nat_fun_int_o(aa_fun8216954948888934460_int_o(aTP_Lamm_wm(Uu),Uub),Uuc))) ) ).

% ATP.lambda_807
tff(fact_10495_ATP_Olambda__808,axiom,
    ! [Uu: set_nat,Uua: fun_na106483476154090400at_nat,Uub: fun_na8563630686737515764_nat_o,Uuc: set_nat] : ( aa_set_nat_nat(aa_fun255155394271802123at_nat(aa_fun6284103964554420476at_nat(aTP_Lamm_wi(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_wg(Uua),Uuc)),collect_nat(aa_set_nat_fun_nat_o(aa_fun2835368149466334012_nat_o(aTP_Lamm_wh(Uu),Uub),Uuc))) ) ).

% ATP.lambda_808
tff(fact_10496_ATP_Olambda__809,axiom,
    ! [Uu: set_set_nat,Uua: fun_se876191773397849432nt_int,Uub: fun_se7360890175450123928_int_o,Uuc: int] : ( aa_int_int(aa_fun7007818992012977073nt_int(aa_fun7130641361752856810nt_int(aTP_Lamm_wd(Uu),Uua),Uub),Uuc) = groups8292507037921071086at_int(aa_int2448287375063835929at_int(aTP_Lamm_wb(Uua),Uuc),collect_set_nat(aa_int_fun_set_nat_o(aa_fun9086678093774146108_nat_o(aTP_Lamm_wc(Uu),Uub),Uuc))) ) ).

% ATP.lambda_809
tff(fact_10497_ATP_Olambda__810,axiom,
    ! [Uu: set_set_nat,Uua: fun_se1179353343636554400at_nat,Uub: fun_se382663106265794036_nat_o,Uuc: nat] : ( aa_nat_nat(aa_fun1153650177998607445at_nat(aa_fun5868473082678171206at_nat(aTP_Lamm_vy(Uu),Uua),Uub),Uuc) = groups8294997508430121362at_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_vw(Uua),Uuc),collect_set_nat(aa_nat_fun_set_nat_o(aa_fun9105140013458882876_nat_o(aTP_Lamm_vx(Uu),Uub),Uuc))) ) ).

% ATP.lambda_810
tff(fact_10498_ATP_Olambda__811,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_vj(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(aTP_Lamm_vi(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_vf(Uu),Uub),Uuc))) ) ).

% ATP.lambda_811
tff(fact_10499_ATP_Olambda__812,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_vc(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_va(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_vb(Uu),Uub),Uuc))) ) ).

% ATP.lambda_812
tff(fact_10500_ATP_Olambda__813,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_int,Uub: fun_int_fun_nat_o,Uuc: nat] : ( aa_nat_int(aa_fun8696964665364114571at_int(aa_fun3875905286069234426at_int(aTP_Lamm_vp(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_nat_fun_int_int(aTP_Lamm_pq(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_vb(Uu),Uub),Uuc))) ) ).

% ATP.lambda_813
tff(fact_10501_ATP_Olambda__814,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_vg(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_pn(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_vf(Uu),Uub),Uuc))) ) ).

% ATP.lambda_814
tff(fact_10502_ATP_Olambda__815,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_int,Uub: fun_int_fun_int_o,Uuc: int] : ( aa_int_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_vt(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_kx(Uua),Uuc),collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_vs(Uu),Uub),Uuc))) ) ).

% ATP.lambda_815
tff(fact_10503_ATP_Olambda__816,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: fun_nat_fun_nat_o,Uuc: nat] : ( aa_nat_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_vn(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ku(Uua),Uuc)),collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_vm(Uu),Uub),Uuc))) ) ).

% ATP.lambda_816
tff(fact_10504_ATP_Olambda__817,axiom,
    ! [Uu: code_natural,Uua: code_natural,Uub: code_natural,Uuc: code_natural] : ( aa_Cod6808739660676082561atural(aa_Cod4292056611156568242atural(aa_Cod8811915439870959971atural(aTP_Lamm_amj(Uu),Uua),Uub),Uuc) = produc4480994950612372183atural(produc3574140220909816553atural(inc_shift(numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),Uu),Uuc),produc3574140220909816553atural(Uub,inc_shift(numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),Uua))) ) ).

% ATP.lambda_817
tff(fact_10505_ATP_Olambda__818,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aix(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_818
tff(fact_10506_ATP_Olambda__819,axiom,
    ! [Uu: set_Pr8057934254988874055nt_int,Uua: set_se6260736226359567993nt_int,Uub: set_Pr958786334691620121nt_int,Uuc: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(aa_set6233473662189019185_int_o(aTP_Lamm_ht(Uu),Uua),Uub),Uuc)
    <=> ( member5325734588016868240nt_int(produc782388162306416855nt_int(Uub,Uuc),Uu)
        & ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uua)
        & ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uuc),Uua) ) ) ).

% ATP.lambda_819
tff(fact_10507_ATP_Olambda__820,axiom,
    ! [Uu: set_Pr5488025237498180813et_nat,Uua: set_set_nat,Uub: set_nat,Uuc: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aa_set550086741393328324_nat_o(aTP_Lamm_hu(Uu),Uua),Uub),Uuc)
    <=> ( member8277197624267554838et_nat(produc4532415448927165861et_nat(Uub,Uuc),Uu)
        & ~ aa_set_set_nat_o(member_set_nat(Uub),Uua)
        & ~ aa_set_set_nat_o(member_set_nat(Uuc),Uua) ) ) ).

% ATP.lambda_820
tff(fact_10508_ATP_Olambda__821,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_set8149993881734264422_nat_o(aTP_Lamm_hv(Uu),Uua),Uub),Uuc)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uub),Uuc)),Uu)
        & ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua)
        & ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uua) ) ) ).

% ATP.lambda_821
tff(fact_10509_ATP_Olambda__822,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_set5204721951351315026_int_o(aTP_Lamm_hx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uub),Uuc)),Uu)
        & ~ aa_set_int_o(member_int(Uub),Uua)
        & ~ aa_set_int_o(member_int(Uuc),Uua) ) ) ).

% ATP.lambda_822
tff(fact_10510_ATP_Olambda__823,axiom,
    ! [Uu: set_Pr2890355626087285447_nat_o,Uua: set_fu3555266933746557007_nat_o,Uub: fun_Pr6460944699146269935_nat_o,Uuc: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(aa_set2556502855246081627at_o_o(aTP_Lamm_hw(Uu),Uua),Uub),Uuc)
    <=> ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(Uub,Uuc),Uu)
        & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uub),Uua)
        & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uuc),Uua) ) ) ).

% ATP.lambda_823
tff(fact_10511_ATP_Olambda__824,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : ( aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_rt(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_824
tff(fact_10512_ATP_Olambda__825,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : ( aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rv(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_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_825
tff(fact_10513_ATP_Olambda__826,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ru(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(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Uu),Uuc))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ) ).

% ATP.lambda_826
tff(fact_10514_ATP_Olambda__827,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : ( aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_rq(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_827
tff(fact_10515_ATP_Olambda__828,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : ( aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rs(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_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_828
tff(fact_10516_ATP_Olambda__829,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_rj(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_829
tff(fact_10517_ATP_Olambda__830,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rr(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(aa_int_fun_nat_int(power_power_int,Uu),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ) ).

% ATP.lambda_830
tff(fact_10518_ATP_Olambda__831,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: code_integer,Uuc: nat] : ( aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_aed(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_Code_integer(power_8256067586552552935nteger(Uu),Uuc)) ) ).

% ATP.lambda_831
tff(fact_10519_ATP_Olambda__832,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : ( aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_aee(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_rat(power_power_rat(Uu),Uuc)) ) ).

% ATP.lambda_832
tff(fact_10520_ATP_Olambda__833,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : ( aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_aef(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_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(aa_int_fun_nat_int(power_power_int,Uu),Uuc)) ) ).

% ATP.lambda_833
tff(fact_10521_ATP_Olambda__834,axiom,
    ! [Uu: set_list_nat,Uua: fun_na7234455742697417626_nat_o,Uub: nat,Uuc: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aa_fun6745918892298368316_nat_o(aTP_Lamm_aii(Uu),Uua),Uub),Uuc)
    <=> ( member_list_nat(Uuc,Uu)
        & aa_list_nat_o(aa_nat5883842100386203267_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_834
tff(fact_10522_ATP_Olambda__835,axiom,
    ! [Uu: set_o,Uua: fun_nat_fun_o_o,Uub: nat,Uuc: $o] :
      ( aa_o_o(aa_nat_fun_o_o(aa_fun3555063939605388254un_o_o(aTP_Lamm_ahz(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & aa_o_o(aa_nat_fun_o_o(Uua,Uub),(Uuc)) ) ) ).

% ATP.lambda_835
tff(fact_10523_ATP_Olambda__836,axiom,
    ! [Uu: set_set_nat,Uua: fun_na8563630686737515764_nat_o,Uub: nat,Uuc: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aa_fun1792963693083279932_nat_o(aTP_Lamm_we(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_set_nat_o(member_set_nat(Uuc),Uu)
        & aa_set_nat_o(aa_nat_fun_set_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_836
tff(fact_10524_ATP_Olambda__837,axiom,
    ! [Uu: set_set_nat,Uua: fun_in7136481858371106712_nat_o,Uub: int,Uuc: set_nat] :
      ( aa_set_nat_o(aa_int_fun_set_nat_o(aa_fun7992546631809917244_nat_o(aTP_Lamm_wj(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_set_nat_o(member_set_nat(Uuc),Uu)
        & aa_set_nat_o(aa_int_fun_set_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_837
tff(fact_10525_ATP_Olambda__838,axiom,
    ! [Uu: set_nat,Uua: fun_Pr6984107991879650211_nat_o,Uub: product_prod_nat_nat,Uuc: nat] :
      ( aa_nat_o(aa_Pro4733806262894120092_nat_o(aa_fun5503210535175347332_nat_o(aTP_Lamm_ail(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_nat_o(aa_Pro4733806262894120092_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_838
tff(fact_10526_ATP_Olambda__839,axiom,
    ! [Uu: set_nat,Uua: fun_li2421049240157457050_nat_o,Uub: list_nat,Uuc: nat] :
      ( aa_nat_o(aa_lis1686035672126679939_nat_o(aa_fun6275404141584987452_nat_o(aTP_Lamm_aie(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_nat_o(aa_lis1686035672126679939_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_839
tff(fact_10527_ATP_Olambda__840,axiom,
    ! [Uu: set_nat,Uua: fun_o_fun_nat_o,Uub: $o,Uuc: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(aa_fun9138341750119017558_nat_o(aTP_Lamm_ahv(Uu),Uua),(Uub)),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_nat_o(aa_o_fun_nat_o(Uua,(Uub)),Uuc) ) ) ).

% ATP.lambda_840
tff(fact_10528_ATP_Olambda__841,axiom,
    ! [Uu: set_nat,Uua: fun_se382663106265794036_nat_o,Uub: set_nat,Uuc: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aa_fun924172432987161148_nat_o(aTP_Lamm_vu(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_nat_o(aa_set_nat_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_841
tff(fact_10529_ATP_Olambda__842,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_vk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_842
tff(fact_10530_ATP_Olambda__843,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_uy(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_843
tff(fact_10531_ATP_Olambda__844,axiom,
    ! [Uu: set_int,Uua: fun_se7360890175450123928_int_o,Uub: set_nat,Uuc: int] :
      ( aa_int_o(aa_set_nat_fun_int_o(aa_fun87714373998387516_int_o(aTP_Lamm_vz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_set_nat_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_844
tff(fact_10532_ATP_Olambda__845,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_vd(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_845
tff(fact_10533_ATP_Olambda__846,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_vq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_846
tff(fact_10534_ATP_Olambda__847,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr6984107991879650211_nat_o,Uub: nat,Uuc: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_fun2489310642977492936_nat_o(aTP_Lamm_aex(Uu),Uua),Uub),Uuc)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uuc),Uu)
        & aa_nat_o(aa_Pro4733806262894120092_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_847
tff(fact_10535_ATP_Olambda__848,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr4738963024209204295_int_o,Uub: int,Uuc: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_int6101145382336329604_nat_o(aa_fun1144309370076579528_nat_o(aTP_Lamm_aey(Uu),Uua),Uub),Uuc)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uuc),Uu)
        & aa_int_o(aa_Pro2912763917461432128_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_848
tff(fact_10536_ATP_Olambda__849,axiom,
    ! [Uu: set_list_nat,Uua: fun_li2421049240157457050_nat_o,Uub: nat,Uuc: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aa_fun1865438607270172220_nat_o(aTP_Lamm_aid(Uu),Uua),Uub),Uuc)
    <=> ( member_list_nat(Uuc,Uu)
        & aa_nat_o(aa_lis1686035672126679939_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_849
tff(fact_10537_ATP_Olambda__850,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_nat_o,Uub: nat,Uuc: $o] :
      ( aa_o_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_ahu(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & aa_nat_o(aa_o_fun_nat_o(Uua,(Uuc)),Uub) ) ) ).

% ATP.lambda_850
tff(fact_10538_ATP_Olambda__851,axiom,
    ! [Uu: set_set_nat,Uua: fun_se382663106265794036_nat_o,Uub: nat,Uuc: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aa_fun9105140013458882876_nat_o(aTP_Lamm_vx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_set_nat_o(member_set_nat(Uuc),Uu)
        & aa_nat_o(aa_set_nat_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_851
tff(fact_10539_ATP_Olambda__852,axiom,
    ! [Uu: set_set_nat,Uua: fun_se7360890175450123928_int_o,Uub: int,Uuc: set_nat] :
      ( aa_set_nat_o(aa_int_fun_set_nat_o(aa_fun9086678093774146108_nat_o(aTP_Lamm_wc(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_set_nat_o(member_set_nat(Uuc),Uu)
        & aa_int_o(aa_set_nat_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_852
tff(fact_10540_ATP_Olambda__853,axiom,
    ! [Uu: set_nat,Uua: fun_na3970208099681795815_nat_o,Uub: product_prod_nat_nat,Uuc: nat] :
      ( aa_nat_o(aa_Pro4733806262894120092_nat_o(aa_fun5389978208584994248_nat_o(aTP_Lamm_aev(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_853
tff(fact_10541_ATP_Olambda__854,axiom,
    ! [Uu: set_nat,Uua: fun_na7234455742697417626_nat_o,Uub: list_nat,Uuc: nat] :
      ( aa_nat_o(aa_lis1686035672126679939_nat_o(aa_fun1932512389758407740_nat_o(aTP_Lamm_aih(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_list_nat_o(aa_nat5883842100386203267_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_854
tff(fact_10542_ATP_Olambda__855,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_o_o,Uub: $o,Uuc: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(aa_fun2937962278134185242_nat_o(aTP_Lamm_ahy(Uu),Uua),(Uub)),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_o_o(aa_nat_fun_o_o(Uua,Uuc),(Uub)) ) ) ).

% ATP.lambda_855
tff(fact_10543_ATP_Olambda__856,axiom,
    ! [Uu: set_nat,Uua: fun_na8563630686737515764_nat_o,Uub: set_nat,Uuc: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aa_fun2835368149466334012_nat_o(aTP_Lamm_wh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_set_nat_o(aa_nat_fun_set_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_856
tff(fact_10544_ATP_Olambda__857,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_vm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_857
tff(fact_10545_ATP_Olambda__858,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_vf(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_858
tff(fact_10546_ATP_Olambda__859,axiom,
    ! [Uu: set_int,Uua: fun_in2375804543815556931_nat_o,Uub: product_prod_nat_nat,Uuc: int] :
      ( aa_int_o(aa_Pro2912763917461432128_int_o(aa_fun8623471109766342856_int_o(aTP_Lamm_aew(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_Pro6150637604445964947_nat_o(aa_int6101145382336329604_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_859
tff(fact_10547_ATP_Olambda__860,axiom,
    ! [Uu: set_int,Uua: fun_in7136481858371106712_nat_o,Uub: set_nat,Uuc: int] :
      ( aa_int_o(aa_set_nat_fun_int_o(aa_fun8216954948888934460_int_o(aTP_Lamm_wm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_set_nat_o(aa_int_fun_set_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_860
tff(fact_10548_ATP_Olambda__861,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_vb(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_861
tff(fact_10549_ATP_Olambda__862,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_vs(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_862
tff(fact_10550_ATP_Olambda__863,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: code_integer,Uuc: nat] : ( aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_aeg(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Uu),Uuc)),aa_nat_Code_integer(power_8256067586552552935nteger(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ) ).

% ATP.lambda_863
tff(fact_10551_ATP_Olambda__864,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : ( aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_aeh(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ) ).

% ATP.lambda_864
tff(fact_10552_ATP_Olambda__865,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : ( aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_aei(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uuc)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ) ).

% ATP.lambda_865
tff(fact_10553_ATP_Olambda__866,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_rf(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_866
tff(fact_10554_ATP_Olambda__867,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_ajd(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_867
tff(fact_10555_ATP_Olambda__868,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_ajk(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_868
tff(fact_10556_ATP_Olambda__869,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_ajf(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_869
tff(fact_10557_ATP_Olambda__870,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_aji(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_870
tff(fact_10558_ATP_Olambda__871,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ajm(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_871
tff(fact_10559_ATP_Olambda__872,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ajo(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_872
tff(fact_10560_ATP_Olambda__873,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_ali(Uu),Uua),Uub),Uuc)
    <=> ( 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_873
tff(fact_10561_ATP_Olambda__874,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aa_fun2745573882733218024_nat_o(aTP_Lamm_yj(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uuc)),aa_nat_Code_integer(Uub,Uuc)) != one_one_Code_integer ) ) ) ).

% ATP.lambda_874
tff(fact_10562_ATP_Olambda__875,axiom,
    ! [Uu: set_nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      ( aa_nat_o(aa_fun7183233715851567790_nat_o(aa_fun8595640264602358482_nat_o(aTP_Lamm_yn(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Uua,Uuc)),aa_nat_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_875
tff(fact_10563_ATP_Olambda__876,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_yr(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_876
tff(fact_10564_ATP_Olambda__877,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aa_fun6348600740168882050_nat_o(aTP_Lamm_yv(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uua,Uuc)),aa_nat_nat(Uub,Uuc)) != one_one_nat ) ) ) ).

% ATP.lambda_877
tff(fact_10565_ATP_Olambda__878,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      ( aa_nat_o(aa_fun2180201864171524266_nat_o(aa_fun8566290670789036234_nat_o(aTP_Lamm_yz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uua,Uuc)),aa_nat_int(Uub,Uuc)) != one_one_int ) ) ) ).

% ATP.lambda_878
tff(fact_10566_ATP_Olambda__879,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: fun_int_Code_integer,Uuc: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aa_fun8818272713621706452_int_o(aTP_Lamm_yl(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uuc)),aa_int_Code_integer(Uub,Uuc)) != one_one_Code_integer ) ) ) ).

% ATP.lambda_879
tff(fact_10567_ATP_Olambda__880,axiom,
    ! [Uu: set_int,Uua: fun_int_assn,Uub: fun_int_assn,Uuc: int] :
      ( aa_int_o(aa_fun202081646129719286_int_o(aa_fun871127997271750846_int_o(aTP_Lamm_yp(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(Uua,Uuc)),aa_int_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_880
tff(fact_10568_ATP_Olambda__881,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_yt(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_881
tff(fact_10569_ATP_Olambda__882,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aa_fun8703205213358249326_int_o(aTP_Lamm_yx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(Uua,Uuc)),aa_int_nat(Uub,Uuc)) != one_one_nat ) ) ) ).

% ATP.lambda_882
tff(fact_10570_ATP_Olambda__883,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: fun_int_int,Uuc: int] :
      ( aa_int_o(aa_fun8155382727227203058_int_o(aa_fun1697523107123627702_int_o(aTP_Lamm_zb(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uua,Uuc)),aa_int_int(Uub,Uuc)) != one_one_int ) ) ) ).

% ATP.lambda_883
tff(fact_10571_ATP_Olambda__884,axiom,
    ! [Uu: set_set_nat,Uua: fun_se7586849273513198688nteger,Uub: fun_se7586849273513198688nteger,Uuc: set_nat] :
      ( aa_set_nat_o(aa_fun5627232717963835469_nat_o(aa_fun4072113896377786438_nat_o(aTP_Lamm_yf(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_set_nat_o(member_set_nat(Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_set871585252387937313nteger(Uua,Uuc)),aa_set871585252387937313nteger(Uub,Uuc)) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_884
tff(fact_10572_ATP_Olambda__885,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_rat,Uub: fun_set_nat_rat,Uuc: set_nat] :
      ( aa_set_nat_o(aa_fun3672760304936660370_nat_o(aa_fun4148231721690063568_nat_o(aTP_Lamm_yh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_set_nat_o(member_set_nat(Uuc),Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(Uua,Uuc)),aa_set_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_885
tff(fact_10573_ATP_Olambda__886,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aa_fun2745573882733218024_nat_o(aTP_Lamm_xp(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(Uua,Uuc)),aa_nat_Code_integer(Uub,Uuc)) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_886
tff(fact_10574_ATP_Olambda__887,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_xt(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_887
tff(fact_10575_ATP_Olambda__888,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aa_fun6348600740168882050_nat_o(aTP_Lamm_xx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uua,Uuc)),aa_nat_nat(Uub,Uuc)) != zero_zero_nat ) ) ) ).

% ATP.lambda_888
tff(fact_10576_ATP_Olambda__889,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      ( aa_nat_o(aa_fun2180201864171524266_nat_o(aa_fun8566290670789036234_nat_o(aTP_Lamm_yb(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uua,Uuc)),aa_nat_int(Uub,Uuc)) != zero_zero_int ) ) ) ).

% ATP.lambda_889
tff(fact_10577_ATP_Olambda__890,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: fun_int_Code_integer,Uuc: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aa_fun8818272713621706452_int_o(aTP_Lamm_xr(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_int_Code_integer(Uua,Uuc)),aa_int_Code_integer(Uub,Uuc)) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_890
tff(fact_10578_ATP_Olambda__891,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_xv(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_891
tff(fact_10579_ATP_Olambda__892,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aa_fun8703205213358249326_int_o(aTP_Lamm_xz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Uua,Uuc)),aa_int_nat(Uub,Uuc)) != zero_zero_nat ) ) ) ).

% ATP.lambda_892
tff(fact_10580_ATP_Olambda__893,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: fun_int_int,Uuc: int] :
      ( aa_int_o(aa_fun8155382727227203058_int_o(aa_fun1697523107123627702_int_o(aTP_Lamm_yd(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Uua,Uuc)),aa_int_int(Uub,Uuc)) != zero_zero_int ) ) ) ).

% ATP.lambda_893
tff(fact_10581_ATP_Olambda__894,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_lb(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uuc)) ) ).

% ATP.lambda_894
tff(fact_10582_ATP_Olambda__895,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int,Uuc: int] : ( aa_int_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_ld(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uuc)) ) ).

% ATP.lambda_895
tff(fact_10583_ATP_Olambda__896,axiom,
    ! [Uu: set_int,Uua: fun_se876191773397849432nt_int,Uub: fun_se7360890175450123928_int_o,Uuc: set_nat] : ( aa_set_nat_int(aa_fun5327250345464651019at_int(aa_fun3157859007690329412at_int(aTP_Lamm_xh(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_set3358596244511904025nt_int(Uua,Uuc),collect_int(aa_set_nat_fun_int_o(aa_fun87714373998387516_int_o(aTP_Lamm_vz(Uu),Uub),Uuc))) ) ).

% ATP.lambda_896
tff(fact_10584_ATP_Olambda__897,axiom,
    ! [Uu: set_nat,Uua: fun_se1179353343636554400at_nat,Uub: fun_se382663106265794036_nat_o,Uuc: set_nat] : ( aa_set_nat_nat(aa_fun6238879321244136459at_nat(aa_fun1036415571424436988at_nat(aTP_Lamm_xc(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_set6537093552120046177at_nat(Uua,Uuc)),collect_nat(aa_set_nat_fun_nat_o(aa_fun924172432987161148_nat_o(aTP_Lamm_vu(Uu),Uub),Uuc))) ) ).

% ATP.lambda_897
tff(fact_10585_ATP_Olambda__898,axiom,
    ! [Uu: set_nat,Uua: fun_se7912642235559603452at_int,Uub: fun_se382663106265794036_nat_o,Uuc: set_nat] : ( aa_set_nat_int(aa_fun2061028301734939751at_int(aa_fun8609273137425597436at_int(aTP_Lamm_xe(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_set2359242532610849469at_int(Uua,Uuc)),collect_nat(aa_set_nat_fun_nat_o(aa_fun924172432987161148_nat_o(aTP_Lamm_vu(Uu),Uub),Uuc))) ) ).

% ATP.lambda_898
tff(fact_10586_ATP_Olambda__899,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_int,Uub: fun_int_fun_int_o,Uuc: int] : ( aa_int_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_xa(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_int_fun_int_int(Uua,Uuc),collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_vq(Uu),Uub),Uuc))) ) ).

% ATP.lambda_899
tff(fact_10587_ATP_Olambda__900,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: fun_nat_fun_nat_o,Uuc: nat] : ( aa_nat_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_ws(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uua,Uuc)),collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_vk(Uu),Uub),Uuc))) ) ).

% ATP.lambda_900
tff(fact_10588_ATP_Olambda__901,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_int,Uub: fun_nat_fun_nat_o,Uuc: nat] : ( aa_nat_int(aa_fun4679828425598226919at_int(aa_fun5734951013428723122at_int(aTP_Lamm_wu(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(Uua,Uuc)),collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_vk(Uu),Uub),Uuc))) ) ).

% ATP.lambda_901
tff(fact_10589_ATP_Olambda__902,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_wq(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_vd(Uu),Uub),Uuc))) ) ).

% ATP.lambda_902
tff(fact_10590_ATP_Olambda__903,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_ww(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_nat_fun_int_int(Uua,Uuc),collect_int(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_vd(Uu),Uub),Uuc))) ) ).

% ATP.lambda_903
tff(fact_10591_ATP_Olambda__904,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_wo(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_uy(Uu),Uub),Uuc))) ) ).

% ATP.lambda_904
tff(fact_10592_ATP_Olambda__905,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_int,Uub: fun_int_fun_nat_o,Uuc: int] : ( aa_int_int(aa_fun472946340410393319nt_int(aa_fun6062826860762256214nt_int(aTP_Lamm_wy(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(Uua,Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_uy(Uu),Uub),Uuc))) ) ).

% ATP.lambda_905
tff(fact_10593_ATP_Olambda__906,axiom,
    ! [Uu: set_set_nat,Uua: fun_in6555721293707029592at_int,Uub: fun_in7136481858371106712_nat_o,Uuc: int] : ( aa_int_int(aa_fun1461796484577889457nt_int(aa_fun1283830802835827434nt_int(aTP_Lamm_wk(Uu),Uua),Uub),Uuc) = groups8292507037921071086at_int(aa_int2448287375063835929at_int(Uua,Uuc),collect_set_nat(aa_int_fun_set_nat_o(aa_fun7992546631809917244_nat_o(aTP_Lamm_wj(Uu),Uub),Uuc))) ) ).

% ATP.lambda_906
tff(fact_10594_ATP_Olambda__907,axiom,
    ! [Uu: set_set_nat,Uua: fun_na106483476154090400at_nat,Uub: fun_na8563630686737515764_nat_o,Uuc: nat] : ( aa_nat_nat(aa_fun7187098123162309461at_nat(aa_fun585184524491904070at_nat(aTP_Lamm_wf(Uu),Uua),Uub),Uuc) = groups8294997508430121362at_nat(aa_nat6910189468943142497at_nat(Uua,Uuc),collect_set_nat(aa_nat_fun_set_nat_o(aa_fun1792963693083279932_nat_o(aTP_Lamm_we(Uu),Uub),Uuc))) ) ).

% ATP.lambda_907
tff(fact_10595_ATP_Olambda__908,axiom,
    ! [Uu: set_int,Uua: fun_se876191773397849432nt_int,Uub: fun_se7360890175450123928_int_o,Uuc: set_nat] : ( aa_set_nat_int(aa_fun5327250345464651019at_int(aa_fun3157859007690329412at_int(aTP_Lamm_wa(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_set3358596244511904025nt_int(Uua,Uuc),collect_int(aa_set_nat_fun_int_o(aa_fun87714373998387516_int_o(aTP_Lamm_vz(Uu),Uub),Uuc))) ) ).

% ATP.lambda_908
tff(fact_10596_ATP_Olambda__909,axiom,
    ! [Uu: set_nat,Uua: fun_se1179353343636554400at_nat,Uub: fun_se382663106265794036_nat_o,Uuc: set_nat] : ( aa_set_nat_nat(aa_fun6238879321244136459at_nat(aa_fun1036415571424436988at_nat(aTP_Lamm_vv(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(Uua,Uuc)),collect_nat(aa_set_nat_fun_nat_o(aa_fun924172432987161148_nat_o(aTP_Lamm_vu(Uu),Uub),Uuc))) ) ).

% ATP.lambda_909
tff(fact_10597_ATP_Olambda__910,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_int,Uub: fun_int_fun_int_o,Uuc: int] : ( aa_int_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_vr(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(Uua,Uuc),collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_vq(Uu),Uub),Uuc))) ) ).

% ATP.lambda_910
tff(fact_10598_ATP_Olambda__911,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: fun_nat_fun_nat_o,Uuc: nat] : ( aa_nat_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_vl(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uua,Uuc)),collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_vk(Uu),Uub),Uuc))) ) ).

% ATP.lambda_911
tff(fact_10599_ATP_Olambda__912,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_vh(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(Uua,Uuc),collect_int(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_vd(Uu),Uub),Uuc))) ) ).

% ATP.lambda_912
tff(fact_10600_ATP_Olambda__913,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_ve(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_vd(Uu),Uub),Uuc))) ) ).

% ATP.lambda_913
tff(fact_10601_ATP_Olambda__914,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_uz(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_uy(Uu),Uub),Uuc))) ) ).

% ATP.lambda_914
tff(fact_10602_ATP_Olambda__915,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_int,Uub: fun_int_fun_nat_o,Uuc: int] : ( aa_int_int(aa_fun472946340410393319nt_int(aa_fun6062826860762256214nt_int(aTP_Lamm_vo(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(Uua,Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_uy(Uu),Uub),Uuc))) ) ).

% ATP.lambda_915
tff(fact_10603_ATP_Olambda__916,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_hy((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_916
tff(fact_10604_ATP_Olambda__917,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_hd(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_917
tff(fact_10605_ATP_Olambda__918,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_gz(Uu),Uua),Uub),Uuc)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_918
tff(fact_10606_ATP_Olambda__919,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_ha(Uu),Uua),Uub),Uuc) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_919
tff(fact_10607_ATP_Olambda__920,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_hb(Uu),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_920
tff(fact_10608_ATP_Olambda__921,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_gv(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_921
tff(fact_10609_ATP_Olambda__922,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_gw(Uu),Uua),Uub),Uuc) = aa_o_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_922
tff(fact_10610_ATP_Olambda__923,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_hc(Uu),Uua),Uub),Uuc) = aa_int7462565939943104719nt_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_923
tff(fact_10611_ATP_Olambda__924,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_gx(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_924
tff(fact_10612_ATP_Olambda__925,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_gy(Uu),Uua),Uub),Uuc) = aa_int_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_925
tff(fact_10613_ATP_Olambda__926,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_he(Uu),Uua),Uub),Uuc) = aa_fun359341089595923442_nat_o(Uu,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_926
tff(fact_10614_ATP_Olambda__927,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_mc(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)),aa_nat_nat(suc,Uuc))) ) ).

% ATP.lambda_927
tff(fact_10615_ATP_Olambda__928,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_qp(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)),aa_nat_nat(suc,Uuc))) ) ).

% ATP.lambda_928
tff(fact_10616_ATP_Olambda__929,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lv(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_929
tff(fact_10617_ATP_Olambda__930,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ql(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_930
tff(fact_10618_ATP_Olambda__931,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aiz(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_931
tff(fact_10619_ATP_Olambda__932,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aju(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_932
tff(fact_10620_ATP_Olambda__933,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ajs(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_933
tff(fact_10621_ATP_Olambda__934,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ajq(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_934
tff(fact_10622_ATP_Olambda__935,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: code_integer,Uud: $o] : ( aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aTP_Lamm_akb((Uu)),(Uua)),(Uub)),Uuc),(Uud)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aTP_Lamm_aka((Uu)),(Uua)),(Uub)),(Uud)),code_bit_cut_integer(Uuc)) ) ).

% ATP.lambda_935
tff(fact_10623_ATP_Olambda__936,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_nz(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Uud)),Uua)),one_one_rat)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ) ).

% ATP.lambda_936
tff(fact_10624_ATP_Olambda__937,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_nx(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_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_937
tff(fact_10625_ATP_Olambda__938,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_ny(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(plus_plus_rat,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ) ).

% ATP.lambda_938
tff(fact_10626_ATP_Olambda__939,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: code_integer,Uue: $o] : ( aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aTP_Lamm_aka((Uu)),(Uua)),(Uub)),(Uuc)),Uud),(Uue)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aTP_Lamm_ajz((Uu)),(Uua)),(Uub)),(Uuc)),(Uue)),code_bit_cut_integer(Uud)) ) ).

% ATP.lambda_939
tff(fact_10627_ATP_Olambda__940,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: code_integer,Uuf: $o] : ( aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aTP_Lamm_ajz((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),Uue),(Uuf)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aTP_Lamm_ajy((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uuf)),code_bit_cut_integer(Uue)) ) ).

% ATP.lambda_940
tff(fact_10628_ATP_Olambda__941,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o] : ( aa_o_f2266643866902735003t_char(aa_o_f5688925798504779150t_char(aa_o_f2285770652090220913t_char(aa_o_f2419597222644285284t_char(aa_o_f1772367729240163399t_char(aa_o_f5510610115356171066t_char(aa_o_f4853929186051764509t_char(aTP_Lamm_amd,(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)) = cons_char(aa_o_char(char2((Uu),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)),$false)) ) ).

% ATP.lambda_941
tff(fact_10629_ATP_Olambda__942,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: code_integer] : ( aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aa_o_f3038258172971468011o_char(aTP_Lamm_ajx((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),Uug) = char2((Uu),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)) ) ).

% ATP.lambda_942
tff(fact_10630_ATP_Olambda__943,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: code_integer,Uug: $o] : ( aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aTP_Lamm_ajy((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),Uuf),(Uug)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aa_o_f3038258172971468011o_char(aTP_Lamm_ajx((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uug)),code_bit_cut_integer(Uuf)) ) ).

% ATP.lambda_943
tff(fact_10631_ATP_Olambda__944,axiom,
    ! [Uu: $o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_hm((Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_944
tff(fact_10632_ATP_Olambda__945,axiom,
    ! [Uu: $o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hn((Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_945
tff(fact_10633_ATP_Olambda__946,axiom,
    ! [Uu: code_integer,Uua: $o] : ( aa_o_Code_integer(aTP_Lamm_agm(Uu),(Uua)) = Uu ) ).

% ATP.lambda_946
tff(fact_10634_ATP_Olambda__947,axiom,
    ! [Uu: code_integer,Uua: nat] : ( aa_nat_Code_integer(aTP_Lamm_agn(Uu),Uua) = Uu ) ).

% ATP.lambda_947
tff(fact_10635_ATP_Olambda__948,axiom,
    ! [Uu: code_integer,Uua: int] : ( aa_int_Code_integer(aTP_Lamm_ago(Uu),Uua) = Uu ) ).

% ATP.lambda_948
tff(fact_10636_ATP_Olambda__949,axiom,
    ! [Uu: $o,Uua: product_unit] :
      ( aa_Product_unit_o(aTP_Lamm_amn((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_949
tff(fact_10637_ATP_Olambda__950,axiom,
    ! [Uu: $o,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_ao((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_950
tff(fact_10638_ATP_Olambda__951,axiom,
    ! [Uu: $o,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ap((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_951
tff(fact_10639_ATP_Olambda__952,axiom,
    ! [Uu: $o,Uua: $o] :
      ( aa_o_o(aa_o_fun_o_o(aTP_Lamm_aq,(Uu)),(Uua))
    <=> (Uu) ) ).

% ATP.lambda_952
tff(fact_10640_ATP_Olambda__953,axiom,
    ! [Uu: $o,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_al((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_953
tff(fact_10641_ATP_Olambda__954,axiom,
    ! [Uu: $o,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_am((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_954
tff(fact_10642_ATP_Olambda__955,axiom,
    ! [Uu: $o,Uua: rat] :
      ( aa_rat_o(aTP_Lamm_ar((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_955
tff(fact_10643_ATP_Olambda__956,axiom,
    ! [Uu: $o,Uua: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(aTP_Lamm_as,(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_956
tff(fact_10644_ATP_Olambda__957,axiom,
    ! [Uu: $o,Uua: int] :
      ( aa_int_o(aa_o_fun_int_o(aTP_Lamm_at,(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_957
tff(fact_10645_ATP_Olambda__958,axiom,
    ! [Uu: $o,Uua: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aTP_Lamm_an((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_958
tff(fact_10646_ATP_Olambda__959,axiom,
    ! [Uu: rat,Uua: $o] : ( aa_o_rat(aTP_Lamm_agp(Uu),(Uua)) = Uu ) ).

% ATP.lambda_959
tff(fact_10647_ATP_Olambda__960,axiom,
    ! [Uu: rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_agq(Uu),Uua) = Uu ) ).

% ATP.lambda_960
tff(fact_10648_ATP_Olambda__961,axiom,
    ! [Uu: rat,Uua: int] : ( aa_int_rat(aTP_Lamm_agr(Uu),Uua) = Uu ) ).

% ATP.lambda_961
tff(fact_10649_ATP_Olambda__962,axiom,
    ! [Uu: nat,Uua: list_nat] : ( aa_list_nat_nat(aTP_Lamm_ev(Uu),Uua) = Uu ) ).

% ATP.lambda_962
tff(fact_10650_ATP_Olambda__963,axiom,
    ! [Uu: nat,Uua: $o] : ( aa_o_nat(aTP_Lamm_en(Uu),(Uua)) = Uu ) ).

% ATP.lambda_963
tff(fact_10651_ATP_Olambda__964,axiom,
    ! [Uu: nat,Uua: set_nat] : ( aa_set_nat_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_agl,Uu),Uua) = Uu ) ).

% ATP.lambda_964
tff(fact_10652_ATP_Olambda__965,axiom,
    ! [Uu: nat,Uua: rat] : ( aa_rat_nat(aTP_Lamm_ep(Uu),Uua) = Uu ) ).

% ATP.lambda_965
tff(fact_10653_ATP_Olambda__966,axiom,
    ! [Uu: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_er,Uu),Uua) = Uu ) ).

% ATP.lambda_966
tff(fact_10654_ATP_Olambda__967,axiom,
    ! [Uu: nat,Uua: int] : ( aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_et,Uu),Uua) = Uu ) ).

% ATP.lambda_967
tff(fact_10655_ATP_Olambda__968,axiom,
    ! [Uu: int,Uua: list_nat] : ( aa_list_nat_int(aTP_Lamm_ew(Uu),Uua) = Uu ) ).

% ATP.lambda_968
tff(fact_10656_ATP_Olambda__969,axiom,
    ! [Uu: int,Uua: $o] : ( aa_o_int(aTP_Lamm_eo(Uu),(Uua)) = Uu ) ).

% ATP.lambda_969
tff(fact_10657_ATP_Olambda__970,axiom,
    ! [Uu: int,Uua: set_nat] : ( aa_set_nat_int(aa_int2448287375063835929at_int(aTP_Lamm_agk,Uu),Uua) = Uu ) ).

% ATP.lambda_970
tff(fact_10658_ATP_Olambda__971,axiom,
    ! [Uu: int,Uua: rat] : ( aa_rat_int(aTP_Lamm_eq(Uu),Uua) = Uu ) ).

% ATP.lambda_971
tff(fact_10659_ATP_Olambda__972,axiom,
    ! [Uu: int,Uua: nat] : ( aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_es,Uu),Uua) = Uu ) ).

% ATP.lambda_972
tff(fact_10660_ATP_Olambda__973,axiom,
    ! [Uu: int,Uua: int] : ( aa_int_int(aa_int_fun_int_int(aTP_Lamm_eu,Uu),Uua) = Uu ) ).

% ATP.lambda_973
tff(fact_10661_ATP_Olambda__974,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hz,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_974
tff(fact_10662_ATP_Olambda__975,axiom,
    ! [Uu: code_integer] : ( aa_Cod1783974652533396728nteger(aTP_Lamm_cx,Uu) = Uu ) ).

% ATP.lambda_975
tff(fact_10663_ATP_Olambda__976,axiom,
    ! [Uu: product_prod_nat_nat] : ( aa_Pro262675715343573244at_nat(aTP_Lamm_alw,Uu) = Uu ) ).

% ATP.lambda_976
tff(fact_10664_ATP_Olambda__977,axiom,
    ! [Uu: assn] : ( aa_assn_assn(aTP_Lamm_cy,Uu) = Uu ) ).

% ATP.lambda_977
tff(fact_10665_ATP_Olambda__978,axiom,
    ! [Uu: list_char] : ( aa_lis7614194356200183040t_char(aTP_Lamm_amb,Uu) = Uu ) ).

% ATP.lambda_978
tff(fact_10666_ATP_Olambda__979,axiom,
    ! [Uu: rat] : ( aa_rat_rat(aTP_Lamm_cz,Uu) = Uu ) ).

% ATP.lambda_979
tff(fact_10667_ATP_Olambda__980,axiom,
    ! [Uu: nat] : ( aa_nat_nat(aTP_Lamm_da,Uu) = Uu ) ).

% ATP.lambda_980
tff(fact_10668_ATP_Olambda__981,axiom,
    ! [Uu: int] : ( aa_int_int(aTP_Lamm_db,Uu) = Uu ) ).

% ATP.lambda_981
tff(fact_10669_ATP_Olambda__982,axiom,
    ! [Uu: code_integer] : ( aa_Cod1783974652533396728nteger(aTP_Lamm_fi,Uu) = zero_z3403309356797280102nteger ) ).

% ATP.lambda_982
tff(fact_10670_ATP_Olambda__983,axiom,
    ! [Uu: rat] : ( aa_rat_rat(aTP_Lamm_fj,Uu) = zero_zero_rat ) ).

% ATP.lambda_983
tff(fact_10671_ATP_Olambda__984,axiom,
    ! [Uu: nat] : ( aa_nat_nat(aTP_Lamm_fk,Uu) = zero_zero_nat ) ).

% ATP.lambda_984
tff(fact_10672_ATP_Olambda__985,axiom,
    ! [Uu: int] : ( aa_int_int(aTP_Lamm_fl,Uu) = zero_zero_int ) ).

% ATP.lambda_985
tff(fact_10673_ATP_Olambda__986,axiom,
    ! [Uu: nat] : ( aa_nat_nat(aTP_Lamm_pc,Uu) = one_one_nat ) ).

% ATP.lambda_986
tff(fact_10674_ATP_Olambda__987,axiom,
    ! [Uu: nat] : ( aa_nat_int(aTP_Lamm_pd,Uu) = one_one_int ) ).

% ATP.lambda_987
tff(fact_10675_ATP_Olambda__988,axiom,
    ! [Uu: int] : ( aa_int_int(aTP_Lamm_pe,Uu) = one_one_int ) ).

% ATP.lambda_988
tff(fact_10676_ATP_Olambda__989,axiom,
    ! [Uu: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aTP_Lamm_ad,Uu)
    <=> $false ) ).

% ATP.lambda_989
tff(fact_10677_ATP_Olambda__990,axiom,
    ! [Uu: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_bh,Uu)
    <=> $false ) ).

% ATP.lambda_990
tff(fact_10678_ATP_Olambda__991,axiom,
    ! [Uu: $o] :
      ( aa_o_o(aTP_Lamm_bi,(Uu))
    <=> $false ) ).

% ATP.lambda_991
tff(fact_10679_ATP_Olambda__992,axiom,
    ! [Uu: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_be,Uu)
    <=> $false ) ).

% ATP.lambda_992
tff(fact_10680_ATP_Olambda__993,axiom,
    ! [Uu: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_bf,Uu)
    <=> $false ) ).

% ATP.lambda_993
tff(fact_10681_ATP_Olambda__994,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_bj,Uu)
    <=> $false ) ).

% ATP.lambda_994
tff(fact_10682_ATP_Olambda__995,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_bk,Uu)
    <=> $false ) ).

% ATP.lambda_995
tff(fact_10683_ATP_Olambda__996,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aTP_Lamm_bg,Uu)
    <=> $false ) ).

% ATP.lambda_996
tff(fact_10684_ATP_Olambda__997,axiom,
    ! [Uu: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_df,Uu)
    <=> $true ) ).

% ATP.lambda_997
tff(fact_10685_ATP_Olambda__998,axiom,
    ! [Uu: $o] :
      ( aa_o_o(aTP_Lamm_dg,(Uu))
    <=> $true ) ).

% ATP.lambda_998
tff(fact_10686_ATP_Olambda__999,axiom,
    ! [Uu: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_dc,Uu)
    <=> $true ) ).

% ATP.lambda_999
tff(fact_10687_ATP_Olambda__1000,axiom,
    ! [Uu: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_dd,Uu)
    <=> $true ) ).

% ATP.lambda_1000
tff(fact_10688_ATP_Olambda__1001,axiom,
    ! [Uu: rat] :
      ( aa_rat_o(aTP_Lamm_dh,Uu)
    <=> $true ) ).

% ATP.lambda_1001
tff(fact_10689_ATP_Olambda__1002,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_di,Uu)
    <=> $true ) ).

% ATP.lambda_1002
tff(fact_10690_ATP_Olambda__1003,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_dj,Uu)
    <=> $true ) ).

% ATP.lambda_1003
tff(fact_10691_ATP_Olambda__1004,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aTP_Lamm_de,Uu)
    <=> $true ) ).

% ATP.lambda_1004

% Helper facts (24)
tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X: $o,Y: $o] :
      ( ( (X) != (Y) )
      | aa_o_o(aa_o_fun_o_o(fequal_o,(X)),(Y)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X: $o,Y: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fequal_o,(X)),(Y))
      | ( (X) = (Y) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
      | aa_num_o(aa_num_fun_num_o(fequal_num,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(fequal_num,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__String__Ochar_T,axiom,
    ! [X: char,Y: char] :
      ( ( X != Y )
      | aa_char_o(aa_char_fun_char_o(fequal_char,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__String__Ochar_T,axiom,
    ! [X: char,Y: char] :
      ( ~ aa_char_o(aa_char_fun_char_o(fequal_char,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X != Y )
      | aa_set_nat_o(aa_set7861427066675100071_nat_o(fequal_set_nat,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(fequal_set_nat,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__String__Ochar_J_T,axiom,
    ! [X: list_char,Y: list_char] :
      ( ( X != Y )
      | aa_list_char_o(aa_lis7814493268959572977char_o(fequal_list_char,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__String__Ochar_J_T,axiom,
    ! [X: list_char,Y: list_char] :
      ( ~ aa_list_char_o(aa_lis7814493268959572977char_o(fequal_list_char,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: list_Code_integer,Y: list_Code_integer] :
      ( ( X != Y )
      | aa_lis2813719629158071341eger_o(aa_lis7224287298985949145eger_o(fequal5224421480335545304nteger,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: list_Code_integer,Y: list_Code_integer] :
      ( ~ aa_lis2813719629158071341eger_o(aa_lis7224287298985949145eger_o(fequal5224421480335545304nteger,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( X != Y )
      | aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ~ aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
      ( ( X != Y )
      | aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(fequal1127456680179375854nt_int,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
      ( ~ aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(fequal1127456680179375854nt_int,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_T,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( ( X != Y )
      | aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(fequal1890233700291625924_nat_o,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_T,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( ~ aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(fequal1890233700291625924_nat_o,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_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_T,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( X != Y )
      | aa_Pro5947980084102686253_nat_o(aa_Pro1894297876208389727_nat_o(fequal7949547063978738364at_nat,X),Y) ) ).

tff(help_fequal_1_1_fequal_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_T,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ~ aa_Pro5947980084102686253_nat_o(aa_Pro1894297876208389727_nat_o(fequal7949547063978738364at_nat,X),Y)
      | ( X = Y ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    ( ( ( x = aa_fun5526454687433275355o_assn(abs_assn,aTP_Lamm_a) )
      & ( x != y ) )
  <=> ( ( x = aa_fun5526454687433275355o_assn(abs_assn,aTP_Lamm_a) )
      & ( y != aa_fun5526454687433275355o_assn(abs_assn,aTP_Lamm_aa) ) ) ) ).

%------------------------------------------------------------------------------